Search for Efficient Formulations for Hamiltonian Simulation of non-Abelian Lattice Gauge Theories
UUMD-PP-020-6
Search for Efficient Formulations for Hamiltonian Simulationof non-Abelian Lattice Gauge Theories
Zohreh Davoudi,
1, 2
Indrakshi Raychowdhury, and Andrew Shaw Maryland Center for Fundamental Physics and Department of Physics,University of Maryland, College Park, MD 20742, USA RIKEN Center for Accelerator-based Sciences, Wako 351-0198, Japan (Dated: September 25, 2020)Hamiltonian formulation of lattice gauge theories (LGTs) is the most natural framework for thepurpose of quantum simulation, an area of research that is growing with advances in quantum-computing algorithms and hardware. It, therefore, remains an important task to identify the mostaccurate, while computationally economic, Hamiltonian formulation(s) in such theories, consideringthe necessary truncation imposed on the Hilbert space of gauge bosons with any finite computingresources. This paper is a first step toward addressing this question in the case of non-Abelian LGTs,which further require the imposition of non-Abelian Gauss’s laws on the Hilbert space, introducingadditional computational complexity. Focusing on the case of SU(2) LGT in 1+1 dimensions coupledto one flavor of fermionic matter, a number of different formulations of the original Kogut-Susskindframework are analyzed with regard to the dependence of the dimension of the physical Hilbertspace on boundary conditions, system’s size, and the cutoff on the excitations of gauge bosons.The impact of such dependencies on the accuracy of the spectrum and dynamics obtained froma Hamiltonian computation is examined, and the (classical) computational-resource requirementsgiven these considerations are studied. Besides the well-known angular-momentum formulation ofthe theory, the cases of purely fermionic and purely bosonic formulations (with open boundaryconditions), and the Loop-String-Hadron formulation are analyzed, along with a brief discussionof a Quantum-Link-Model formulation of the same theory. Clear advantages are found in workingwith the Loop-String-Hadron framework which implements non-Abelian Gauss’s laws a priori usinga complete set of gauge-invariant operators. Although small lattices are studied in the numericalanalysis of this work, and only the simplest algorithms are considered, a range of conclusions will beapplicable to larger systems and potentially to higher dimensions. Future studies will extend thisinvestigation to the analysis of resource requirements for quantum simulations of non-Abelian LGT,with the goal of shedding light on the most efficient Hamiltonian formulation of gauge theories ofrelevance in nature.
CONTENTS
I. Introduction 1II. An overview of the Kogut-Susskind SU(2) LGTand its various formulations 4A. Angular-momentum formulation 5B. Purely fermionic formulation 6C. Purely bosonic formulation 7D. Loop-String-Hadron formulation 8III. Physical Hilbert-space analysis 11A. Gauge-invariant angular-momentum basis 12B. Purely fermionic formulation 14C. Purely bosonic formulation 15D. Loop-String-Hadron formulation 16IV. Realization of Global Symmetries 17V. Comparative (classical) cost analysis 19A. Physical Hilbert-space construction 19B. Hamiltonian generation 21C. Observable computation 21D. Total cost and comparisons 23VI. Spectrum and dynamics in truncated theory 24 A. Spectrum analysis 26B. Dynamics analysis 27VII. Conclusions and outlook 29Acknowledgments 31References 32A. Quantum Link Model 35B. Physical Hilbert-space dimensionality 37C. Observables with open boundary conditions 37
I. INTRODUCTION
Gauge field theories are at the core of our modern un-derstanding of nature, from the descriptions of quantumHall effect and superconductivity in condensed-matterphysics [1, 2], to the mechanisms underlying the inter-actions of sub-atomic particles at the most fundamentallevel within the Standard Model of particle physics [3],to emerging in the context of high-energy models pro-posed for new physics that are yet to be discovered [4].Strongly interacting gauge theories coupled to matter, a r X i v : . [ h e p - l a t ] S e p such as the quantum theory of strong interactions orquantum chromodynamics (QCD), are notoriously hardto simulate, and often demand applications of nonper-turbative numerical strategies. Lattice-based methods,in which quantum fields are placed on a finite spacetimelattice, provide a natural regulation of ultraviolet modesand along with Renormalization Group (RG) methods,recover the continuum limit of the theory. Such a dis-cretized theory also provides the framework for nonper-turbative numerical studies, such as those based in pathintegral (Lagrangian) and Hamiltonian (canonical) for-mulations. While these two approaches are intrinsicallyequivalent, symmetries and constraints are manifesteddifferently in each case [5].
Path integral vs. Hamiltonian formulation of lat-tice gauge theories.
From a computational standpoint,the path-integral formulation has emerged as the pri-mary tool in the LGT program given its parallels withthe quantum statistical physics [6, 7], and its relianceon efficient state-of-the-art quantum Monte Carlo sam-pling techniques given this connection [8]. However, inorder for such an analogy with statistical mechanics tobe established, a Wick rotation to Euclidean spacetimeis performed such that only imaginary-time correlationfunctions are directly accessed in this numerical program.This feature introduces two limitations: firstly the con-nection to real-time quantities is lost, and except in lim-ited cases (e.g., Refs. [9–13]), practical proposals are lack-ing for mapping a generic Euclidean correlation functionthat is obtained numerically on a spacetime lattice to dy-namical amplitudes as measured in experiments. Second,a non-zero fermionic chemical potential introduces a signproblem by making the sampling weight in the Euclideanpath integral imaginary [14], along with an inherentlyrelated signal-to-noise problem observed in correlationfunctions at zero chemical potential but with non-zerobaryonic number [15]. On the other hand, the Hamilto-nian formulation lacks the manifestly Lorentz covarianceof the path-integral formulation, and further requiresgauge fixing. In particular, in the most common gauge inwhich the temporal component of the gauge field is set tozero, the information about ‘constants of motion’ is lostand the related constraint must be imposed a posteriori on the Hilbert space. From a computational perspec-tive, a Hamiltonian formulation enables both real-timeand imaginary-time simulations. However, the most effi-cient numerical approach in Hamiltonian-based studies isno longer stochastic as in the path-integral formulation,and the cost of a typical numerical simulation scales withpowers of the dimension of the Hilbert space (which it-self grows exponentially with the size of the system). Ex-tremely efficient Hamiltonian-simulation algorithms havebeen developed and implemented in recent years for low-dimensional LGTs using tensor-network methods [16], In asymptotically free theories like QCD. but they rely on strict assumptions on the rate of entan-glement growth in the physical system [17], assumptionsthat generally break down as system evolves arbitrarilyin time.Despite the drawbacks encountered in a Hamiltonianapproach to LGTs, there has been revived interest inthe Hamiltonian-simulation program given the improv-ing prospects of quantum simulation and quantum com-putation. A plethora of ideas, proposals, and imple-mentations have emerged for simulations of quantummany-body systems in general [18–22], and quantum fieldtheories and LGTs in particular (see e.g., Refs. [23–62]), in recent years, in light of advances in existingand upcoming digital and analog quantum-simulationtechnologies [63–74]. Generally speaking, a quantum-simulating/computing hardware will reduce the exponen-tial cost of encoding the Hilbert space of a LGT onto theclassical bits down to a polynomial cost, by storing infor-mation onto the quantum-mechanical wavefunctions ofqubits. Nonetheless, quantum hardware will continue toexhibit small capacity and noise-limited capability for theforeseeable future. As a result, the search for an ultimateefficient formulation of LGTs for the quantum-simulationprogram is a crucial first step toward harnessing thepower of quantum-simulating/computing platforms. Inthe meantime, as the classical Hamiltonian-simulation al-gorithms advance, such efficient formulation(s) can facil-itate classical studies as well. In what follows, we elab-orate on the meaning of an efficient formulation in thecase of non-Abelian LGTs, and will make such efficiencyconsiderations explicit by analyzing in depth the caseof the SU(2) LGT in 1+1 Dimensions (D) coupled tomatter. The focus is exclusively on the cost analysis ofexact Hamiltonian-simulation algorithms using classical-computing hardware. Nonetheless, this study lays thegroundwork for an analysis of resource requirements forsimulating the same theory on quantum hardware, to bepresented in future work. A summary of the pros and cons of variousrepresentations of Kogut-Susskind theory.
In1980s, Kogut and Susskind formulated a lattice Hamilto-nian for Yang-Mills gauge-field theories coupled to mat-ter [83] that recovered the continuum limit, and wasshown to be equivalent to Wilson’s path-integral formula-tion [6] of the same lattice theory. The Kogut-Susskind(KS) Hamiltonian further made it possible to performquantum-mechanical perturbation theory around the A number of other proposals for general boson(gauge)-field dig-itizations exist as can be found in e.g., Refs. [50, 75–80]. Thesewill not be analyzed in the current work. Instead the focus is onthe representation of the gauge theory itself in terms of the cho-sen basis states for fermions and bosons, and the gauge groupwill be kept exact despite the imposed truncation on the highexcitations. One exception to this trend is the discussion of theQuantum Link Model [81, 82] of the SU(2) LGT that, given itspopularity in the context of quantum simulation, is discussed insome length in Appendix A. strong-coupling vacuum, i.e., the ground state of the the-ory in the limit where the mass term and the electric-fieldterm in the Hamiltonian dominate, see Sec. II. Spectraand dynamics of both U(1) and SU(2) LGTs using the KSHamiltonian in low dimensions were later studied towardthe continuum limit, using high-order strong-coupling ex-pansions, as well as non-perturbative numerical methods,see e.g., Refs. [84–89]. Such investigations have continuedto date using state-of-the-art algorithms such as thosebased on Matrix Product States [16, 90–92]. Given therequirement of generating and storing the Hamiltonianmatrix whose size is determined by the size of the Hilbertspace, and given the infinite dimensionality of the Hilbertspace of a gauge theory, the truncation on excitations ofthe gauge degrees of freedom (DOF) is a necessity. Fur-thermore, only a small but still exponentially large (inthe system’s size) portion of the space of all possible basisstates are those satisfying the gauge constraints, i.e., thelocal Gauss’s laws. As will be demonstrated, exact andinexact Hamiltonian methods are not capable of simulat-ing the full Hilbert space of a gauge theory even for smallsystems and small cutoff values on the gauge-field exci-tations. As a result, as a first step in the computation,a mechanism to construct the physical Hilbert space andits corresponding Hamiltonian must be implemented.The representation chosen for fermionic and gaugeDOE in a given LGT dictates the way the Hilbert-spacetruncation is performed and whether gauge invarianceremains intact. It also determines the complexity of theconstruction of the physical Hilbert space and its asso-ciated Hamiltonian matrix for the sake of computation.For non-Abelian LGTs, in particular, the Gauss’s lawconstraints are more complex to impose. In the case ofthe SU(2) LGT coupled to matter in the original proposalof Kogut and Susskind, the Hilbert space of the gaugeDOF are expressed in terms of local angular-momentumbasis states that mimic those associated with rigid-bodyrotations in fixed and body frames [83]. To constructthe physical Hilbert space, these on-site basis states arecombined with the fermionic DOF expressed in the fun-damental representation of SU(2) in such a way that thenet angular momentum is zero at each site. Furthermore,an Abelian constraint is satisfied such that the total an-gular momenta on the left and right of a link connectingtwo lattice sites are equal. As it will be demonstrated,even in 1+1 D, the computational complexity of impos-ing such constraints grows quickly with the size of thesystem, and with the cutoff imposed on the total angu-lar momentum. The physical states, in general, becomelinear combinations of a large and growing number ofterms in the basis chosen, adding to the complexity ofHamiltonian generation. In the Quantum-Link-Model formulation of the SU(2) LGT,the gauge DOF are chosen to be fermionic, yielding a finite-dimensional Hilbert space [81, 82]. However, unless the contin-uum limit is taken through a dimensional-reduction procedure,the theory is not equivalent to the KS LGT.
With open boundary conditions (OBC) in 1+1 D, it ispossible to eliminate any dependence on the gauge DOEwith the use of a gauge transformation, along with theapplication of Gauss’s laws at the level of the Hamil-tonian operator itself, as already proposed as a viableefficient basis for Hamiltonian simulation of the SU(2)LGT in Refs. [90, 91]. Such a trick significantly re-duces the computational cost of imposing Gauss’s lawson the Hilbert space and eliminates the need for a cut-off on the gauge DOF. However, as will be shown,there remain redundancies in this representation com-pared with the physical Hilbert space of the KS the-ory in the angular-momentum basis that grows slowlywith the system’s size. Furthermore, this formulationcan not be extended to higher dimensions since thereare not sufficient Gauss’s laws present to eliminate allgauge DOF. So it will be beneficial to examine a formu-lation with better generalizability prospects. A recentbosonization proposal [93, 94] for LGTs is explored aswell, in which a subset of Gauss’s laws correspondingto the Cartan sub-algebra of the SU(N) group coupledto one flavor of fundamental fermions are augmentedwith an additional U(1) Abelian Gauss’s law to allowthe elimination of fermionic DOF. While this procedureworks in any dimension, it trades the finite-dimensionalHilbert space of the fermions with intrinsically infinite-dimensional Hilbert space of the gauge DOF, includingan additional U(1) field. The Hilbert space of thesebosonic fields must be cut off for practical purposes,which could lead to systematic uncertainties in computa-tions with finite resources. Furthermore, constructing theHamiltonian matrix in the physical Hilbert space remainscomputationally involved in the bosonic formulation.The complexity associated with non-Abelian LGTs inthe KS theory in its original formulation is the moti-vation behind the development of a recent frameworkcalled the Loop-String-Hadron (LSH) formulation for theSU(2) LGT coupled to matter, which is valid in anydimensions [40, 95]. It is founded upon the prepoten-tial formalism of pure LGTs [96–100], which is funda-mentally a representation that re-expresses the angular-momentum basis in terms of the harmonic-oscillator basisof Schwinger bosons [101]. As a result, the SU(2) gauge-link and electric-field operators are expressed in termsof harmonic-oscillator creation and annihilation opera-tors and allow gauge-invariant operators to be formedout of gauge and fermionic DOF at each site. Theseoperators, therefore, excite only the states in the phys-ical sector of the Hilbert space, as long as an AbelianGauss’s law is satisfied, which requires the number ofoscillators at the left and right of the link to be equal. Prepotential formulation for SU(3) [102] as well as SU(N) [103]LGTs have also been constructed in terms of irreducibleSchwinger bosons [104, 105] in any dimension. These exhibitthe same features as the SU(2) theory discussed in the presentpaper.
The LSH formulation constructs a complete set of prop-erly normalized gauge-invariant operators and expressesthe Hamiltonian in terms of this complete basis [95]. Aswill be shown, the Hilbert space of the KS theory in theangular-momentum basis after imposing the Abelian andnon-Abelian Gauss’s laws, and for a given cutoff on thegauge DOF, is identical to that of the LSH Hamiltonian.Nonetheless, the computational cost of generating theassociated Hamiltonian is far less in the LSH frameworkgiven its already gauge-invariant physical basis states.Consequently, the simplicity of the fermionic represen-tation (with OBC) is enjoyed by the LSH formulationas well but without associated redundancies, and withthe prospects of straightforward applications to higherdimensions.
Outline of the paper.
While all the different formu-lations studied here have been introduced, and to someextent implemented, in literature, the conclusions brieflystated above and those that will follow, are new and haveresulted from a thorough comparative analysis that isconducted in this work. In particular, an analysis of thesize of the full and physical Hilbert spaces as a functionof the system’s size and, when applicable, the cutoff onthe gauge DOF is presented in Sec. III for all the for-mulations of SU(2) LGT in 1+1 D enumerated above(and reviewed in Sec. II). Here, empirical relations areobtained from a numerical study with small lattice sizes.These results lead to a discussion of the time complexityof exact classical Hamiltonian-simulation algorithms inSec. V. Section VI contains an analysis of the impact ofthe cutoff on the spectrum and dynamics of the theory.A detailed discussion of the global symmetries of SU(2)LGT in 1+1 D is presented in Sec. IV, which allows thedecomposition of the physical Hilbert space of the theoryto even smaller decoupled sectors, hence simplifying thecomputation. While not a focus of this work, a brief com-parative study of the KS SU(2) theory in 1+1 D with aQLM formulation [82] is presented in Appendix A. Giventhe extent of discussions, and the spread of observationsand conclusions made throughout this paper, Sec. VIIwill summarize the main points of the study more crisply,along with presenting an outlook of this work.In summary, the results presented here should offer aclear path to the practitioner of Hamiltonian-simulationtechniques to evaluate the pros and cons of a given for-mulation of the SU(2) LGT in 1+1 D in connection tothe simulation algorithm used. A similar study for the2+1-dimensional theory can shed light on the validityof the conclusions made for higher-dimensional cases. Moreover, the conclusions of this work will guide futurestudies of non-Abelian LGTs in the context of quantumsimulation. See a recent work on the efficient Hamiltonian simulation of theU(1) LGT in 2+1 D in Ref. [60].
II. AN OVERVIEW OF THEKOGUT-SUSSKIND SU(2) LGT AND ITSVARIOUS FORMULATIONS
Within the Hamiltonian formulation of LGTs introducedby Kogut and Susskind, the temporal direction is contin-uous while the spatial direction is discretized. Each sitealong the spatial direction is split into two staggered sites,as shown in Fig. 1, such that matter and anti-matterfields occupy even and odd sites, respectively. The num-ber of sites along this direction is denoted by N and iscalled the lattice size throughout. The spacing betweenadjacent sites after staggering is denoted as a . For theSU(2) LGT in 1+1 D, the KS Hamiltonian can be writtenas: H (KS) = H (KS) I + H (KS) E + H (KS) M . (1)Here, H (KS) I denotes interactions among the fermionicand gauge DOF H (KS) I = 12 a N (cid:88) x =0 (cid:104) ψ † ( x ) ˆ U ( x ) ψ ( x + 1) + h . c . (cid:105) , (2)where N = N − N = N − ψ is in the fundamental representationof SU(2) and consists of two components, i.e., ψ = (cid:0) ψ ψ (cid:1) .The gauge link ˆ U ( x ) is a 2 × x along the spatial directionand ends at point x + 1, as shown in Fig. 1. A tempo-ral gauge is chosen which sets the gauge link along thetemporal direction equal to unity. H (KS) E corresponds to the energy stored in the electricfield, H (KS) E = g a N (cid:88) x =0 ˆ E ( x ) . (3)Here, N = N − N = N − g is a coupling. Further, ˆ E = ( ˆ E ) + ( ˆ E ) + ( ˆ E ) ≡ ˆ E L = ˆ E R . ˆ E L and ˆ E R are the left and the right electric-field operators, respectively, as shown in Fig. 1. Thesesatisfy the SU(2) Lie algebra at each site,[ ˆ E aL , ˆ E bL ] = − i(cid:15) abc ˆ E cL , [ ˆ E aR , ˆ E bR ] = i(cid:15) abc ˆ E cR , [ ˆ E aL , ˆ E bR ] = 0 , (4)where (cid:15) abc is the Levi-Civita tensor and the spatial de-pendence of the fields is suppressed in these relations.Further, the electric fields on different sites commute. Here and throughout, the position argument of the functionsand the superscript of state vectors are assumed to be an index.A multiplication by the lattice spacing a converts these to theabsolute position. FIG. 1. A physical site along the spatial direction is split to two staggered sites in the KS Hamiltonian. These sites areconnected by a gauge link. Corresponding to each staggered site, there is a two-component fermion field, a left electric field,and a right electric field, as indexed in the figure.
The electric fields and the gauge link satisfy the canoni-cal commutation relations at each site,[ ˆ E aL , ˆ U ] = T a ˆ U , [ ˆ E aR , ˆ U ] = ˆ U T a , (5)where T a = τ a , and τ a is the a th Pauli matrix. Thecorresponding commutation relations for fields with dif-ferent site indices vanish.Finally, H (KS) M in Eq. (1) is a staggered mass term H (KS) M = m N (cid:88) x =0 ( − x ψ † ( x ) ψ ( x ) . (6)Here, N = N − m denotesthe mass of each component of the fermions.In this theory, a fermion SU(2)-charge-density operatordefined at each site,ˆ ρ a ( x ) ≡ ψ † ( x ) T a ψ ( x ) , (7)which satisfies the SU(2) Lie algebra. It further satisfiesthe following commutation relation at each site,[ˆ ρ a , ψ ] = − T a ψ. (8)Such a commutation relation vanishes for fields at differ-ent sites. This SU(2)-charge-density operator also com-mutes with the electric fields and the gauge link. Withthese commutation relations, and those given in Eqs. (4)and (5), one can show that the Hamiltonian in Eq. (1)commutes with the following operator,ˆ G a ( x ) = − ˆ E aL ( x ) + ˆ E aR ( x −
1) + ˆ ρ a ( x ) . (9)As a result, the Hilbert space of the theory is classifiedinto sectors corresponding to each of the eigenvalues ofthe Gauss’s law operators G a , and these eigenvalues arethe ‘constants of motion’. The physical sector of thisHilbert space is that corresponding to the zero eigenvalueof this operator. A. Angular-momentum formulation
The first step in forming the Hilbert space of a LGT forthe sake of computation is to map the vacuum and theexcitations of the fields to a state basis. In the absence of the magnetic Hamiltonian, which is the case in 1+1 DLGTs, the most efficient basis is formed out of eigen-states of the electric-field operator. The direct product ofthe fermionic eigenstates and the electric-field eigenstatesforms the full Hilbert space. This is called the electric-field basis, or the strong-coupling basis, i.e., in the g → ∞ limit, the interaction terms in Eq. (2) that involves transi-tions between different eigenvalues of the electric-field op-erator becomes insignificant compared with the electric-field term, Eq. (3), and the Hamiltonian becomes diag-onal in the electric-field basis. Since the electric fieldssatisfy the SU(2) algebra, a familiar representation is theangular-momentum representation. In fact, as pointedout by Kogut and Susskind, the left and right electricfield can be mapped to the body-frame ( ˆ J b ) and space-frame ( ˆ J s ) angular momenta of a rigid body. Explicitly,ˆ E L = − ˆ J b ( ≡ − ˆ J L ) and ˆ E R = ˆ J s ( ≡ ˆ J R ), satisfyingˆ J L = ˆ J R on each link.Given this correspondence, one may write the electric-field basis states for the KS formulation as | Φ (cid:105) ( x )(KS) = | J R , m R (cid:105) ( x − ⊗| f , f (cid:105) ( x ) ⊗| J L , m L (cid:105) ( x ) , (10)for each site x . Here, f and f quantum numbers re-fer to the occupation number of the two components ofthe (anti)matter field, ψ and ψ , each taking values0 and 1, corresponding to the absence and presence of(anti)matter, respectively: ψ | f , f (cid:105) = (1 − δ f , ) | f − , f (cid:105) , (11) ψ † | f , f (cid:105) = (1 − δ f , ) | f + 1 , f (cid:105) , (12)at each site, and similarly for the other component of ψ . Here, δ denotes the Kronecker-delta symbol. Further-more, the angular-momentum basis states satisfyˆ J R | J R , m R (cid:105) = J R ( J R + 1) | J R , m R (cid:105) (13)ˆ J L | J L , m L (cid:105) = J L ( J L + 1) | J L , m L (cid:105) (14)and ˆ J R | J R , m R (cid:105) = m R | J R , m R (cid:105) (15)ˆ J L | J L , m L (cid:105) = m L | J L , m L (cid:105) (16)at each site x , where for brevity the site indices aresuppressed. Here, J L , J R = 0 , , , , · · · , and m R and m L quantum numbers satisfy − J R ≤ m R ≤ J R and − J L ≤ m L ≤ J L , as dictated by the angular-momentumgroup algebra. The action of the gauge-link operator onthis basis can be written as:ˆ U ( α,β ) ( x ) (cid:104) · · · | J R , m R (cid:105) ( x − ⊗ | f , f (cid:105) ( x ) ⊗ | J L , m L (cid:105) ( x ) ⊗ | J R , m R (cid:105) ( x ) ⊗ | f , f (cid:105) ( x +1) ⊗ | J L , m L (cid:105) ( x +1) · · · (cid:105) = · · · | J R , m R (cid:105) ( x − ⊗ | f , f (cid:105) ( x ) ⊗ (cid:88) j = { , , ,... } (cid:115) J + 12 j + 1 (cid:104) J, m L ; 12 , α | j, m L + α (cid:105)(cid:104) J, m R ; 12 , β | j, m R + β (cid:105) | j, m L + α (cid:105) ( x ) ⊗ | j, m R + β (cid:105) ( x ) (cid:21) ⊗ | f , f (cid:105) ( x +1) ⊗ | J L , m L (cid:105) ( x +1) · · · , (17)where α, β = ± . Note that J L and J R on each linkare equal, and as such we have defined J ≡ J ( x ) L = J ( x ) R in this relation. Ellipses denote states that precede andfollow those shown at site x and x + 1, respectively.The physical states | φ (cid:105) (KS) can be formed by identi-fying proper linear combinations of the basis states inEq. (10) such that the Gauss’s laws are satisfied at eachsite, and by constructing the direct product of these com-binations for adjacent sites along the lattice, followingadditional gauge and boundary conditions as is detailedbelow. First, given the Gauss’s law operators defined inEq. (9), the physical states | φ (cid:105) (KS) are required to satisfy G a ( x ) | φ (cid:105) (KS) = 0. Explicitly, (cid:20) ˆ J aL ( x ) + ˆ J aR ( x −
1) + 12 ψ † ( x ) τ a ψ ( x ) (cid:21) | φ (cid:105) (KS) = 0 , (18)for a = 1 , ,
3, and for every x where x = 0 , , · · · , N − J L (corresponding to ˆ E L ), ˆ J f with ˆ J f = (correspondingto the presence of one and only one fermion), and ˆ J R (corresponding to − ˆ E L ) should add to zero at each site.When there is no fermion present or two fermions arepresent, ˆ J f = 0 and the left and right angular momentaare the same. Moreover, as mentioned before, J L and J R quantum numbers need to be equal on each link. Thesetwo requirements, in addition to the boundary conditionsimposed on the J R value at site x = 0 and the J L value atsite x = N −
1, constrain the Hilbert space to a physicalgauge-invariant one, as analyzed in Sec. III A Note that: U = U ( , − ) , U = U ( − , − ) , U = U ( , ) , U = U ( − , ) . B. Purely fermionic formulation
The KS Hamiltonian in Eq. (1) combined with theGauss’s law constraints on the Hilbert space, in essence,leaves no dynamical gauge DOF in 1+1 D beyond possi-ble boundary modes. In particular, with OBC where theincoming flux of the (right) electric field is set to a fixedvalue, the value of electric-field excitations throughoutthe lattice is fixed. This, in fact, is a general feature ofLGTs in 1+1 D, as is evident from the proof outlined be-low. As a result, the KS Hamiltonian acting on the phys-ical Hilbert space can be brought to a purely fermionicform, in which the identification of (anti)fermion config-urations is sufficient to construct the Hilbert space. Thiseliminates the need for adopting a state basis for thegauge DOF, and for solving the complex (non-diagonal)Gauss’s laws locally which is the case in an angular-momentum basis. Such an elimination of gauge DOFin LGTs in 1+1 D was first discussed in Ref. [85] andis used in recent tensor-network simulations of the SU(2)LGT in Ref. [91]. Here, we present a generic derivation ofsuch a purely fermionic representation, before analyzingits Hilbert space in the following section.Consider the following gauge transformation on thefermion fields in the KS Hamiltonian: ψ ( x ) → ψ (cid:48) ( x ) = (cid:34) (cid:89) y 1) = (cid:15) a for a = 1 , , 3, and the Gauss’s laws G a | φ (cid:105) (KS) = 0with G a defined in Eq. (9), fully fixes the values of E L and E R at all sites on the one-dimensional lattice in thephysical Hilbert space: E aL ( x ) = (cid:15) a + x (cid:88) y =0 ρ a ( y ) = E aR ( x ) , (27)with ρ a defined in Eq. (7). Consequently, the electric-field Hamiltonian H (KS)E becomes H (F) E = g a N (cid:88) x =0 3 (cid:88) a =1 (cid:34) (cid:15) a + x (cid:88) y =0 ψ † (cid:48) ( y ) T a ψ (cid:48) ( y ) (cid:35) , (28)where N = N − H (F) M = m N (cid:88) x =0 ( − x ψ † (cid:48) ( x ) ψ (cid:48) ( x ) , (29)where N = N − | Φ (cid:105) (KS , F) = N − (cid:89) x =0 | f , f (cid:105) ( x ) , (30)where as before, f and f refer to the occupation numberof the two components of the (anti)matter field, ψ and ψ , respectively, each taking values 0 or 1. Note that ψ † ( x ) ψ ( x ) = ψ † (cid:48) ( x ) ψ (cid:48) ( x ). C. Purely bosonic formulation Gauge transformation, along with the imposition of thelocal Gauss’s laws with OBC, led to the elimination of thegauge DOF in the previous section. Unfortunately, thisprocedure can obtain a purely fermionic theory only in1+1 D, as in higher dimensions the number of constraintsat each lattice site is not sufficient to eliminate the gaugeDOF in all spatial directions. One could reversely con-sider eliminating the fermionic DOF with the use of theGauss’s laws, as proposed in Ref. [93], to obtain a fullybosonic theory. This protocol works in all dimensions,but in the case of SU (2 N ) theories, requires enlargingthe gauge group to U (2 N ) to accommodate a sufficientnumber of constraints needed to eliminate the fermions. One further needs to keep track of the fermionic statisticsby encoding in the purely bosonic interactions, the non-trivial signs associated with the anti-commuting natureof the fermions [94]. The extended theory can be shownto be equivalent to the original theory for all physical pur-poses, as long as the cutoff on the new gauge DOF of theextended symmetry is set sufficiently high, see Sec. III C.In the following, the bosonized form of the SU(2) LGTin 1+1 D is derived, following the procedure outlined inRef. [93] for general dimensions.Consider the Gauss’s laws in the KS formulation ofthe SU(2) LGT in 1+1 D, given in Eq. (18). Althoughthere exist three Gauss’s laws at each site, only theGauss’s law corresponding to the a = 3 component ofGauss’s law operator in Eq. (9) provides a diagonal re-lation in the angular momentum/fermionic basis. Inother words, two of the Gauss’s laws mix basis stateswith different quantum numbers, and only one of theGauss’s laws leads to an algebraic relation among thegauge and fermionic DOF. Explicitly, for a basis state | J R , m R (cid:105) ( x − ⊗ | f , f (cid:105) ( x ) ⊗ | J L , m L (cid:105) ( x ) at site x , thisrelation reads m L ( x ) + m R ( x − 1) = − 12 ( f ( x ) − f ( x )) . (31)However, in order to fully express the { f , f } quantumnumbers at each site in terms of the { J R , m R , J L , m L } quantum numbers surrounding the site, at least one moreindependent relation is needed. Such a relation can beobtained by adding an extra U(1) symmetry to enlargethe gauge group, effectively modifying each link on thelattice by a U(1) link U , i.e., U ( x ) → U ( x ) × U ( x ),where U is the SU(2) link. This introduces a staggered As shown in Ref. [93] for the case of the SU (2 N + 1) theory,the introduction of an auxiliary Z gauge field on each link onthe lattice is sufficient to eliminate the fermions, without theneed to enlarge the group to U (2 N + 1). This enhancement alsotakes care of the fermionic statistics when fermions are replacedwith the hardcore bosons and are subsequently eliminated. Sincethe focus of this work is the SU(2) theory, this case will not beanalyzed here further. U(1) charge, ˆ ρ ( x ) ≡ (cid:2) ψ † ( x ) ψ ( x ) − (1 − ( − x ) (cid:3) , (32)along with a corresponding U(1) electric field E ( x ) oneach link emanating from site x .The physical Hilbert space of the extended U(2) the-ory is the direct product of the physical Hilbert spacesof the KS SU(2) theory and the U(1) theory, i.e., | φ (cid:105) U (2) = | φ (cid:105) (KS) ⊗ | φ (cid:105) U (1) , where | φ (cid:105) (KS) was introducedin Sec. II A, and | φ (cid:105) U (1) = | E (cid:105) (0) ⊗ | E (cid:105) (1) · · · ⊗ | E (cid:105) ( N − , (33)with ˆ E ( x ) | E (cid:105) ( x ) = E ( x ) | E (cid:105) ( x ) , E ( x ) ∈ Z , (34)for all values of the electric field that satisfy the U(1)Gauss’s law ˆ G | φ (cid:105) U (2) = 0 withˆ G ( x ) = 12 (cid:104) − ˆ E ( x ) + ˆ E ( x − (cid:105) + ˆ ρ ( x ) . (35)Explicitly, the U(1) gauss’s law acting on the new basisstate | J R , m R (cid:105) ( x − ⊗ | f , f (cid:105) ( x ) ⊗ | J L , m L (cid:105) ( x ) ⊗ | E (cid:105) ( x ) atsite x gives E ( x ) − E ( x − 1) = f ( x ) + f ( x ) − (1 − ( − x ) . (36)Here, OBC is considered with E ( − 1) = (cid:15) , where (cid:15) is aconstant integer. From Eqs. (32) and (36), the { f , f } quantum numbers at each site become redundant, as theycan be written as f i ( x ) = 12 [ E ( x ) − E ( x − 1) + (1 − ( − x )] + s i [ m L ( x ) + m R ( x − , (37)for i = 1 , 2. Here, s = − 1, and s = 1.As a consequence of Eq. (37), the action of the massHamiltonian ∝ ψ † ( x ) ψ ( x ) on | f , f (cid:105) ( x ) can be writtenas the action of the corresponding operators in the set { ˆ J R ( x − , ˆ J L ( x ) , ˆ E ( x − , ˆ E ( x ) } on the bosonic basisstates in the physical Hilbert space, effectively renderinga purely bosonic term. However, the action of the matter-gauge interaction Hamiltonian ∝ ψ † ( x ) U ( x ) ψ ( x +1)+h . c . will be non-trivial due to the non-commuting nature ofthe fermions, and this feature must be built in the purelybosonic formulation explicitly. In other words, the gauge-matter interaction Hamiltonian must carry the informa-tion regarding fermionic signs in its purely bosonic form.While there are a number of protocols for transform-ing the fermions to hardcore bosons (spins) such that the The staggered term in the U(1) charge ensures that filled evenand odd sites have opposite charges, corresponding to the pres-ence of matter and antimatter at the site, respectively. fermionic anti-commutation relations are preserved, suchas the familiar Jordan-Wigner transformation [107], analternative protocol that preserves the locality of interac-tions is presented in Ref. [94]. For a generic SU ( N ) the-ory, this protocol amounts to augmenting the theory withan additional Z gauge symmetry, where the new localGauss’s law corresponding to the auxiliary gauge keepstrack of the fermionic signs, see also Refs. [108, 109].Back to the case of bosonized SU(2) theory, the extendedU(2) theory (necessitated by the need for an extra diago-nal Gauss’s law) includes the Z symmetry as a subgroup.Therefore, the protocol of Ref. [94] does not require intro-ducing an additional Z gauge group. In other words, thetransformation from fermions to hardcore bosons can al-ready proceed by exploiting the local U(1) electric fieldsintroduced above. Nonetheless, since the Hamiltonianterms are either local or nearest neighbor, and that onlya 1+1 D theory is considered in the current work, allsuch transformation, local or non-local variants, are ofcomparable (low) complexity. Distinctions among differ-ent transformations become more relevant in the contextof quantum simulation, in which the fermions need tobe mapped to qubit DOF. Such considerations will bestudied in future work.Besides their coupling to the fermions in the modifiedmatter-gauge interaction Hamiltonian, which guaranteesthe U(1) Gauss’s law constraint, no further dynamics isintroduced for the U(1) gauge DOF. As a result, apartfrom the issue of the fermionic statistics that needs tobe dealt with via a separate transformation as discussedabove, the Hamiltonian of the extended U(2) theory isa straightforward extension of the KS Hamiltonian pre-sented in Sec. II A, as shown in Ref. [93]. The extendedHamiltonian involves nearest-link interactions, as a resultof the replacement in Eq. (37) and the transformation tohardcore bosons, but is otherwise local. The physicalHilbert spaces of the SU(2) theory and the U(2) theoryare isomorphic, meaning that in the limit where the U(1)gauge link approaches unity, the Hamiltonian matrix ele-ments in the original theory is recovered from those in theextended theory. This is established straightforwardlyfor OBC, while for PBC, the isomorphism holds only ina given topological sector [93]. In the remainder of thispaper, we analyze the dimensionality of, and the resourcerequirement for constructing, the physical Hilbert spaceof the bosonized theory compared with the KS theory,along with the effect of the U(1) cutoff on the spectrum. D. Loop-String-Hadron formulation An alternate reformulation of Kogut-Susskind Hamilto-nian formalism in terms of Schwinger bosons, known asthe prepotential formalism, has been developed over thepast decade [96–100, 102–105]. In a recent work [95], theprepotential formalism of the SU(2) LGT has been madecomplete to include staggered fermions, explicit Hamil-tonian, and the associated Hilbert space. In this sec- FIG. 2. The KS LGT is illustrated in terms of the DOF of the LSH formulation. The left and right electric fields and thegauge link are replaced by a set of prepotential (Schwinger boson) doublets, (cid:0) a L ( R ) a L ( R ) (cid:1) , at each end of a link. As a consequenceof this construction, the gauge link explicitly breaks into a left part and a right part. The staggered matter remains the sameas in original KS formalism, see Fig. 1. The LSH framework confines the gauge group at and around each site as indexed inthe figure. tion, the LSH formalism in 1+1 D will be introduced.Later on, we demonstrate the advantage of this formu-lation compared with the original KS theory within theangular-momentum basis in the physical sector, and withthe purely fermionic and bosonic formulations.Within the prepotential framework, the original canon-ical conjugate variables of the theory, i.e, electric-fieldand link operators are replaced by a set of harmonic-oscillator doublets, defined at each end of a link as shownin Fig. 2. Both the electric-field as well as the link oper-ators can be re-expressed in terms of Schwinger bosonsto satisfy all properties of these variables spelled out insection II A. However, the most important feature of theprepotential formalism is that the link operators U , orig-inally defined over a link connecting neighboring sites( x, x + 1), are now split into a product of two parts: U ( x ) = U L ( x ) U R ( x ) , (38)where U L ( R ) is left (right) part of the link attached tosite x ( x + 1). As a result of this decomposition, thegauge group is now totally confined to each lattice site,which allows one to define gauge-invariant operators andstates locally. For the pure gauge theory, these localgauge-invariant operators and states can be interpretedas local snapshots of Wilson-loop operators of the orig-inal gauge theory. One can now construct a local loopHilbert space by the action of local loop operators on thestrong-coupling vacuum defined locally at each site. Atthis point, it must be emphasized that mapping the localloop picture to the original loop description of the gaugetheory requires one extra constraint on each link. Thisconstraint demands that the states must satisfy N L ( x ) = N R ( x ) , (39)where N L ( R ) counts the total number of Schwingerbosons residing at the left (right) end of a link connectingsites x and x + 1. This constraint is a consequence ofthe relation E L = E R on the link and is equivalent tothe constraint J L = J R in the angular-momentum basis. In the notation of Ref. [95], these are indicated as N L ( R ) . The inclusion of the staggered fermionic matter in theSU(2) LGT is straightforward, and combines smoothlywith the local loop description obtained in the prepoten-tial framework. The reason is that both the prepotentialSchwinger bosons and the matter fields associated with agiven site transform in the fundamental representation ofthe local SU(2). One can now combine matter and pre-potential to construct local string operators, besides localloop operators. Acting on the strong-coupling vacuum,these build a larger local gauge-invariant Hilbert space,including string and ‘hadron’ states. This complete de-scription is named the LSH formalism in Ref. [95]. TheLSH formalism is briefly described in the following, fo-cusing on necessary steps for working with this formalismin one spatial dimension.Within the LSH framework, a gauge-invariant and or-thonormal basis is chosen, that is defined locally at eachsite and is characterized by a set of three integers: n l ( x ) , n i ( x ) , n o ( x ) , (40)for all x . The three quantum numbers signify loop, in-coming string, and outgoing string at each site, respec-tively. The allowed values of these integers are givenby 0 ≤ n l ( x ) ≤ ∞ , (41) n i ( x ) ∈ { , } , (42) n o ( x ) ∈ { , } . (43)It is clear from the range of the quantum numbers that n l is bosonic, whereas n i and n o are fermionic in nature.In terms of the LSH formalism, the operator buildingthe local string Hilbert space consists of SU(2)-invariant Note that the string quantum numbers were named ‘quark’ quan-tum numbers in Ref. [95] to remove the ambiguity associatedwith the absence of any string when a hadron is present at thesite, see Fig. 3. n i and n o will be called string quantum numbersthroughout but a state with n i = n o = 1 and n l = 0 should beunderstood as a state with no string starting and ending at the‘quarks’. x = 0 x = 1 x = 2 x = 3 x = 4 x = 5 x = 0 x = 1 x = 2 x = 3 x = 4 x = 5 | n l = 0, n i = 0, n o = 1 | n l = 1, n i = 0, n o = 1 | n l = 2, n i = 0, n o = 1 | n l = 2, n i = 1, n o = 0 | n l = 1, n i = 1, n o = 0 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 0, n o = 1 | n l = 1, n i = 0, n o = 1 | n l = 2, n i = 1, n o = 1 | n l = 1, n i = 1, n o = 0 | n l = 1, n i = 0, n o = 0 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 0, n o = 1 | n l = 1, n i = 0, n o = 1 | n l = 1, n i = 1, n o = 0 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 0, n o = 0 | n l = 0, n i = 1, n o = 1 | n l = 0, n i = 0, n o = 1 | n l = 1, n i = 0, n o = 1 | n l = 2, n i = 1, n o = 1 | n l = 1, n i = 1, n o = 0 | n l = 1, n i = 0, n o = 0 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 1, n o = 1 | n l = 0, n i = 0, n o = 1 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 1, n o = 1 | n l = 0, n i = 1, n o = 1 | n l = 0, n i = 1, n o = 1 ⌦ ⌦ ⌦⌦ ⌦ ⌦⌦ ⌦ ⌦ ⌦⌦ ⌦ ⌦ ⌦ ⌦⌦ ⌦ ⌦ ⌦ ⌦⌦ ⌦ ⌦ ⌦ ⌦ FIG. 3. The LSH states are illustrated on a 1D lattice with N = 6, where N denotes the number of staggered sites. Here, agauge-invariant state is characterized by loop quantum numbers, n l , and string quantum numbers: n i for an incoming stringand n o for an outgoing string. Any state may consist any number of open or closed strings (depending on the boundaryconditions as well as the cuto↵ on the boson excitations), and any number of hadrons up to N . A sample of LSH states areshown on this lattice, denoted by a set of strings and/or hadrons. The number of solid thick lines passing through a site denotesthe loop quantum number n l . The number of solid disks (red or green) at a site from (to) which a thick line starts (ends)denotes outgoing (incoming) string quantum number n o ( n i ). If two strings start and end at the same site, as shown e.g., inthe second panel at x = 3, it is equivalent to a hadron sitting on top of a loop, and is denoted as | n l = 1 , n i = 1 , n o = 1 i . Asingle hadron present at a site is denoted by | n l = 0 , n i = 1 , n o = 1 i (e.g., the last panel at x = 0). | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) Within the LSH framework, a gauge-invariant and or-thonormal basis is chosen, that is defined locally at eachsite and is characterized by a set of three integers: n l ( x ) , n i ( x ) , n o ( x ) , (40)for all x . The three quantum numbers signify loop, in-coming string, and outgoing string at each site. The al- lowed values of these integers are given by0 n l ( x ) 1 , (41)0 n i ( x ) , (42)0 n O ( x ) . (43)It is clear from the range of the quantum numbers that n l is bosonic, whereas n i and n o are fermionic in nature.In terms of the LSH formalism, the operator buildingthe local string Hilbert space consists of SU(2)-invariant10 x = 0 x = 1 x = 2 x = 3 x = 4 x = 5 | n l = 0, n i = 0, n o = 1 | n l = 1, n i = 0, n o = 1 | n l = 2, n i = 0, n o = 1 | n l = 2, n i = 1, n o = 0 | n l = 1, n i = 1, n o = 0 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 0, n o = 1 | n l = 1, n i = 0, n o = 1 | n l = 2, n i = 1, n o = 1 | n l = 1, n i = 1, n o = 0 | n l = 1, n i = 0, n o = 0 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 0, n o = 1 | n l = 1, n i = 0, n o = 1 | n l = 1, n i = 1, n o = 0 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 0, n o = 0 | n l = 0, n i = 1, n o = 1 | n l = 0, n i = 0, n o = 1 | n l = 1, n i = 0, n o = 1 | n l = 2, n i = 1, n o = 1 | n l = 1, n i = 1, n o = 0 | n l = 1, n i = 0, n o = 0 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 1, n o = 1 | n l = 0, n i = 0, n o = 1 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 1, n o = 1 | n l = 0, n i = 1, n o = 1 | n l = 0, n i = 1, n o = 1 ⌦ ⌦ ⌦⌦ ⌦ ⌦⌦ ⌦ ⌦ ⌦⌦ ⌦ ⌦ ⌦ ⌦⌦ ⌦ ⌦ ⌦ ⌦⌦ ⌦ ⌦ ⌦ ⌦ FIG. 3. The LSH states are illustrated on a 1D lattice with N = 6, where N denotes the number of staggered sites. Here, agauge-invariant state is characterized by loop quantum numbers, n l , and string quantum numbers: n i for an incoming stringand n o for an outgoing string. Any state may consist any number of open or closed strings (depending on the boundaryconditions as well as the cuto↵ on the boson excitations), and any number of hadrons up to N . A sample of LSH states areshown on this lattice, denoted by a set of strings and/or hadrons. The number of solid thick lines passing through a site denotesthe loop quantum number n l . The number of solid disks (red or green) at a site from (to) which a thick line starts (ends)denotes outgoing (incoming) string quantum number n o ( n i ). If two strings start and end at the same site, as shown e.g., inthe second panel at x = 3, it is equivalent to a hadron sitting on top of a loop, and is denoted as | n l = 1 , n i = 1 , n o = 1 i . Asingle hadron present at a site is denoted by | n l = 0 , n i = 1 , n o = 1 i (e.g., the last panel at x = 0). | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) Within the LSH framework, a gauge-invariant and or-thonormal basis is chosen, that is defined locally at eachsite and is characterized by a set of three integers: n l ( x ) , n i ( x ) , n o ( x ) , (40)for all x . The three quantum numbers signify loop, in-coming string, and outgoing string at each site. The al- lowed values of these integers are given by0 n l ( x ) 1 , (41)0 n i ( x ) , (42)0 n O ( x ) . (43)It is clear from the range of the quantum numbers that n l is bosonic, whereas n i and n o are fermionic in nature.In terms of the LSH formalism, the operator buildingthe local string Hilbert space consists of SU(2)-invariant10 x = 0 x = 1 x = 2 x = 3 x = 4 x = 5 | n l = 0, n i = 0, n o = 1 | n l = 1, n i = 0, n o = 1 | n l = 2, n i = 0, n o = 1 | n l = 2, n i = 1, n o = 0 | n l = 1, n i = 1, n o = 0 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 0, n o = 1 | n l = 1, n i = 0, n o = 1 | n l = 2, n i = 1, n o = 1 | n l = 1, n i = 1, n o = 0 | n l = 1, n i = 0, n o = 0 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 0, n o = 1 | n l = 1, n i = 0, n o = 1 | n l = 1, n i = 1, n o = 0 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 0, n o = 0 | n l = 0, n i = 1, n o = 1 | n l = 0, n i = 0, n o = 1 | n l = 1, n i = 0, n o = 1 | n l = 2, n i = 1, n o = 1 | n l = 1, n i = 1, n o = 0 | n l = 1, n i = 0, n o = 0 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 1, n o = 1 | n l = 0, n i = 0, n o = 1 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 1, n o = 1 | n l = 0, n i = 1, n o = 1 | n l = 0, n i = 1, n o = 1 ⌦ ⌦ ⌦⌦ ⌦ ⌦⌦ ⌦ ⌦ ⌦⌦ ⌦ ⌦ ⌦ ⌦⌦ ⌦ ⌦ ⌦ ⌦⌦ ⌦ ⌦ ⌦ ⌦ FIG. 3. The LSH states are illustrated on a 1D lattice with N = 6, where N denotes the number of staggered sites. Here, agauge-invariant state is characterized by loop quantum numbers, n l , and string quantum numbers: n i for an incoming stringand n o for an outgoing string. Any state may consist any number of open or closed strings (depending on the boundaryconditions as well as the cuto↵ on the boson excitations), and any number of hadrons up to N . A sample of LSH states areshown on this lattice, denoted by a set of strings and/or hadrons. The number of solid thick lines passing through a site denotesthe loop quantum number n l . The number of solid disks (red or green) at a site from (to) which a thick line starts (ends)denotes outgoing (incoming) string quantum number n o ( n i ). If two strings start and end at the same site, as shown e.g., inthe second panel at x = 3, it is equivalent to a hadron sitting on top of a loop, and is denoted as | n l = 1 , n i = 1 , n o = 1 i . Asingle hadron present at a site is denoted by | n l = 0 , n i = 1 , n o = 1 i (e.g., the last panel at x = 0). | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) Within the LSH framework, a gauge-invariant and or-thonormal basis is chosen, that is defined locally at eachsite and is characterized by a set of three integers: n l ( x ) , n i ( x ) , n o ( x ) , (40)for all x . The three quantum numbers signify loop, in-coming string, and outgoing string at each site. The al- lowed values of these integers are given by0 n l ( x ) 1 , (41)0 n i ( x ) , (42)0 n O ( x ) . (43)It is clear from the range of the quantum numbers that n l is bosonic, whereas n i and n o are fermionic in nature.In terms of the LSH formalism, the operator buildingthe local string Hilbert space consists of SU(2)-invariant10 x = 0 x = 1 x = 2 x = 3 x = 4 x = 5 | n l = 0, n i = 0, n o = 1 | n l = 1, n i = 0, n o = 1 | n l = 2, n i = 0, n o = 1 | n l = 2, n i = 1, n o = 0 | n l = 1, n i = 1, n o = 0 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 0, n o = 1 | n l = 1, n i = 0, n o = 1 | n l = 2, n i = 1, n o = 1 | n l = 1, n i = 1, n o = 0 | n l = 1, n i = 0, n o = 0 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 0, n o = 1 | n l = 1, n i = 0, n o = 1 | n l = 1, n i = 1, n o = 0 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 0, n o = 0 | n l = 0, n i = 1, n o = 1 | n l = 0, n i = 0, n o = 1 | n l = 1, n i = 0, n o = 1 | n l = 2, n i = 1, n o = 1 | n l = 1, n i = 1, n o = 0 | n l = 1, n i = 0, n o = 0 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 1, n o = 1 | n l = 0, n i = 0, n o = 1 | n l = 0, n i = 1, n o = 0 | n l = 0, n i = 1, n o = 1 | n l = 0, n i = 1, n o = 1 | n l = 0, n i = 1, n o = 1 ⌦ ⌦ ⌦⌦ ⌦ ⌦⌦ ⌦ ⌦ ⌦⌦ ⌦ ⌦ ⌦ ⌦⌦ ⌦ ⌦ ⌦ ⌦⌦ ⌦ ⌦ ⌦ ⌦ FIG. 3. The LSH states are illustrated on a 1D lattice with N = 6, where N denotes the number of staggered sites. Here, agauge-invariant state is characterized by loop quantum numbers, n l , and string quantum numbers: n i for an incoming stringand n o for an outgoing string. Any state may consist any number of open or closed strings (depending on the boundaryconditions as well as the cuto↵ on the boson excitations), and any number of hadrons up to N . A sample of LSH states areshown on this lattice, denoted by a set of strings and/or hadrons. The number of solid thick lines passing through a site denotesthe loop quantum number n l . The number of solid disks (red or green) at a site from (to) which a thick line starts (ends)denotes outgoing (incoming) string quantum number n o ( n i ). If two strings start and end at the same site, as shown e.g., inthe second panel at x = 3, it is equivalent to a hadron sitting on top of a loop, and is denoted as | n l = 1 , n i = 1 , n o = 1 i . Asingle hadron present at a site is denoted by | n l = 0 , n i = 1 , n o = 1 i (e.g., the last panel at x = 0). | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) | , , i (0) ⌦ | , , i (1) ⌦ | , , i (2) ⌦ | , , i (3) ⌦ | , , i (4) ⌦ | , , i (5) Within the LSH framework, a gauge-invariant and or-thonormal basis is chosen, that is defined locally at eachsite and is characterized by a set of three integers: n l ( x ) , n i ( x ) , n o ( x ) , (40)for all x . The three quantum numbers signify loop, in-coming string, and outgoing string at each site. The al- lowed values of these integers are given by0 n l ( x ) 1 , (41)0 n i ( x ) , (42)0 n O ( x ) . (43)It is clear from the range of the quantum numbers that n l is bosonic, whereas n i and n o are fermionic in nature.In terms of the LSH formalism, the operator buildingthe local string Hilbert space consists of SU(2)-invariant FIG. 3. The LSH states are illustrated on a one-dimensional lattice with N = 6, where N denotes the number of staggeredsites. Here, a gauge-invariant state is characterized by | φ (cid:105) (LSH) = (cid:81) N − x =0 | n l , n i , n o (cid:105) ( x ) , with loop quantum numbers, n l , andstring quantum numbers: n i for an incoming string and n o for an outgoing string. Any state may consist of any number ofopen or closed strings (depending on the boundary conditions as well as the cutoff on the boson excitations), and any numberof hadrons up to N . A sample of LSH states are shown on this lattice, denoted by a set of strings and/or hadrons. The numberof solid thick lines passing through a site denotes the loop quantum number n l . The number of solid disks (red or green) at asite from (to) which a thick line starts (ends) denotes outgoing (incoming) string quantum number n o ( n i ). If two strings startand end at the same site, it is equivalent to a hadron sitting on top of a loop, and is denoted as | n l = 1 , n i = 1 , n o = 1 (cid:105) . Asingle hadron present at a site is denoted by | n l = 0 , n i = 1 , n o = 1 (cid:105) . bilinears of one bosonic prepotential operator and onefermionic matter field, yielding overall fermionic statis-tics, whereas the local loop Hilbert space is constructedby the action of SU(2)-invariant bilinears of two bosonicprepotential operators. Such operators will not be intro-duced in detail, instead the Hamiltonian will be writtenshortly in this operator basis. Characterization of gauge-invariant states on a one-dimensional lattice consisting ofsix staggered sites in terms of the three quantum num-bers is illustrated in Fig. 3 via a few examples coveringany situation that can occur within this theory.Let us define a set of LSH operators consisting of diag-onal and ladder operators locally at each site as follow-ing: ˆ n l | n l , n i , n o (cid:105) = n l | n l , n i , n o (cid:105) , (44)ˆ n i | n l , n i , n o (cid:105) = n i | n l , n i , n o (cid:105) , (45)ˆ n o | n l , n i , n o (cid:105) = n o | n l , n i , n o (cid:105) , (46) In the notation of Ref. [95], ˆ n l , ˆ n i , and ˆ n o are indicated as ˆ N l , ˆ N i ,and ˆ N o , respectively. Further, in that reference ˆ λ ± is indicatedas ˆΛ ± . ˆ λ ± | n l , n i , n o (cid:105) = | n l ± , n i , n o (cid:105) , (47)ˆ χ + i | n l , n i , n o (cid:105) = (1 − δ n i , ) | n l , n i + 1 , n o (cid:105) , (48)ˆ χ − i | n l , n i , n o (cid:105) = (1 − δ n i , ) | n l , n i − , n o (cid:105) , (49)ˆ χ + o | n l , n i , n o (cid:105) = (1 − δ n , ) | n l , n i , n o + 1 (cid:105) , (50)ˆ χ − o | n l , n i , n o (cid:105) = (1 − δ n , ) | n l , n i , n o − (cid:105) . (51)Here, the site index x is made implicit for brevity. Beingan SU(2) gauge-invariant basis, one no longer needs tosatisfy the SU(2) Gauss’s laws at each site. However,the neighboring sites still need to be glued together bythe Abelian Gauss’s law, i.e., Eq. (39). In terms of theLSH operators, the Abelian Gauss’s law reads as:ˆ n l ( x ) + ˆ n o ( x )(1 − ˆ n i ( x ))= ˆ n l ( x + 1) + ˆ n i ( x + 1)(1 − ˆ n o ( x + 1)) . (52)Upon acting on the LSH basis states and comparing withEq. (39), one obtains: N L ( x ) = n l ( x ) + n o ( x )(1 − n i ( x )) , (53) N R ( x ) = n l ( x + 1) + n i ( x + 1)(1 − n o ( x + 1)) , (54)where, N L ( x ) and N R ( x ) count bosonic occupation num-bers at each end of the link connecting site x and x + 1.Pictorially, the left and right sides of Eq. (39) are repre-sented in Fig. 3 by the number of solid lines on the left1and right side of each link, respectively. Another impor-tant relation isˆ N ψ ( x ) ≡ ψ † ( x ) ψ ( x ) = ˆ n i ( x ) + ˆ n o ( x ) , (55)which establishes the relation between the { f , f } quan-tum numbers of the original formulation and the stringquantum number of the LSH formulation.The Hamiltonian of the SU(2) LGT coupled to matterin the LSH formulation can be written in terms of theLSH operators and is given by: H (LSH) = H (LSH) I + H (LSH) E + H (LSH) M . (56)Here, H (LSH) I is the matter-gauge interaction term, H (LSH) E is the electric-energy term, and H (LSH) M is themass term. Explicitly, in terms of the LSH operators de-fined in Eqs. (44)-(51), each part of the Hamiltonian canbe written as [95]: H (LSH) I = 12 a (cid:88) n (cid:40) (cid:112) ˆ n l ( x ) + ˆ n o ( x )(1 − ˆ n i ( x )) + 1 × (cid:104) ˆ S ++ o ( x ) ˆ S + − i ( x + 1) + ˆ S + − o ( x ) ˆ S −− i ( x + 1) (cid:105) × (cid:112) ˆ n l ( x + 1) + ˆ n i ( x + 1)(1 − ˆ n o ( x + 1)) + 1 + h . c . (cid:41) , (57) H (LSH) E = g a (cid:88) n (cid:34) ˆ n l ( x ) + ˆ n o ( x )(1 − ˆ n i ( x ))2 × (cid:18) ˆ n l ( x ) + ˆ n o ( x )(1 − ˆ n i ( x ))2 + 1 (cid:19) (cid:35) , (58) H (LSH) M = m (cid:88) n ( − x (ˆ n i ( x ) + ˆ n o ( x )) , (59)where (57) contains the LSH ladder operators in the fol-lowing combinations (suppressing the site indices):ˆ S ++ o = ˆ χ + o ( λ + ) ˆ n i (cid:112) ˆ n l + 2 − ˆ n i , (60)ˆ S −− o = ˆ χ − o ( λ − ) ˆ n i (cid:112) ˆ n l + 2(1 − ˆ n i ) , (61)ˆ S + − o = ˆ χ + i ( λ − ) − ˆ n o (cid:112) ˆ n l + 2ˆ n o , (62)ˆ S − + o = ˆ χ − i ( λ + ) − ˆ n o (cid:112) ˆ n l + 1 + ˆ n o , (63)and ˆ S + − i = ˆ χ − o ( λ + ) − ˆ n i (cid:112) ˆ n l + 1 + ˆ n i , (64)ˆ S − + i = ˆ χ + o ( λ − ) − ˆ n i (cid:112) ˆ n l + 2ˆ n i , (65)ˆ S −− i = ˆ χ − i ( λ − ) ˆ n o (cid:112) ˆ n l + 2(1 − ˆ n o ) , (66)ˆ S ++ i = ˆ χ + i ( λ + ) ˆ n o (cid:112) ˆ n l + 2 − ˆ n o . (67)The strong-coupling vacuum of the LSH Hamiltonian isgiven by n l ( x ) = 0 , for all x,n i ( x ) = 0 , n o ( x ) = 0 , for x even , (68) n i ( x ) = 1 , n o ( x ) = 1 , for x odd . It is easy to verify that Eq. (68) satisfies the AbelianGauss’s law, Eq. (52).This completes the introduction of the LSH formula-tion for the SU(2) LGT in 1+1 D. In later sections, thefinite-dimensional Hilbert space of the theory will be con-structed by imposing a cutoff on the N L and N R quantumnumbers, and the associated cost of the classical simula-tion within this framework will be analyzed. III. PHYSICAL HILBERT-SPACE ANALYSIS As introduced in the previous section, the naive basisstates in the KS LGTs spans a Hilbert space that ispredominantly unphysical. The physical sector corre-sponds to the zero eigenvalue of the Gauss’s law oper-ator in Eq. (9). As mentioned before, in contrast to theU(1) LGT, in SU(2) LGT the Gauss’s law is not a singlealgebraic constraint on the eigenvalues of the electric-field operator, but instead, it mixes states with differentelectric-field quantum numbers, and is therefore a non-diagonal constraint when expressed in the electric-fieldbasis. A major complexity in the Hamiltonian formu-lation of non-Abelian LGTs is to diagonalize the Gauss’slaw operator locally to form the physical Hilbert space,as otherwise the computation is prohibitively costly evenin small systems.A question worth addressing is how beneficial it is,from a computational perspective, to work with a formu-lation that solves the Gauss’s law at the level of operatorsas opposed to states (e.g., the LSH formulation) com-pared with a formulation that sustains a simple mappingof the Hilbert space to operators in the Hamiltonian butrequires solving the Gauss’s laws for basis states subse-quently (e.g., the KS formulation in the angular momen-tum representation). Such a cost analysis is presentedin Sec. V, but it requires understanding and analyzingin more detail the steps involved in forming the physicalHilbert space in each formulation and the dimensionalityof the Hilbert spaces involved. Another interesting ques-tion is how fast the dimensionality of the Hilbert spaceand its physical subsector grows as a function of the lat-tice size and the cutoff on the electric-field excitations ineach of the formulations considered. Such questions arestudied in various depth in this section for all the formu-lations introduced in Sec. II, and briefly in Appendix Afor the QLM. In d > D , another relevant basis is the magnetic-field basis,in which the magnetic Hamiltonian is diagonal. The Gauss’slaws in such a basis remain non-diagonal conditions as well. In the following for the sake of brevity, the KS formulation in theangular momentum representation may be called KS formulationin short. ▽ ▽ ▽ ▽ ▽ ▽ ▽▽▽▽ ○ ○ ○ ○ ○ ○ ○○○○ ◻ ◻ ◻ ◻ ◻ ◻ ◻◻◻◻◇ ◇ ◇ ◇ ◇ ◇◇◇◇◇ △ △ △ △ △ △ △△△△ - - ( Λ ) l og ( N S t a t e s ) ▽ N = ○ N = ◻ N = ◇ N = △ N = + [ Λ ] + [ Λ ] + [ Λ ] + [ Λ ] + [ Λ ] ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽▽▽ ○ ○ ○ ○ ○ ○ ○ ○ ○○ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻◻◻◇ ◇ ◇ ◇ ◇ ◇ ◇◇◇◇ △ △ △ △ △ △ △ △△△ - - ( Λ ) l og ( N S t a t e s ) ▽ N = ○ N = ◻ N = ◇ N = △ N = + [ Λ ] + [ Λ ] + [ Λ ] + [ Λ ] + [ Λ ] l og ( M ) ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ △ △ △ △ △ △ △ △ △ △ △ Λ l og ( N s t a t e s ) ▽ N = ○ N = ◻ N = ◇ N = △ N = ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ △ △ △ △ △ △ △ △ △ △ △ Λ l og ( N s t a t e s ) ▽ N = ○ N = ◻ N = ◇ N = △ N = (a)(b) l og ( M ) FIG. 4. The dependence of the logarithm of the dimensionof the physical Hilbert space, M , on (the logarithm of) thecutoff on the electric-field excitations, Λ(= 2 J max ), withinthe KS (and LSH) formulation with PBC (a) and OBC (b),for various lattice sizes N . The lines are empirical fits to thepoints, with the fit values presented in Supplemental Material. A. Gauge-invariant angular-momentum basis Despite significant state reduction after imposing phys-ical constraints on the full basis states, without a finitecutoff on the electric-field quantum numbers, the physi-cal Hilbert space will still be infinite-dimensional. In theremainder of this paper, we impose: J L , J R ≤ J max , anddenote the cutoff as Λ = 2 J max . Examining the depen-dence of the dimension of the Hilbert space, as well asthat of observables, on this cutoff is one of the objec-tives of this work. After the imposition of this cutoff,the states in the physical Hilbert space can be obtainedfollowing the procedure outlined in Sec. II A given theboundary conditions specified. For PBC, the J R value atsite x = 0 and the J L value at site N − J R value at site x = 0 is set to a constant (cid:15) smaller than the cutoff J max , while J L value at site N − N − (cid:15) is set to zero, but the conclusionsdrawn can be extended to other values of this incoming‘flux’.The dimension of the physical Hilbert space, called M throughout, for lattices of the size up to N = 10 andcutoffs up to Λ = 10 is provided in Tables III and IV ofAppendix B for PBC and OBC, respectively. There area few interesting features to observe: (a)(b) ▽ ▽ ▽ ▽ ▽ ○ ○ ○ ○ ○ ◻ ◻ ◻ ◻ ◻◇ ◇ ◇ ◇ ◇ △ △ △ △ △ N l og ( N s t a t e s ) ▽ Λ = ○ Λ = ◻ Λ = ◇ Λ = △ Λ = + N + N + N + N + N + N + N + N + N + N ▽ ▽ ▽ ▽ ▽ ○ ○ ○ ○ ○ ◻ ◻ ◻ ◻ ◻◇ ◇ ◇ ◇ ◇ △ △ △ △ △ N l og ( N s t a t e s ) ▽ Λ = ○ Λ = ◻ Λ = ◇ Λ = △ Λ = + N + N + N + N + N + N + N + N + N + N l og ( M ) ▽ ▽ ▽ ▽ ▽ ○ ○ ○ ○ ○ ◻ ◻ ◻ ◻ ◻◇ ◇ ◇ ◇ ◇ △ △ △ △ △ N l og ( N s t a t e s ) ▽ Λ = ○ Λ = ◻ Λ = ◇ Λ = △ Λ = ▽ ▽ ▽ ▽ ▽ ○ ○ ○ ○ ○ ◻ ◻ ◻ ◻ ◻◇ ◇ ◇ ◇ ◇ △ △ △ △ △ N l og ( N s t a t e s ) ▽ Λ = ○ Λ = ◻ Λ = ◇ Λ = △ Λ = l og ( M ) FIG. 5. The dependence of the logarithm of the dimension ofin the physical Hilbert space, M , on the lattice size N , withinthe KS (and LSH) formulation with PBC (a) and OBC (b), forvarious cutoffs on the electric-field excitations, Λ(= 2 J max ).The lines are empirical fits to the points, with the fit valuespresented in Supplemental Material. (cid:46) For PBC, asymptotically the dimension of thephysical Hilbert space grows linearly as a functionof the cutoff for all x . This feature is evident fromthe plot of M as a function of Λ for various N as shown in Fig. 4(a). This is a consequence ofthe observation that as Λ increases, the numberof new allowed states quickly saturates, i.e., intro-ducing an additional possibility for the J R,L quan-tum numbers amounts to only adding (cid:0) NN (cid:1) pos-sible states. For N = 2 , , , , 10, the cutoff Λ atwhich the growth of states become linear afterwardsis 0 , , , , 4, respectively. The best empirical fitsto this linear dependence are shown in the plot.Second, as expected, the dimension of the physicalHilbert space grows exponentially with the system’ssize at a fixed cutoff, as plotted in Fig. 5(a). Thegrowth, up to constant factors and higher-orderterms in the exponent, can be approximated by M ∼ e pN . The coefficient of N in the exponentapproaches a constant value as a function of cutoff,as shown in Fig. 6(a). This value can be obtainedfrom a fit to points shown in the plot, as depictedin the figure. For moderate N values such that thehigher-order terms in the exponent are negligible,this p value can be used to approximate the di-mension of the physical Hilbert space with PBC asΛ → ∞ .3 (a)(b) ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ Λ E xpon e n t p = + - ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ Λ E xpon e n t / N p = + - ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ Λ E xpon e n t / N p = + - ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ Λ E xpon e n t / N p = + - FIG. 6. The dimension of the physical Hilbert space, M ,within the KS (and LSH) formulation with PBC (a) and OBC(b) is approximated by e pN , and the coefficient of the latticesize, N , in the exponent is obtained from fits to the N depen-dence of M for several values of Λ. The exponents approach,with an exponential form, a fixed value, and the empirical fitto this Λ dependence obtains the asymptotic value of p de-noted by the horizontal lines in the plots and shown in theinset boxes. The uncertainty on these values is estimated byvariations in the fit values when each data point is removedfrom the set, one at a time, and the remaining points are refit.The numerical values associated with these plots are listed inSupplemental Material. (cid:46) For OBC, the dimension of the physical Hilbertspace grows as a function of Λ until it becomesa constant for Λ ≥ N (Λ ≥ N + 2 (cid:15) for an arbi-trary (cid:15) ), as depicted in Fig. 4(b). The reason forthis behavior is that the J quantum number onlychanges (by ) from the left to the right side of site x if the site’s total fermionic occupation number isequal to one. If the J R value at site x = 0 is setto (cid:15) , it can become at most J L = (cid:15) + N/ N can be approximated by an exponentialform, M ∼ e q Λ . The coefficient of Λ in the expo-nent for various values of N is plotted in Fig. 7,and is seen to asymptote to a constant value atlarge N . The fit to this asymptotic value is shownin the plot. This value can be used to approximatethe dimension of the physical Hilbert space for anarbitrary large N and any Λ. Similarly, the de-pendence of the dimension of the physical Hilbertspace on the lattice size can be approximated byan exponential form, M ∼ e pN , for a fixed cutoff, ▽▽▽▽▽ ▽▽▽▽▽ N E xpon e n t / Λ p = + - ▽▽▽▽▽ ▽▽▽▽▽ N E xpon e n t / Λ q = + - FIG. 7. The dimension of the physical Hilbert space, M ,within the KS (and LSH) formulation with OBC is approxi-mated by e q Λ , and the coefficient of the cutoff on the electric-field excitations, Λ(= 2 J max ), in the exponent is obtainedfrom fits to the Λ dependence of M for several values of N .The exponents approach, with an exponential form, a fixedvalue, and the empirical fit to this N dependence obtains theasymptotic value of q denoted by the horizontal line in theplot and shown in the inset box. The uncertainty on thisvalue is estimated by variations in the fit values when eachdata point is removed from the set, one at a time, and theremaining points are refit. The numerical values associatedwith these plots are listed in Supplemental Material. and up to constant factors and higher order termsin the exponent. The coefficient of N in the expo-nent asymptotes to a constant value at large Λ, asshown in Fig. 6(b). (cid:46) The size of the full Hilbert space before implement-ing physical constraints can be approximated by M (full) ( N, Λ) = × (cid:88) j (2 j + 1) N , (69)with PBC, where j = { , , , · · · , Λ2 } . To com-pare this with the dimension of the physical Hilbertspace with PBC, one can again write the lattice-sizedependence of the M as e pN . The coefficient of N in this exponent as a function of Λ can be plot-ted for both the full and physical Hilbert space, asis shown in Fig. 8. As is evident, even for smallvalues of the cutoff, the full Hilbert space growsmuch faster with the system’s size than the phys-ical Hilbert space. For example, with Λ = 5, the p values differ by ≈ 7. This means that for a lat-tice size N = 10, for example, the dimension ofthe full Hilbert space is ≈ 30 orders of magnitudelarger than that of the physical Hilbert space. Asa result, it is not plausible to perform a classicalHamiltonian simulation with a manageable cost ifthe physical constraints are not imposed a priori . Considering the Abelian Gauss’s law that allows assigning onlyone J quantum number to each link. p = 1 . . e . ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ Λ E xpon e n t / N p = 1 . 780 + 3 . 787 log(1 + ⇤) FIG. 8. Shown in blue is the same as in Fig. 6-a, i.e., thecoefficient of the lattice size, N , in the exponent of M ∼ e pN for several values of Λ, where M denotes the dimension of thephysical Hilbert space within the KS (and LSH) formulationwith PBC. The same quantity can be plotted for the dimen-sion of the full Hilbert space, as shown in orange, along withan empirical functional form obtained from a fit to the points.The numerical values associated with these plots are listed inSupplemental Material. Implementing the physical constraints, nonetheless,introduces further complexity at the onset of thecalculation and amounts to an additional prepro-cessing cost. We will come back to this point whencomparing the simulation cost between the KS andLSH formulations in Sec. V. (cid:46) A physical basis state is generally a superpositionof the original angular-momentum basis states, seee.g., the example in Eqs. (70) below. The numberof terms in each superposition can become expo-nentially large in system’s size. This creates sig-nificant complexity when generating the Hamilto-nian matrix, due to the need to keep track of theHamiltonian action on each constituent basis state.The maximum number of terms in a physical stateis plotted in Fig. 9 as a function of Λ for PBC,demonstrating this exponential growth. We willcome back to this feature in Sec. V when analyzingthe computational cost of the Hamiltonian simula-tion. B. Purely fermionic formulation As discussed in Sec. II B, the basis states that representthe Hilbert space of the purely fermionic representationof the KS formulation with OBC consist of the directproduct of on-site fermionic states, see Eq. (30), givingrise to M = 4 N basis states, where N denotes the sizeof the lattice in 1+1 D as before. The dimension of theHilbert space of the fermionic theory is larger than thedimension of the physical Hilbert space of the KS for-mulation in the angular momentum (and LSH) basis forcutoff values that allow the full physical Hilbert space tobe constructed with OBC (i.e., Λ ≥ N + 2 (cid:15) ). The ratioof the former to the latter is shown in Fig. 10 for various N , along with an empirical fit form to the ratio as a func- ▽ ○ ○ ○ ○ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ Λ N t e r m s ▽ N = ○ N = ◻ N = ▽ ○ ○ ○ ○ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ ◻ Λ N t e r m s ▽ N = ○ N = ◻ N = N terms = (2⇤) N FIG. 9. Each physical state in the KS formulation in theangular-momentum basis is, in general, a linear combinationof multiple basis states in Eq. (10). Shown is the number ofterms in the longest linear combination formed to representa physical state in the KS theory with PBC, as a function ofthe cutoff, Λ, for various number of lattice sizes N . This num-ber grows polynomially with Λ for a fixed N , while it growsexponentially with N for a fixed Λ, with the form shown. tion of the lattice size. This form shows that the ratioof the dimensions of the two Hilbert spaces asymptotesslowly to a fixed number.To understand this mismatch between the number of(physical) states in both formulations, despite the factthe fermionic formulation is constructed to fully representthe physical Hilbert space, inspecting the following exam-ple will be illuminating. Consider the N = 2 theory inthe ν = 1 sector, where ν denotes the normalized fermionoccupation number on the lattice defined in Eq. (76) be-low. In the KS formulation in the angular-momentumbasis, the only four physical basis states are: 1) [ | , (cid:105) | , (cid:105) | , (cid:105) ] (0) ⊗ [ | , (cid:105) | , (cid:105) | , (cid:105) ] (1) , 2) 12 (cid:20) | , (cid:105) | , (cid:105) | , − (cid:105) (cid:21) (0) ⊗ (cid:20) | , (cid:105) | , (cid:105) | , (cid:105) (cid:21) (1) − (cid:20) | , (cid:105) | , (cid:105) | , − (cid:105) (cid:21) (0) ⊗ (cid:20) | , − (cid:105) | , (cid:105) | , (cid:105) (cid:21) (1) − (cid:20) | , (cid:105) | , (cid:105) | , (cid:105) (cid:21) (0) ⊗ (cid:20) | , (cid:105) | , (cid:105) | , (cid:105) (cid:21) (1) + 12 (cid:20) | , (cid:105) | , (cid:105) | , (cid:105) (cid:21) (0) ⊗ (cid:20) | , − (cid:105) | , (cid:105) | , (cid:105) (cid:21) (1) , Such states are constructed efficiently in Ref. [90] by acting bythe interacting Hamiltonian on the strong-coupling vacuum, i.e.,state 1) shown, but they differ in relative signs with the statespresented here. Nonetheless, only the signs denoted here giverise to gauge-invariant states as can be checked by acting by theGauss’s law operators in Eq. (9) on the states shown. 53) 1 √ (cid:20) | , (cid:105) | , (cid:105) | , − (cid:105) (cid:21) (0) ⊗ (cid:20) | , (cid:105) | , (cid:105) | , − (cid:105) (cid:21) (1) − √ (cid:20) | , (cid:105) | , (cid:105) | , − (cid:105) (cid:21) (0) ⊗ (cid:20) | , (cid:105) | , (cid:105) | , (cid:105) (cid:21) (1) − √ (cid:20) | , (cid:105) | , (cid:105) | , − (cid:105) (cid:21) (0) ⊗ (cid:20) | , − (cid:105) | , (cid:105) | , (cid:105) (cid:21) (1) + 1 √ (cid:20) | , (cid:105) | , (cid:105) | , − (cid:105) (cid:21) (0) ⊗ (cid:20) | , − (cid:105) | , (cid:105) | , (cid:105) (cid:21) (1) − √ (cid:20) | , (cid:105) | , (cid:105) | , (cid:105) (cid:21) (0) ⊗ (cid:20) | , (cid:105) | , (cid:105) | , − (cid:105) (cid:21) (1) + 12 √ (cid:20) | , (cid:105) | , (cid:105) | , (cid:105) (cid:21) (0) ⊗ (cid:20) | , (cid:105) | , (cid:105) | , (cid:105) (cid:21) (1) + 12 √ (cid:20) | , (cid:105) | , (cid:105) | , (cid:105) (cid:21) (0) ⊗ (cid:20) | , − (cid:105) | , (cid:105) | , (cid:105) (cid:21) (1) − √ (cid:20) | , (cid:105) | , (cid:105) | , (cid:105) (cid:21) (0) ⊗ (cid:20) | , − (cid:105) | , (cid:105) | , (cid:105) (cid:21) (1) , 4) [ | , (cid:105) | , (cid:105) | , (cid:105) ] (0) ⊗ [ | , (cid:105) | , (cid:105) | , (cid:105) ] (1) , (70)where each triplet in the square brackets denotes[ | J R , m R (cid:105) ⊗ | f , f (cid:105) ⊗ | J L , m L (cid:105) ] ( x ) at the correspondingsite x , and the direct product symbol is suppressed insuch triplets for brevity. On the other hand, in the purelyfermionic representation of the same theory, the six basisstates are 1) | , (cid:105) (0) ⊗ | , (cid:105) (1) , | , (cid:105) (0) ⊗ | , (cid:105) (1) , | , (cid:105) (0) ⊗ | , (cid:105) (1) , | , (cid:105) (0) ⊗ | , (cid:105) (1) , | , (cid:105) (0) ⊗ | , (cid:105) (1) , | , (cid:105) (0) ⊗ | , (cid:105) (1) . (71)As is seen, while all the six possible fermionic configura-tions in the ν = 1 sector are present in the physical basisstates of the KS formulation in the angular-momentumbasis, only two proper linear combinations of states 2)-5)in the fermionic formulation appear in the KS formula-tion in the angular-momentum basis. Further inspectionof the two representations reveals that the spectrum ofboth theories matches exactly for all values of the cou-plings, but with degeneracies present in the fermioniccase. To conclude, the fermionic representation of theSU(2) LGT in 1+1 D with OBC has redundancies in therepresentation compared with the KS formulation in theangular-momentum basis, however it avoids complex lin-ear combinations of basis states that arise in the latterdue to the imposition of the Gauss’s laws. As will bediscussed in Sec. III D, the LSH formulation of the SU(2)LGT is free from the redundancies of the fermionic for-mulation, while at the same time it does not involve acumbersome physical Hilbert-space construction. ▽ ▽ ▽ ▽ N R a ti o ▽ ▽ ▽ ▽ N R a ti o N ▽ ▽ ▽ ▽ N R a ti o ⇣ ⇢ (F) /⇢ (LSH) ⌘ ⇣ N (F)states /N (LSH)states ⌘ M (F) /M (LSH) ⇣ M (F) /M (LSH) ⌘ ⇢ (F) /⇢ (LSH) . 209 + 0 . N . 508 + 0 . e . N . . e . N (a)(b) (c) FIG. 10. The upper panel depicts the ratio of the dimensionof the Hilbert space in the fully fermionic formulation withOBC to the dimension of the physical Hilbert space within theKS (and LSH) formulation without removing the gauge DOF(but with a sufficiently large cutoff such that the dimension ofthe Hilbert space saturates to a fixed value), for several valuesof the lattice size, N . The middle panel depicts the density ofthe Hamiltonian matrix within the physical Hilbert space foreach theory. The lower panel is the ratio of the Hamiltonianmatrix densities multiplied by the square of the ratio of thesize of the Hilbert spaces in each theory. This latter quantityenters the analysis of the computational complexity of matrixmanipulation in Sec. V. The numerical values associated withthese plots are listed in Supplemental Material. C. Purely bosonic formulation The physical basis states of the bosonized SU(2) the-ory with OBC are, at the first sight, the direct productof the physical basis states of the KS theory discussedin Sec. III A and the electric-field basis states satisfyingthe extra U(1) Gauss’s law. Recall that the U(1) sym-metry was introduced in the bosonized form to allow theelimination of fermionic DOF in favor of bosonic DOEin the SU(2) theory. The statement above is only trueif the cutoff on the U(1) electric field is set sufficientlyhigh such that all fermionic configurations allowed in thephysical Hilbert space of the SU(2) theory can be real-ized. To make this statement more explicit, consider the6example studied in Sec. III B, where N = 2 and ν = 1in the KS theory with OBC, and the incoming fluxes ofthe SU(2) and U(1) electric fields are set to zero. Aswas shown, while there are six allowed fermionic con-figurations in the purely fermionic representation, thesereduce to four linear combinations of basis states forΛ ≥ ν = 1). Note that with Λ ≥ 2, the physical Hilbertspace of the SU(2) theory is complete. Now consider thepurely bosonic formulation, with Λ denoting the cut-off on the U(1) electric-field excitations. Obviously forΛ = 0, the only state allowed is the strong-couplingvacuum states, i.e., state 1) in Eq. (70), and the phys-ical Hilbert space of the bosonized U(2) theory has di-mension one in the specified sector. For Λ = 1, thereare three states contributing, corresponding to states1), 2), and 3) in Eq. (70), times the U(1) electric-fieldstates | E (cid:105) (0) ⊗ | E (cid:105) (1) = | (cid:105) (0) ⊗ | (cid:105) (1) for state 1) and | (cid:105) (0) ⊗ | (cid:105) (1) for states 2) and 3). Finally, for Λ ≥ 2, allfour states in Eq. (70) are allowed, and the correspondingU(1) electric-field states are those given above for states1), 2), and 3), and | (cid:105) (0) ⊗ | (cid:105) (1) for state 4).In general, the dimension of the physical Hilbert spaceof the extended U(2) theory approaches that of the orig-inal SU(2) theory with Λ ≥ N , and reaches a saturationvalue at Λ = N . This trend has been depicted in Fig. 11for N = 2 , , , D. Loop-String-Hadron formulation As described in Sec. II D, the Hilbert space of the LSHformulation is spanned by basis states | n l , n i , n o (cid:105) ( x ) , (72)for x = 0 , , , ..., N − 1, subject to the Abelian Gauss’slaw constraint N L ( x ) = N R ( x ) along each link connect-ing sites x and x + 1. N L ( x ) and N R ( x ) quantum num-bers are expressed in terms of the LSH quantum num-bers according to Eqs. (53) and (54). In the following,an efficient procedure for generating the physical Hilbertspace of the LSH formulation will be presented for both N = 2 , ⇤ = 2 N = 4 , ⇤ = 4 N = 6 , ⇤ = 6 N = 8 , ⇤ = 8 ⇤ ▽ ▽ ▽ Λ N S t a t e s ▽ ▽ ▽ ▽ ▽ Λ N S t a t e s ▽ ▽ ▽ ▽ ▽ ▽ ▽ Λ N S t a t e s ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ Λ N S t a t e s M FIG. 11. The dimension of the physical Hilbert space, M ,of the bosonized formulation of the KS theory as a functionof Λ , the cutoff on the electric field excitations of the U(1)subgroup of the extended U(2) theory. The value of Λ, thecutoff on the electric field excitations of the SU(2) subgroup,is fixed to the smallest value at which the physical Hilbertspace of the theory saturates to its full size. The dimensionof the full physical Hilbert space is denoted by the dashedline in each plot. The numerical values associated with theseplots are listed in Supplemental Material. OBC and PBC. It can be shown that this Hilbert spaceis in one-to-one correspondence with the physical Hilbertspace of the KS theory in the angular-momentum repre-sentation, and that LSH formulation is a more economi-cal encoding of such a Hilbert space given its reliance onfully gauge-invariant DOF. (cid:46) OBC fixes the incoming electric flux into the lat-tice. In the language of LSH quantum numbers,this condition reads: N R ( − 1) = (cid:15) . Given this,and using Eqs. (39), (53), and (54) consecutively,it is straightforward to show that n l ( x ) = (cid:15) + x − (cid:88) y =0 ( n o ( y ) − n i ( y )) − n i ( x ) (1 − n o ( x )) . (73)7 FIG. 12. The LSH Hilbert space with OBC on a lattice ofsize N = 2 and for the case of ν = 1 and Λ = 1. The states aredenoted as (cid:81) N − i =0 | n l , n i , n o (cid:105) ( i ) . The four basis states shownare in one-to-one correspondence with the four physical basisstates in the angular-momentum basis presented in Eq. (70). Eq. (73) implies that for the one-dimensional lat-tice with OBC, n l quantum number at any site iscompletely fixed by the boundary condition andstring quantum numbers n i and n o at all the sitesto its left. In other words, once the string quan-tum numbers are specified throughout the lattice,all LSH quantum numbers are known, and hence aparticular gauge-invariant state is specified. Notethat for an N -site lattice, fixing the fermionic quan-tum numbers give rise to 4 N basis states, that is thesame as the number of basis states one obtains withthe purely fermionic formulation of the KS theoryas described in Sec. III B. However, at this point,it should be noted that there exist certain stringconfigurations (for a fixed value of (cid:15) ) that makethe right-hand side of Eq. (73) negative on one ormore sites on the lattice. Such spurious string con-figurations must be discarded, and thus the Hilbertspace of the LSH formulation is smaller in size thanthe purely fermionic formulation. In fact, the di-mension of the LSH Hilbert space comes out to beof exactly the same as that of the physical Hilbertspace of the KS theory in the angular-momentumbasis. However, the cost of generating the phys-ical Hilbert space is much less than that of theKS formulation in the angular momentum basis, asthe states no longer need to satisfy SU(2) Gauss’slaws at each site (since this has already been takencare of by the LSH construction), and the only re- maining Gauss’s law that is Abelian in nature issolved analytically. Moreover, while working witha cutoff Λ, the LSH Hilbert space is constrained toonly contain those string configurations that yield0 ≤ N L/R ( x ) ≤ Λ for all x . An example of the LSHHilbert space with OBC on a lattice of size N = 2and with Λ = 2 and ν = 1 is given in Fig. 12. Thesebasis states are in one-to-one correspondence withthe physical basis states in the angular-momentumbasis, i.e., the states enumerated in Eq. (70). (cid:46) PBC implies that0 ≤ N R ( − 1) = N L ( N − ≤ Λ , (74)yielding (Λ + 1)4 N states to start with. Identifying (cid:15) ≡ N R ( − 1) and following the same prescriptionoutlined above for OBC, all possible states in theLSH Hilbert space can be constructed subject tocutoff Λ. Note that with PBC, one obtains manycopies of the same { n i , n o } configurations corre-sponding to different winding numbers, i.e., thenumber of closed loops that go around the lattice.A detailed account of the global symmetries of theKS theory will be presented in the next section.The LSH Hilbert space, both for OBC and PBC, isidentical to the physical Hilbert space of the KS theory,in the sense that each state in the LSH basis correspondsto one and only one state in the KS physical Hilbertspace and vice versa. Therefore, all discussions regard-ing the scaling of the physical Hilbert space presentedin Sec. III A are valid for the LSH formulation as well.The only distinction is that the LSH Hilbert space andthe associated Hamiltonian can be generated with far lesscomputational complexity, as will be further discussed inSec. V. IV. REALIZATION OF GLOBAL SYMMETRIES The physical Hilbert space, projected out by imposingthe Gauss’s laws as studied in the previous sections, canbe further characterized by global symmetry sectors aswell as topological configurations. Identification of thesesymmetries can further simplify the Hamiltonian simula-tion, as the Hilbert space that needs to be studied canbe further divided to smaller sectors. These symmetriesare manifested differently in the case of PBC and OBC,and are discussed in the following.The total fermion occupation number is conserved, asthe operator ˆ Q ≡ N − (cid:88) x =0 ψ † ( x ) ψ ( x ) , (75)commutes with the Hamiltonian with both boundaryconditions. Note that in the LSH framework, this quan-tum number is simply Q = (cid:80) x [ n i ( x ) + n o ( x )]. With8 Q q N = 2 and Λ ≥ OBC, Q can be any integer in the interval [0 , N ]. WithPBC, Q can only be an even integer in the same inter-val, as an odd total fermionic occupation number createsan imbalance between the net flux of electric field intosite x = 0 and out of site x = N − 1, which contradictsPBC. For convenience, the global charge associated withthe total number of fermions can be normalized by thelattice size as ν ≡ QN , (76)with ν ∈ [0 , ν = 1 sector of the Hilbert space.Besides the conservation of the total number offermions, there is an additional quantum number thatdivides the Hilbert space of each Q sector to multipledisjoint sectors in general. This quantum number in theLSH language can be written as q ≡ N − (cid:88) x =1 [ n o ( x ) − n i ( x )] , (77)which using the identities in Eq. (54) can be written as q = N L ( N − − N R ( − q = 2 [ J L ( N − − J R ( − q = 2 in Eqs. (70)and Fig. 12 does not evolve to states 1), 2), and 4) with q = 0. This global charge is conserved with OBC since J R ( − 1) is fixed and there is no operator in the Hamil-tonian to affect the J L ( N − 1) quantum number. WithOBC and J R ( − 1) = 0 , q can be any integer in the in-terval [0 , min( N, Λ)]. With PBC, only the q = 0 sectorexists as the net electric field fluxes into and out of theone-dimensional lattice are equal, as mentioned above.An example of the breakdown of the physical Hilbertspace with OBC into the Q and q sectors is given in Ta-ble IV for N = 2 and Λ ≥ N . A larger Hilbert spacecorresponding to N = 10 is analyzed in Appendix B.In addition to the total fermionic number and the netflux, the Hamiltonian and the associated Hilbert spaceare symmetric under charge conjugation. While there isno U (1) charge associated with the fields in the SU(2)LGT, the system is invariant, up to µ → − µ , if the pres-ence of fermions on the lattice is exchanged with theirabsence, i.,e, corresponding to a particle-hole exchange FIG. 13. The physical Hilbert space within the LSH formu-lation with PBC with N = 2, Λ = 3, and Q = 2. symmetry. One can verify that the associated charge con-jugation operator, ˆ C , commutes with the Hamiltonian ofboth PBC and OBC, [ ˆ C, ˆ H ] = 0 , (78)and that ˆ C = 1 , { ˆ Q , ˆ C } = 0 , (79)where ˆ Q ≡ ˆ Q − N ˆ I , with ˆ I being an identity operator.Therefore, for any given state in the Hilbert space withcharge Q : ˆ Q ˆ C | ψ (cid:105) = − ˆ C ˆ Q| ψ (cid:105) + N ˆ C | ψ (cid:105) = (2 N − Q ) ˆ C | ψ (cid:105) . (80)This implies that the charge-conjugated state exhibits acharge 2 N − Q . For an even Q -charge sector, the result-ing spectrum is invariant under µ → − µ , and hence thepair of charge-conjugated Hilbert spaces for { Q, N − Q } are physically identical. For odd Q -charge sectors, thecharge-conjugated Hilbert space is only physically equiv-alent to the original Hilbert space once µ is replaced with − µ .Finally, for PBC it is useful to characterize thestates by a winding number variable l , such that forany state | ψ (cid:105) in the Hilbert space with a given cut-off Λ, [Tr( U (0) U (1) U (2) . . . U ( N − l | ψ (cid:105) is also avalid state of the physical Hilbert space with cutoffsup to Λ + l , where 0 ≤ l < ∞ . Since the opera-tor [Tr( U (0) U (1) U (2) . . . U ( N − l does not commutewith the Hamiltonian, it is not a symmetry of the the-ory, nonetheless, it provides a useful characterization ofthe states. As was explicitly realized in the previoussection, the dimension of the Hilbert space for OBC isfinite for arbitrary Λ. However with PBC, the Hilbert-space dimension grows linearly with the cutoff (once a9 Q l N = 2 and Λ = 3 in terms of the Q quantum number (all states satisfy q = 0). The windingquantum number l , while not a conserved quantity, is alsospecified. saturating value of the cutoff is reached to accommodateall the fermionic configurations). In the linear region,the slope is obtained from the number of all fermionicconfigurations with charge Q = 0, which is (cid:0) NN (cid:1) . Asa result, the complete physical Hilbert space with PBCcontains copies of the particular gauge-invariant Hilbertspace with winding numbers varying from 0 to Λ. Sucha winding-number characterization of PBC Hilbert spaceis evident in the example shown in Fig. 13, where basisstates 1), 2), 3), and 6) are repeated for different valuesof the winding numbers, l = 0 , , , 3. The breakdownof the PBC Hilbert space in terms of the fermionic oc-cupation quantum number Q and the winding number l is worked out for a related example in Table IV. Finally,it should be noted that with PBC, the theory exhibitsa discrete translational symmetry, and the eigenstates ofa discrete momentum operator can be formed as well,see e.g., Ref. [36] for such a classification of momentumeigenstates in the case of the lattice Schwinger model. V. COMPARATIVE (CLASSICAL) COSTANALYSIS A classical algorithm for Hamiltonian simulation, in gen-eral, involves three steps: I ) Hilbert-space construction, II ) Hamiltonian generation, and III ) observable compu-tation. The Hilbert space can be constructed by iden-tifying the theory’s DOF, symmetries, and a convenientbasis to express the states. In the case of LGTs, wherea major portion of the Hilbert space is irrelevant, to re-duce the computational cost, one needs to project to thephysical Hilbert space. This entails one of the follow-ing. One may impose the (non-trivial) Gauss’s laws byreformulating DOF as is the case in the LHS formula-tion. Alternatively, the Gauss’s law constraints can beimposed a posteriori on convenient basis states. Anotheroption is to generate the physical states by the consecu-tive action of the Hamiltonian on a trivial physical statesuch as the strong-coupling vacuum [89, 90]. The associ-ated computational cost of this step, therefore, dependslargely on the Hamiltonian formulation used, as well asthe algorithm itself. In the following, we analyze the firsttwo approaches, noting the third approach is of compa- rable cost as it requires the Hamiltonian matrix to beacted on states by a number of times that grows expo-nentially with the system’s size. After the basis states inthe (physical) Hilbert space are identified, the next stepof the simulation is to generate the Hamiltonian matrix.This step, obviously, depends on the formulation used aswell. For example, some formulations may provide sim-pler operator structures, which could affect the sparsityof the matrix generated. Finally, the Hamiltonian matrixcan be used to compute observables, such as spectrum,and static or dynamical expectation values of operators.This step often entails matrix diagonalization and matrixexponentiation, which can be sped up by efficient sparse-matrix algorithms especially when acted on a sparse statevector.Having introduced various formulations of the SU(2)LGT in 1+1 D and analyzed their physical Hilbert-spacedimensionality with regard to the system’s size, electric-field cutoff, and boundary conditions, one can now an-alyze the classical-simulation cost within each formula-tion. For this purpose, we focus on the KS formula-tion in the physical Hilbert space, the LSH formulation,and the purely fermionic formulation, all with OBC, andwill briefly comment on the case of PBC and the purelybosonic formulation in the end. A. Physical Hilbert-space construction Purely fermionic formulation Within the purely fermionic formulation, redundantgauge symmetries are removed algebraically by an ap-propriate gauge transformation and after applying theGauss’s law repeatedly, as explained in Sec. II B. As such,the projection to the physical Hilbert space is essentiallyfree, and the time complexity is: T (F) I ∼ O (1) . (81)Here, in principle, there is an additional cost as-sociated with generating 4 N fermionic configurations (cid:81) Ni =1 | f , f (cid:105) ( i ) with f , ∈ { , } . Nonetheless, with anefficient Kronecker-product algorithm introduced in thenext subsection, the Hamiltonian can be generated with-out the need to generate and store these fermionic con-figurations. Loop-String-Hadron formulation An efficient algorithm and its associated cost for gener-ating physical Hilbert space of the LSH formulation withOBC goes as follows. One first generates 4 N string con-figurations. The cost of generating each configuration Throughout this paper, computational cost, number of opera-tions, and time complexity are used interchangeably, and are allmeant to convey the same meaning. k associ-ated with each configuration to a binary number, whichgoes as log( k ). This step can therefore be conducted withthe time complexity O ( N ) for a lattice of N sites. Thebinary digits in each generated configuration are then la-beled by n i and n o string numbers, by e.g. assigningthem to the even and odd digits, respectively. This stepis essentially free. Next comes the generation of the n l quantum numbers. As mentioned before, with OBC, theGauss’s law is used to fix this number for any given stringconfiguration. Since n l must be fixed at all links, O ( N )number of operations is needed. There are additional O ( N ) operations required to pick each generated n l andcheck it against the requirement of not exceeding the cut-off as well as being a non-negative integer, as explainedin Sec. III D. However, this step can be simultaneouslyperformed as generating n l quantum numbers consecu-tively, to reduce the cost. The total cost of generatingthe physical Hilbert space with the LSH formulation istherefore: T (LSH) I ∼ O ( N N ) . (82)Note that in order to reduce the dimensionality ofthe Hilbert space, one could additionally restrict thestates to a given global-symmetry sector. For example,if only interested in the charge ν = 1 sector, there are O ( N ) operations involved to check the ν -number of eachstring configuration, reducing the number of configura-tions needed for generation of n l quantum numbers from4 N to (cid:0) NN (cid:1) ≈ N √ N . Since this is not an exponential speedup as a function of the size of the system, such finer de-compositions of the physical Hilbert space will not beconsidered in the rough estimate of the computationalcost in the remainder of this section. Such symmetryconsiderations, however, will be advantageous in practi-cal implementations. Angular-momentum representation An efficient algorithm for the generation of the physicalHilbert space of the KS in the angular-momentum basiswith OBC starts by making a gauge-invariant state atsite x , i.e., one that satisfies the non-Abelian Gauss’slaws in Eq. (18). If there is one and only one fermion atsite x , then a gauge-invariant state is obtained from therelation | ( J R , ) J Rf J L ; 00 (cid:105) ( x ) = (cid:88) m R ,m f ,m Rf ,m L (cid:104) J R , m R ; , m f | J Rf , m Rf (cid:105) (cid:104) J Rf , m Rf ; J L , m L | , (cid:105)| J R , m R (cid:105) ( x − ⊗ | , m f (cid:105) ( x ) ⊗ | J L , m L (cid:105) ( x ) , (83)with the notation defined in Sec. II A. While m f onlytakes values ± in these sums, each sum over m R , m Rf ,and m L involves of the order of J R operations. This isbecause the value of J R fixes the value of J Rf to be equalto J R ± , and that in order for the final total angularmomentum to be zero, the value of J L needs to be equal to J R ± . Now to generate a set of all possible physicalstates, such construction at the site should be repeatedfor all possible values of J R , i.e., 0 ≤ J R ≤ Λ. As a result,the number of operations required to generate a completeset of physical states at a given site is O (Λ ). If, however,there is either no fermion or there are two fermions at site x , the above relation is modified by setting → 0, theexpression simplifies to only two summations, and thefinal number of operations required to generate a gauge-invariant set of states is O (Λ ), which is subdominantcompared with the first case and can be ignored in thelimit Λ (cid:29) 1. Note that in both cases, there is an addi-tional cost involved amounting to checking and removingthe generated J L values that violate 0 ≤ J L ≤ Λ, but thisstep can be checked simultaneously in the sum above, andthe total asymptotic cost in the limit Λ (cid:29) N -fold Kronecker prod-uct of states in each set to connect states at adjacentsites throughout the lattice. This adds a cost with thetime complexity O (cid:0) Λ N (cid:1) . Finally, the boundary con-dition on J R at site x = − J R and J L belonging to the same link are set equal. This can beachieved by a search and elimination algorithm, and in-volves an additional cost that scales as O (cid:0) Λ N (cid:1) , whichis the conservative scaling of the number of basis statesformed in the previous step. As a result, the total timecomplexity of generating the physical Hilbert space is: T (J) I ∼ O (cid:0) Λ + Λ N + Λ N (cid:1) ≈ O (cid:0) Λ N (cid:1) , where ≈ signin this section is meant the approximate scaling in thelimit: N (cid:29) 1. The time complexity if Λ is fixed to a con-stant much smaller than N is: T (J) I ∼ O (cid:0) Λ N + Λ N (cid:1) .An alternative algorithm can be realized by first gener-ating 4 N fermionic configurations throughout the lattice.Each configuration generation involves a time complexitythat scales as O ( N ). Next, given the value of J R at theboundary x = − 1, all J L and J R values can be producedthroughout the lattice, given the known fermionic occu-pation at each site and the Abelian Gauss’s law. Thisinvolves a maximum number of operations that goes as O ( N + N N ), since at each site and given a J R value,there may be two possibilities for the J L value as thefermion occupation may be equal to one. Now given theset of configurations for fermions and angular momentagenerated, the non-Abelian Gauss’s law can be imple-mented following the relation in Eq. (83), introducingan additional cost O (Λ ). The Kronecker-product costis the same as before but there will be no need to im-pose the boundary condition and Abelian Gauss’s lawanymore as these are already implemented. In summary,this algorithm involves a time complexity that scales as T (J) I ∼ O (cid:0) N N + (2Λ) N (cid:1) . (84)For Λ ∼ N and N (cid:29) 1, this later algorithm thereforeis asymptotically faster than what was described earlier.The cost, however, remains super-exponential in N thislimit.1 B. Hamiltonian generation Purely fermionic formulation In the fermionic formulation of the KS theory withOBC, the Hamiltonian becomes non-local with O ( N )terms in the Hamiltonian, see e.g., the electric Hamilto-nian in Eq. (28). However, a considerable advantage isthat the operator structures encountered are only of thetype ψ † ( x ) ψ ( y ), which make the Hamiltonian generationamenable to Kronecker-product algorithms, eliminatingthe need to generate and store the fermionic Hilbert space a priori . Explicitly, each ψ † ( x ) ψ ( y ) operator can be writ-ten as at the Kronecker product of I × at all sites but x and y , A × at site x and B × at site y , in an orderedmanner, where I is the identity matrix, and A and B arematrices formed by the action of ψ † ( x ) and ψ ( y ), respec-tively, on the four allowed fermionic configurations at therespective sites. Since the on-site matrices are sparse andinvolve O ( d i ) elements (with d i = 4 being the dimension-ality of the matrices), performing the Kronecker productalong a lattice of length N comes with the time complex-ity O (4 N ). The total time complexity of this algorithmfor generating the Hamiltonian is therefore: T (F) II ∼ O (cid:0) N N (cid:1) . (85)Note that there are additional O ( N ) operations involvedin finding the position of x and y along the chain, butthat is a subdominant cost compared with the subsequentKronecker-product operation. Loop-String-Hadron formulation In order to generate the LSH Hamiltonian, first note thatthe dimensionality of the physical Hilbert space can beapproximated by 4 N − at large N , estimated by com-paring the asymptotic ratio of the physical Hilbert spacedimension in the fermionic formulation to that of theLSH formulation, as shown in Fig. 10. On the otherhand, in the LSH formulation, the Hamiltonian remainslocal, with the total number of operators scaling as O ( N ).Generating the Hamiltonian matrix elements amounts topicking one state out of the Hilbert space at a time, act-ing by the Hamiltonian operator on the state to arrive atanother state (which requires O ( N ) operations), and findthe position of the state’s assigned index in a previouslyproduced look-up table of states indices, with a time com-plexity that scales as O (4 N − ), and is therefore domi-nant compared with Hamiltonian operation cost. Thisstep identifies the position of the element in the Hamil-tonian matrix and its value. The total time complexity ofHamiltonian generation for the LSH formulation, there-fore, scales as T (LSH) II ∼ O (cid:0) N N (cid:1) . (86)Note that the more efficient Kronecker-product algorithmthat was applied in the fermionic case could not be takenadvantage of in the LSH formulation as the one-to-one mapping between the Kronecker product of on-site phys-ical states and the global physical state is lost given theimposition of the boundary condition and the AbelianGauss’s laws. Such a convenient feature is lost in theangular momentum representation of the physical statesas well. Angular-momentum representation Similar to the LSH formulation, the dimension of thephysical Hilbert space in the angular-momentum basisscales as O (4 N − ), and there are O ( N ) operators in theHamiltonian. The Hamiltonian matrix elements can begenerated by picking one physical state at a time andfind the resulting state after the operation of each termin the Hamiltonian on the chosen state. Despite the LSHstates though, the physical states are linear combina-tions of a multitude of basis states in general. Whilein principle, there are of the order of O (Λ N ) terms foreach physical state, as discussed before, many of thesestates have a vanishing contribution due to the corre-sponding vanishing Clebsch-Gordan coefficients. Empir-ically, the maximum number of terms obtained in a phys-ical state with OBC is seen to scale as O (cid:0) (2Λ) N/ − (cid:1) . Furthermore, the action of the interaction Hamiltonianon each basis state involves O (Λ) operations as is ev-ident from Eq. (17). Finally, the obtained state itselfis a linear combination of other physical states, and re-quires O (4 N − Λ(2Λ) N/ − ) to find its overlap to otherstates in the physical Hilbert space. As a result, thetotal cost of generating the Hamiltonian matrix in theangular-momentum representation scales as T (J) II ∼ O (cid:0) N N Λ N (cid:1) . (87)In the limit Λ ∼ N , this step introduces another super-exponential cost to the Hamiltonian simulation in thisbasis, in addition to the cost of generating the physicalHilbert space as derived in the previous subsection. C. Observable computation General scaling relations The cost of Hamiltonian matrix manipulation requiredto evaluate observables depends upon the dimensional-ity of the Hamiltonian, its sparsity, and the sparsity ofthe state vector in computing expectation values. For asquare matrix with dimensions M × M , the matrix den-sity is defined as the ratio of the number of non-zero ele-ments in the matrix to M . Computation of spectrum inthe Hamiltonian formulation amounts to evaluating the(first m ) eigenvalues, while time-dependent expectation Compare this with the scaling of the maximum number of termsin a physical state for PBC shown in Fig. 9. A | v a (cid:105) = λ a | v a (cid:105) . The power series algorithm forextremal-eigenvalue estimation requires choosing a ran-dom initial unit vector | b (cid:105) = (cid:80) a c a | v a (cid:105) , | c a | = 1. Onerepeats applying A to | b (cid:105) , A k | b (cid:105) = (cid:80) n c n λ ka | v a (cid:105) , andnormalizes the output state after each iteration. Thisprocedure suppresses the contribution of smaller eigen-values and quickly converges to the maximal eigenvalue.The Arnoldi algorithm [110] enhances this procedure bygenerating a Krylov subspace with the set of output vec-tors (cid:8) | b (cid:105) , A | b (cid:105) , A | b (cid:105) , ..., A m − | b (cid:105) (cid:9) that are orthogonal-ized against each other. To generate the lowest eigenval-ues, A is replaced with A − in the procedure outlined.The limiting feature of this approach is the cost of ma-trix multiplication. As the procedure is repeated, it be-comes more costly to expand the Krylov subspace, dueto the growing density of the output vectors. Apply-ing the Lanczos algorithm [111–114], a generalization ofthe Arnoldi algorithm to Hermitian matrices, requires O ( ξρ H M ) operation for an M × M Hamiltonian withmatrix density ρ H . Empirical observations have shownthat ξ ∼ m , where m is the number of desired extremaleigenvalues [112].Among methods to compute matrix exponentiation isthe truncated Taylor series method [115]. For the unitarytime-evolution operator, the k -th order truncated seriesis defined as: e − iHt (cid:12)(cid:12) k = (cid:80) k − l =0 ( − iHt ) l /l !. The numberof terms, k , needed in the Taylor expansion to reach thedesired accuracy depends on time, Hamiltonian’s dimen-sion, and on some form of a Hamiltonian norm, such asthe size of its largest eigenvalue. With identical eigenval-ues and similar Hilbert space sizes in all formulations con-sidered, the number of terms to be computed will be ofthe same order in all cases and will not be constrained anyfurther. The time complexity of multiplying an M × M Hamiltonian with density ρ H and an M -dimensional vec-tor with density ρ v goes as O ( ρ H ρ v M ) [116], hence a k -th order truncation of e − iHt on state vector | v (cid:105) scales as: O ( kρ H ρ v M ). Although the initial state can be chosento be sparse, the time evolution will eventually saturatethe sparsity bound determined by the action of the k -thorder Taylor series on the state, which results in mixingthe initial state with states in the same symmetry sector.To simplify the cost analysis, in the following the time complexity T III ∼ O ( ρ H M ) (88)will be used as a general measure of the spectral anddynamical observables and the companying factors men-tioned above will be dropped. Once again, it should benoted that due to the increase in ρ v as the system evolves,the time-evolution cost will ultimately approach the timecomplexity O ( M ). Computing cost in different formulations Since the Hamiltonian in the angular-momentum basisin the physical Hilbert space is the same as that in theLSH basis, the cost analysis of the observable computa-tion is the same in both formulations. What makes adifference in computational costs of one versus the otheris clearly the Hilbert-space and Hamiltonian-generationcost, as will be studied more closely in the next subsec-tion. As a result, only a cost comparison is presentedhere for the fermionic and LSH formulations. For theLGT formulations of SU(2) theory in 1+1 D consideredin this work, the Hamiltonians are sparse and their den-sity goes roughly as 1 /M p with p > 0, given the localityof the interactions in the original KS formulation. Evenin the fermionic formulation, in which the gauge DOFare traded with the locality of interactions, the numberof operators in the Hamiltonian remains small comparedwith the size of the Hilbert space, rendering the Hamil-tonian matrix extremely sparse. A comparison of boththe dimension of the Hilbert space and the density of theHamiltonian matrix between the fermionic and LSH for-mulations of SU(2) LGT in 1+1 D with OBC is shown inFig. 10. As the plots demonstrate, while the dimension ofthe LSH Hilbert space remains smaller than that of thefermionic representation for all values of N , the density ofthe Hamiltonian matrix in the fermionic formulation de-creases compared with that of the LSH formulation as N increases. The determining factor in the time complex-ity of the eigenvalue computation and matrix exponenti-ation is, nonetheless, ρ H M , and the ratio of this quan-tity between the two formulations is plotted in the lowerpanel of Fig. 10. As is observed, the ratio grows slowlyas O ( N ) as N → ∞ , demonstrating the slightly highercost of observable computation in the fermionic formula-tion. In the comparative analysis of the next subsection,the asymptotic cost of observable computation for theLSH formulation (hence the angular-momentum formu-lation in the physical sector) will be estimated simply as O ( ρ H M ) with M = 4 N and ρ H = 2 − N , where the den-sity is approximated from an empirical fit of the densityof the LSH Hamiltonian for small lattice sizes. For thefermionic Hamiltonian, the cost of the LSH Hamiltonianshould be multiplied by N .3 N l og ( T ) 10 15 20 25 3010100200300400 log( T (F) )log( T (J) ) , ⇤ = 4log( T (J) ) , ⇤ = N log( T (LSH) ) FIG. 14. The asymptotic cumulative cost of the three stepsof given classical numerical algorithms for Hamiltonian simu-lation of the KS SU(2) LGT in 1+1 D with the fermionic for-mulation (F), LSH formulation, and the angular-momentumformulation in the physical sector (J), as a function of latticesize N for large N . D. Total cost and comparisons Given the time complexity of various steps of a Hamil-tonian simulation outlined above for the three formula-tions of SU(2) LGT in 1+1 D studied, the cumulativecost of the simulation can be estimated by adding T I , T II , and T III for each formulation from the previous sub-section. Figure 14 plots this asymptotic scaling cost, i.e.,in the limit where N (cid:29) 1. For the angular-momentumformulation, two scenarios are considered: The cutoff onthe gauge excitations is set to a constant value, and thecutoff is set to N , corresponding to the saturating valueof the cutoff with OBC. As already evident from the an-alytical scaling relations, the least time complexity is of-fered by the fermionic representation, while the angular-momentum formulation provides the worst scaling andis unfit for even small-scale simulations. As an explicitcomparison, for N = 20, angular-momentum formulation(with Λ = N ) requires 160 orders of magnitude largercomputing resources than the LSH formulation, whilethe fermionic formulation requires 20 orders of magni-tude lesser resources than the LSH formulation.It is also interesting to examine the cost of each stepof the simulation, i.e., Hilbert-space constructions (I),Hamiltonian generation (II), and observable computation(III), for the algorithms outlined. While the conclusionsare already evident from the analytic scaling forms pro-vided in the previous subsection, they can be observedmore crisply from the plots shown in Fig. 15. For boththe fermionic and LSH formulations, the least costly stepis constructing the physical Hilbert space, while interest-ingly for the angular-momentum formulation this step isthe most costly step involved. Hamiltonian generation isthe most costly step of the simulation for the fermionicand LSH formulations. Once the Hamiltonian is gen-erated, as already explained, the observable computa-tion costs roughly equal computing resources in all cases.In other words, in the angular-momentum formulation N l og ( T ) ( II ) ( III )( I )( II ) ( III )( I ) ( II )( III ) FermionicLSHAngular-momentum basis 10 15 20 25 30102030405060 10 15 20 25 301020406080 10 15 20 25 30100100200300400 FIG. 15. The asymptotic cost of each step of given classical al-gorithms for Hamiltonian simulation of the KS SU(2) LGT in1+1 D with the fermionic formulation (F), LSH formulation,and the angular-momentum basis in the physical sector (J),as a function of the lattice size N for large N . Step ( I ) refersto Hilbert-space construction, step ( II ) refers to Hamiltoniangeneration, and step ( III ) denotes observable computationassuming a generic scaling for sparse matrix manipulations,see the text. The step ( I ) for the fermionic formulation is of O (1) with the chosen algorithm and is not shown. much of the computing effort is put into generating aHamiltonian that is of much smaller dimensionality thanthe naive construction and is the same as that of LSH.That is the ultimate reason behind why the LSH formula-tion is developed and promoted, as it simplifies a priori the construction of the Hamiltonian and its associatedHilbert space without requiring the implementation ofthese steps on the inefficient angular-momentum basisstates.Before completing this section, a few comments are inorder. First, it should be noted that the algorithms de-scribed can be made far more efficient by parallelizing op-erations when possible. The degree of parallelization willdepend on the formulation considered, as well as the algo-rithm devised for each step of the computation. Second,the benefit of fermionic representation over the LSH for-mulation is weakened by the fact that only the LSH for-mulation can be generalized to other boundary conditionsand to higher dimensions, hence its great benefit overthe other formulations when the gauge DOF are present4 FIG. 16. The spectra of the KS Hamiltonian in the physical Hilbert space with PBC for N = 6, ν = 1, and various values of x and Λ. More precisely, the quantity plotted is E (cid:48) Nx , where E (cid:48) is the scaled energy corresponding to the scaled Hamiltonianin Eq. (89). The numerical values associated with these plots are provided in Supplemental Material. is a significant result. In particular, if the fully bosonicformulation of Sec. II C is considered within the angular-momentum basis, its computational cost will be greaterthan that of the angular-momentum representation withboth bosons and fermions present, given the introductionof an extra U(1) cutoff in constructing the Hilbert spaceand the associated Hamiltonian, with a saturation valuethat scales as N . Last but not least, it must be nowobvious that the extreme advantage offered by the LSHformulation over the angular-momentum formulation willpersist with PBC as the complexity of Hilbert-space con-struction and the Hamiltonian generation is even moresignificant with PBC. While in the LSH formulation withPBC, the Hamiltonian operator still projects one basisstate to another, in the angular-momentum formulation,not only the maximum number of basis states in a givenphysical state can scale as O (Λ N ), as shown in Fig. 9, butalso the Hamiltonian operator, in general, transforms abasis state to a linear combination of many distinct basisstates. VI. SPECTRUM AND DYNAMICS INTRUNCATED THEORY The analysis of the previous section revealed that themost suitable formulation to construct the Hilbert space,generate the Hamiltonian, and analyze system’s prop-erties is the LSH formulation. It reproduces the sameHamiltonian as the KS case in the angular-momentumbasis in the physical sector, and it does so by eliminatingthe need to diagonalize the Gauss’s law operator in theangular-momentum basis at the level of states. As was discussed in Sec. III A, this latter step generally leads toa proliferation of terms in a given physical state whenexpressed in the angular-momentum basis, see Fig. 9, in-creasing the cost of the Hamiltonian-matrix generation.The fermionized form leads to the same Hilbert spacewith OBC as the KS formulation in the physical Hilbertspace and the LSH formulation but involves redundan-cies in the spectrum as mentioned above, making it lesssuitable for computation than the LSH framework. Itfurther introduces non-local fermionic interactions whichmake it less efficient to simulate with certain classicaland quantum-simulation algorithms. The bosonic formu-lation, once augmented by the U(1) gauge fields with suf-ficiently large truncation on their excitations, reproducesthe KS theory exactly, as discussed in Sec. III C. As a re-sult, it is less favored compared with the LSH frameworkwhen it comes to computational cost. Finally, the QLMdoes not correspond to the KS Hamiltonian unless cer-tain limits are implemented. As a result, the comparisonbetween the KS and QLM spectrum and dynamics willnot be meaningful away from those limits, as is the casein this study. Given these considerations, in this section,the LSH formulation (or identically KS formulation inthe physical Hilbert space) will be mostly considered forthe analysis of the dependence of the spectrum and dy-namics on the cutoff on the gauge-field excitations, butan example of the effect of the extra gauge DOF on thespectrum of the purely bosonic formulation will be ana-lyzed as well.For convenience, the KS Hamiltonian in Eq. (1) can bemultiplied by ag to yield a dimensionless scaled Hamil-5 ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - Λ Δ E n / E n ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - Λ Δ E n / E n ▽ ◦ ◇ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - Λ Δ E n / E n ▽ ◦ ◇ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - Λ Δ E n / E n ▽ ◦ ◇ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - - - - - Λ Δ E n / E n ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - - - - - Λ Δ E n / E n ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - - - - - Λ Δ E n / E n x = 1 x = 25 x = 100 x = 400 E / E ⇤ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - Λ Δ E n / E n FIG. 17. The quantity ∆ E (cid:48) E (cid:48) ≡ E (cid:48) (Λ) − E (cid:48) (Λ=8) E (cid:48) (Λ) as a functionof Λ for various values of x , and for the 1st, 21st, and 283rdlowest-lying states in the spectrum of the KS Hamiltonianin the physical Hilbert space with N = 6 and ν = 1 withPBC. E (cid:48) (Λ) is the scaled energy corresponding to the scaledHamiltonian in Eq. (89). The dashed lines denote the firstΛ values at which the corresponding scaled energies becomeequal or less than 10% of their values at Λ = 8 (which areapproximated as the Λ → ∞ values). When needed for pre-sentational clarity, the points are artificially displaced alongthe horizontal axis by a small amount. The numerical val-ues associated with these plots are provided in SupplementalMaterial. ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - Λ Δ E n / E n ▽ ◦ ◇ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - Λ Δ E n / E n ▽ ◦ ◇ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - Λ Δ E n / E n ▽ ◦ ◇ E / E p x ⇤ = 0⇤ = 1 ▽▽▽▽ ▽ ▽ ▽ ▽ ▽ ▽ - - - - - x Δ E / E ◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ - - - - - x Δ E / E r ⇡ . r ⇡ . ⇤ = 2 r ⇡ . ⇤ = 2 ◇◇◇◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - - - - - x Δ E / E E / E p x ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ 20 40 60 80 100 - - - - - - - - x Δ E / E p x r ⇡ . FIG. 18. The quantity ∆ E (cid:48) E (cid:48) ≡ E (cid:48) (Λ) − E (cid:48) (Λ=8) E (cid:48) (Λ) as a function of √ x for given values of Λ as denoted in the plots, and for the1st and 21st states in the spectrum of the KS Hamiltonianin the physical Hilbert space with N = 6 and ν = 1 withPBC. E (cid:48) (Λ) is the scaled energy corresponding to the scaledHamiltonian in Eq. (89). The asymptotic ( x → ∞ ) values ofthe quantity, r , are obtained from the fits to data points ineach case with an exponentially varying function of √ x , andare denoted in the plots. The colored regions denote the √ x values excluded from the fits. The numerical values associatedwith these plots are provided in Supplemental Material. tonian H (cid:48) (KS) : H (cid:48) (KS) ≡ ag H (KS) = x N (cid:88) x =0 (cid:2) ψ † ( x ) U ( x ) ψ ( x + 1) + h . c . (cid:3) + N (cid:88) x =0 E ( x ) + µ N (cid:88) n =0 ( − n ψ † ( x ) ψ ( x ) , (89)where x = a g , µ = mg a , and N , N , and N are definedin Sec. II. The limit x → x → ∞ at afixed mg provides a trajectory in parameter space alongwhich the continuum limit can be taken. The matrix el-ements of this Hamiltonian can be formed using the KSangular-momentum or LSH bases, giving rise to identicalresults in the physical sector, which serves as a strongcheck of the newly-developed LSH formulation for the1+1 D case. While efficient classical simulations such asthose based on tensor networks have enabled studies ofSU(2) lattice gauge theories with a large number (hun-dreds) of sites [16, 90–92], enabling the continuum limitof the results to be taken systematically, such considera-tions are not the focus of this work. Instead, the aim is6to arrive at qualitative conclusions regarding the cutoffdependence of several quantities of interest in small lat-tices where exact diagonalization is achievable on a typ-ical desktop computer. A comprehensive study of SU(2)physics within the LSH formulation using state-of-the-artclassical Hamiltonian algorithms is underway and will bepresented elsewhere. Note that such methods still en-counter the issues stemming from inefficient formulationsof DOF and implementation of non-Abelian symmetriesespecially toward higher dimensions, and hence the anal-ysis of this work is relevant in optimizing such studies. A. Spectrum analysis Since the gauge-matter interaction term, i.e., the termproportional to x in Eq. (89), mixes the electric-fieldeigenstates, it is anticipated that for large x (small ga ),the truncation of the Hamiltonian matrix, imposed bythe cutoff on the electric-field excitations, will introduce amore significant deviation from the exact spectrum com-pared with the small- x values. The ratio of the scaled en-ergy, E (cid:48) , obtained from diagonalizing the scaled Hamilto-nian in Eq. (89) to the corresponding x value, nonethe-less, will approach a constant value as x → ∞ , whichrepresents the continuum limit once N → ∞ . The sameis true for the asymptotic large- x behavior of the ratiosof energies at a given value of x . As a result, these ra-tios provide a more suitable means to analyze the scalingbehavior of spectral quantities. Figure 16 plots the di-mensionless quantity E (cid:48) Nx for the ν = 1 sector of the KSHamiltonian with N = 6 and with PBC, for several val-ues of x and Λ. As is visually evident, Λ = 2 appears suf-ficient to obtain almost cutoff-independent energy eigen-values for a large fraction of the spectrum associated witha given Λ when x < 1. The higher-energy states exhibita more significant cutoff dependence, as is expected, andsuch cutoff dependence becomes more prominent as x increases. Nonetheless, as x → ∞ , the shown quantityasymptotes to fixed values for all states in the spectrum,hence stabilizing the effect of the cutoff.To make these features more explicit, three eigenstatesof this systems are selected, corresponding to the 1st,21st, and 283rd lowest eigenenergies. The 1st eigenen-ergy exists for all Λ values considered. The 21st eigenen-ergy is the first eigenenergy that does not appear in thespectrum of the Λ = 0 theory, as the Λ = 0 theory hasa 20-dimensional physical Hilbert space given the quan-tum numbers specified. Finally, the 283rd eigenenergydoes not exist in the spectrum of the Λ = 1 theory, asthe Λ = 1 theory has a 282-dimensional physical Hilbertspace given the quantum numbers considered. Figure 17plots the quantity ∆ E (cid:48) E (cid:48) ≡ E (cid:48) (Λ) − E (cid:48) (Λ=8) E (cid:48) (Λ) as a function ofΛ for various x values and for the three states identified. For x (cid:28) As is seen for all x , as Λ increases, the quantity plottedexponentially approaches the exact value, which is takento be the value at Λ = 8. This scaling behavior may onlyexhibit itself for sufficiently large Λ, as is seen clearly inthe case of the 21st state at x = 1. This indicates that asufficiently large Λ is required to be able to approximatethe Λ → ∞ values using these asymptotic relations. Thisis similar, in spirit, to the numerical procedure outlinedin Ref. [117], where an iterative process is outlined to in-clude higher-energy ‘shells’ in the Hilbert space in orderto drive the quantities to the scaling regime. In this re-gion, analytic relations based on RG can then be appliedto arrive at cutoff-independent observables. In the sameplots, the values of Λ at which the scaled energy deviatesby 10% or less from the E (cid:48) (Λ = 8) value are denoted asdashed vertical lines. As is seen, the required Λ values toreach this moderate accuracy increases with increasing x but stabilizes at fixed values toward the continuum limit.For higher-energy states in the theory, as is evident fromthe behavior of the 283rd eigenenergy, the energy corre-sponding to the Λ = 8 value is not a good approximationfor the exact value and higher cutoffs must be included inthe analysis. The scaling of the shown quantity, nonethe-less, is expected to follow the same exponential form forsufficiently large Λ adjusted to the large eigenenergy con-sidered.Finally, to demonstrate the convergence of the quantity ∆ E (cid:48) E (cid:48) to a constant value at large x , the same eigenener-gies picked in Fig. 17 are taken for the smallest Λ valuesfor which those eigenenergies exist, and are plotted as afunction of x in Fig. 19. After a threshold x value (thatis larger for higher-energy states), the dependence of theratio on x can be approximated by r + ae − ax , where r, a, b are constants to be fit. The values of the constant r ob-tained from the fits shown correspond to the asymptoticlarge- x value of the ratios, and are provided in the plotsfor each of the eigenenergies selected.For OBC, similar qualitative features are seen for thecutoff dependence of the spectrum, as are evident fromFigs. 26-28 in Appendix C. Such cutoff dependence, how-ever, is less severe than the PBC case in general, con-sistent with the fact that the gauge fields are not trulydynamical DOF in gauge theories in 1+1 D with OBC.In particular, once the value Λ = N is reached, the spec-trum of the theory is the same as that corresponding toΛ → ∞ . Nonetheless, for large lattice sizes, it is not com-putationally viable to implement this saturation value,since the Hilbert space increases as e q Λ with the q valueobtained from a fit to results at small lattice sizes. Asa result, the scaling behavior presented for select spec-tral quantities in Appendix C will still serve to guide ap-proaching cutoff-independent results from computationsperformed at smaller values of Λ. imally affected by the truncation cutoff and the scale dependencefor Λ > ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - Λ Δ E n / E n ▽ ◦ ◇ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - Λ Δ E n / E n ▽ ◦ ◇ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - Λ Δ E n / E n ▽ ◦ ◇ E / E p x ⇤ = 0⇤ = 1 ▽▽▽▽ ▽ ▽ ▽ ▽ ▽ ▽ - - - - - x Δ E / E ◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ - - - - - x Δ E / E r ⇡ . r ⇡ . ⇤ = 2 r ⇡ . ⇤ = 2 ◇◇◇◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - - - - - x Δ E / E E / E p x ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ 20 40 60 80 100 - - - - - - - - x Δ E / E p x r ⇡ . FIG. 19. The same quantity as in Fig. 19 corresponding to the 283rd state in the spectrum for Λ = 2. With coarser resolution,the asymptotic ( x → ∞ ) value of the function can be obtained from a fit to the exponential form shown in the left panel. Byzooming into the large- x region of the plot, a finer structure can be observed in the data as a function of √ x , revealing anexponential asymptote to the continuum value that starts at a much larger value of √ x as shown in the right panel, with anasymptotic value that is within sub-percent of that obtained in the left plot. The numerical values associated with these plotsare provided in Supplemental Material. As discussed in Sec. III C, the bosonized KS theoryexhibits a physical Hilbert space that is identical to thatof the KS theory with fermions in the limit where thecutoff on the U(1) electric-field excitations of the ex-tended U(2) theory is equal or larger than the satura-tion value Λ = N . To examine the effect of the extraU(1) cutoff, consider, as an example, the bosonized the-ory with OBC and with N = 8 and ν = 1. In this sector,the full physical Hilbert space is reached once the cutoffon the SU(2) electric-field excitations is set to Λ ≥ ≥ , , , , , 783 for Λ = 0 , , , , , 5, re-spectively. The values of the scaled energies obtainedfrom H (cid:48) = ag H for a small and a large value of thecoupling x (divided by 2 N x ) are plotted in Fig. 20 forΛ = 2 , , , , 6, demonstrating a strong Λ dependencein the spectrum. For comparison, the total number ofstates whose scaled energies in the shown unit are lessthan or equal to 10% of the exact value (correspondingto Λ = 6) are shown in the figure. B. Dynamics analysis Dynamical quantities, i.e., those obtained from time-dependent expectation values, exhibit cutoff dependenceas well. In fact, intuitively one expects that when thesystem evolves, the unitary evolution operator e − iHt ul-timately mixes the initial state with all states within theHilbert space with the same (conserved) quantum num-bers [118], where H denotes the Hamiltonian of the sys-tem and t is time. In other words, various parts of thesystem entangle quickly as the system evolves. This fea-ture is at the heart of the inefficiency of tensor-networkmethods in dynamical studies as the evolution time in-creases. As a result, quantities evolved to large times compared with 1 / || H || suffer from Hilbert-space trunca-tion errors more significantly. Here, || H || is some formof a Hamiltonian norm, such as the absolute value of thelargest energy eigenvalue.To demonstrate this point quantitatively in the case ofthe KS SU(2) LGT in 1+1 D, consider a select quantityobtained from N ( t ) = 12 µ (cid:104) vac( x = 0) | e iH (cid:48) (KS) t (cid:48) × ( H (cid:48) (KS) M + N µ ) e − iH (cid:48) (KS) t (cid:48) | vac( x = 0) (cid:105) , (90)where | vac( x = 0) (cid:105) denotes the strong-coupling vacuum, t (cid:48) is a dimensionless time: t (cid:48) ≡ t x , and all other quanti-ties are defined after Eq. (89). Here t is time in units of a .The quantity in Eq. (90), therefore, counts the occupa-tion number of particles (as opposed to antiparticles) onthe lattice as a function of time, and is identically zerofor the strong-coupling vacuum. Figure 21 plots N ( t )as a function of Λ for various values of x and t in theKS Hamiltonian within the physical Hilbert space with N = 6 and ν = 1 with PBC. Note that t is the abso-lute time, and hence N ( t ) at different x values can bedirectly compared at a fixed t . At small x values, themixing among states expressed in the electric-field basisis small, and the dependence of the time-evolved quantityon the cutoff can be approximated by Ae − B Λ + C evenfor rather small Λ. As x becomes larger, larger valuesof Λ are needed to enter this asymptotic scaling region,at which point the Λ → ∞ value of the quantity can beestimated based on the exponential scaling. This featurestabilizes as the function of x for large x to ensure con-vergence to a continuum limit (once N → ∞ is taken).Not surprisingly though, at larger values of t , it takesa large value of Λ to enter the asymptotic region. Forexample, at t = 100 no convergence is seen in the quan-tity considered for x = 100 and x = 400 in the rangeof cutoffs considered. This example demonstrates thatstudying dynamics in this theory with high precision will8 FIG. 20. The spectra of the bosonized KS Hamiltonian in thephysical Hilbert space with OBC for N = 8 and ν = 1 at twovalues of x and with Λ = 8, and for several values of the cutoffon the U(1) electric field of the extended U(2) gauge theory,Λ . At Λ = 6, the Hilbert space in this sector coincideswith that of the SU(2) theory. The quantity plotted is E (cid:48) Nx ,where E (cid:48) is the scaled energy corresponding to the scaledHamiltonian in Eq. (89). N (10%)states denotes the total numberof eigenstates in the physical Hilbert space of the bosonizedtheory with a given Λ whose scaled energies in the unitsconsidered are ≤ 10% of the exact energy (corresponding tothe saturated value Λ = 6). The numerical values associatedwith these plots are provided in Supplemental Material. be computationally demanding, given the extreme sensi-tivity to the cutoff considered at long evolution times.Another feature of dynamical quantities is that theircontinuum limit seems to be achieved more slowly, requir-ing more computational resources compared with staticquantities. Although such a limit should be consideredas an ordered double limit in which N → ∞ and x → ∞ ,this point can be demonstrated at a fixed N and for var-ious x values. Examples of this feature are shown inFig. 22, in which the quantity N ( t ) is plotted as a func-tion of x for the KS theory with PBC and Λ = 4 ata select time, t = 100. This observation is consistentwith the slow convergence of high-lying energies in thespectrum as a function of x (see Fig. 19), consideringthat time evolution leads to contributions to observablesfrom all states in the same symmetry sector. Nonethe-less, if percent-level precision is needed in this quantity,one would not need computations to be performed at ex-treme values of x . ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ Λ Δ N p ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ Λ Δ N p ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ Λ Δ N p ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ Λ Δ N p ⇤ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ Λ Δ N p ▽ t = ◦ t = ◇ t = ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ Λ Δ N p ▽ t = ◦ t = ◇ t = ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ Λ Δ N p ▽ t = ◦ t = ◇ t = x = 1 x = 25 x = 100 x = 400 N ( t ) FIG. 21. The quantity N ( t ) defined in Eq. (90) as a functionof Λ for various values of x and t in the KS Hamiltonian inthe physical Hilbert space with N = 6 and ν = 1 with PBC. t is the absolute time as defined in the text in units of a .When possible, the points are fit to N = Ae − B Λ + C andthe colored regions associated with each t are excluded fromsuch fits. The numerical values associated with these plotsare provided in Supplemental Material. It is worth noting that the point at which the quanti-ties can be described by exponentially converging func-tions as Λ → ∞ or as x → ∞ , depends on the size ofthe system, the choice of m/g , and the quantity consid-ered. To formally understand the underlying mechanismfor such an scaling and its breakdown, beyond the em-pirical observations of this work in small systems, is thesubject of a future investigation and potentially has itsroot in the Nyquist-Shannon sampling theorem [77]. Amore systematic understanding of Hamiltonian renormal-ization can shed light on this question, as we have alluded9 p x p x ⇤ = 4 ⇤ = 4 ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ Λ Δ N p ▽ t = ◦ t = ◇ t = N ( t ) ◇◇◇◇◇◇◇◇◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ x N r ⇡ . r ⇡ . ◇◇◇◇◇◇◇◇◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ x N FIG. 22. The left panel shows the quantity N ( t ) defined in Eq. (90) as a function of √ x for Λ = 4 and t = 100 in the KSHamiltonian in the physical Hilbert space with N = 6 and ν = 1 with PBC. The points are fit to N ∼ e − r √ x and the coloredregions are excluded from such fits. By zooming into the large- x region of the plot, a finer structure can be observed in the dataas a function of √ x , revealing an exponential asymptote to the continuum value that starts at a much larger value of √ x asshown in the right panel, with an asymptotic value that is within sub-percent of that obtained in the left plot. The numericalvalues associated with these plots are provided in Supplemental Material. to in Sec. VII. VII. CONCLUSIONS AND OUTLOOK The present paper provides a stepping stone to estab-lishing the most efficient Hamiltonian formulation(s) forthe case of a non-Abelian gauge field theories in 1+1 Di-mensions, with implications for other non-Abelian gaugetheories and for higher dimensions. The theory studiedin this paper is the SU(2) lattice gauge theory coupledto matter. Its various forms, characterized by the re-tained degrees of freedom and the basis states employed,are thoroughly analyzed and scrutinized. These formsinclude the original Kogut-Susskind formulation withinthe angular-momentum basis, the purely fermionic andpurely bosonic formulations (possible only with openboundary conditions), and the recently developed Loop-String-Hadron formulation, with both periodic and openboundary conditions. A schematic of these formulationsand the relation among them are shown in Fig. 23. Giventhe extent and the scope of the conclusions reached, asummary of the main findings of the work is presentedhere, along with a number of comments on the outlookof this work and future directions: (cid:66) The dimension of the Hilbert space grows quicklywith the size of the lattice, N , as ∼ e pN . In theangular-momentum basis, a naive counting of thefull Hilbert space reveals that the exponent growslogarithmically with the cutoff on the electric-fieldexcitations. With open boundary conditions, a sat-uration value of Λ = N + 2 (cid:15) ensures that the fulluntruncated theory is achieved, but at the cost ofexpediting the growth of the full Hilbert space by ∼ e N log N . The physical Hilbert space in the samerepresentation still grows exponentially with thesystem’s size, but with a p value that approaches aconstant of O (1) as a function of the cutoff. The empirical values are obtained in Sec. III. Addition-ally, as a function of the cutoff on the electric-fieldexcitations, the dimension of the physical Hilbertspace approaches a constant with OBC but growslinearly with PBC. The considerable reduction inthe Hilbert-space dimension by restricting compu-tations to the physical sector comes at a significantcost, since imposing the non-Abelian Gauss’s lawslocally amounts to a super-exponential cost in gen-erating the Hilbert space. There is a similar costassociated with generating the Hamiltonian matrix,simply due to the fact that each physical state inthe original angular-momentum basis is a linearcombination of many basis states, with the num-ber of terms in given physical states growing expo-nentially with the cutoff. Furthermore, the actionof Hamiltonian on such states generates other pop-ulated linear combination of states. Given thesefeatures, the angular-momentum basis of the KStheory appears the least appealing framework forHamiltonian simulation. (cid:66) In 1+1 D, the gauge DOF are non-dynamical withOBC and can be fully fixed by the boundary valueof the electric field. This means that restricting tothe physical Hilbert space comes as no computa-tional cost. Nonetheless, the fermionic basis stateslead to redundancies, and while the spectra of boththe original theory in the physical sector and thefermionic theory are identical, there are degenera-cies in the spectrum in the latter case. The differ-ence in the dimension of the physical Hilbert spacesin these formulations is not significant though, andthe ratio remains constant as the system’s sizegrows. The larger Hilbert space is compensatedby a slightly faster decline in the Hamiltonian-matrix density of the fermionic formulation, andthe cost of computing observables, i.e., eigenvalueevaluation or matrix exponentiation, is only O ( N )0 I IIIII IV S o l v e G a u ss ’ s l a w s , ⇤ !1 R e m o v e ga u g e fi e l d s R e m o v e f e r m i o n s S o l v e G a u ss ’ s l a w s Kogut-Susskind formulation of SU(2) LGT in 1+1 DKogut-Susskind formulation of SU(2) LGT in 1+1 D Isomorphic Hilbert spaces, identical Hamiltonians and spectra, identical global-symmetry propertiesIsomorphic Hilbert spaces, identical Hamiltonians and spectra, identical global-symmetry properties R e du c t i o n o f ph y s i c a l H il b e r t s p a ce w i t h l oo p a nd s t r i n g D O F ,i d e n t i c a l s p ec t r a f o r O B C R e du c t i o n o f ph y s i c a l H il b e r t s p a ce w i t h l oo p a nd s t r i n g D O F ,i d e n t i c a l s p ec t r a f o r O B C R e f o r m u l a t e t h e p r e p o t e n t i a l f r a m e w o r k R e f o r m u l a t e t h e p r e p o t e n t i a l f r a m e w o r k R e f o r m u l a t e t h e p r e p o t e n t i a l f r a m e w o r k Purely bosonic formulationPurely bosonic formulation Purely fermionic formulationPurely fermionic formulationLoop-String-Hadron formulationLoop-String-Hadron formulationAngular-momentum formulation in the physical sectorAngular-momentum formulation in the physical sectorAngular-momentum formulation in the physical sector FIG. 23. Various formulations of the KS SU(2) LGT in 1+1 D studied in this work, and the connection among them. higher in the fermionic formulation compared withthe angular-momentum formulation in the physi-cal sector. Nonetheless, the purely fermionic for-mulation is the least costly among all the formula-tions considered in the limit of large lattice sizes,as expected. The drawback is that this formulationhas no analog in higher dimensions as the bound-ary conditions and Gauss’s law constraints are notsufficient to fully remove the gauge DOF. As a re-sult, other formulations with better generalizabilityperspective and similar computational-resource re-quirements are favorable. (cid:66) Alternatively, the fermionic DOF can be removedat the expense of enlarging the gauge group toU(2) and introducing additional complexity due tothe need for sufficiently large cutoffs on the addi-tional U(1) electric field excitations, and for ap-propriate encoding of the fermionic statistics. Thistheory, once the saturating value of the U(1) cutoffis imposed with OBC, and the Gauss’s laws associ-ated with both the U(1) and SU(2) symmetries aresolved, has an identical physical Hilbert space tothat in the original KS theory. All the complexitiesarising from imposing the non-Abelian constraintsin the angular-momentum basis remain relevant tothis formulation, and hence no particular advantageis gained by the purely bosonic formulation withclassical Hamiltonian-simulation algorithms. Sucha formulation, nonetheless, is argued to be benefi-cial in digital quantum simulations as it avoids non-local fermion-to-hard-core-boson transformations. (cid:66) The complexities with the angular-momentum rep-resentation of the KS theory and the increasingmixing of the local basis states in the physicalHilbert space are avoided altogether in the LSH for-mulation. Here, the building blocks of the Hamilto-nian are operators that act directly on local gauge-invariant basis states, namely strings and loops,and only an Abelian Gauss’s law on the link be-tween the lattice sites is left to be imposed a pos-teriori . Furthermore, the action of each opera-tor in the Hamiltonian maps a given state to oneand only one state, a feature that is absent in theangular-momentum basis in the physical Hilbertspace. While the string quantum numbers canonly take values 0 or 1, the loop quantum num-bers are non-negative integers, and their spectrummust be truncated in practical applications. Withthe same truncation cutoff, the Hamiltonian matri-ces in the physical Hilbert space with the angular-momentum and LSH bases are identical, nonethe-less the cost of generating such a matrix is expo-nentially suppressed in the LSH formulation. Sim-ilarly, while in the original angular-momentum ba-sis, the construction of Hilbert space is the mostcostly step in the simulation algorithm, this stepis the least costly with the LSH formulation, andoffers up to hundreds of orders of magnitude reduc-tion in the cost of the simulation compared with theoriginal formulation for lattices with tens of sites,as demonstrated in Sec. V. Although the purelyfermionic formulation remains slightly less complexand computationally less costly to simulate (de-1spite its all-to-all fermionic interactions), the LSHframework is already generalizable to higher dimen-sions, and is shown to retain the same crucial fea-tures as the 1+1 D case. Namely, a complete basisof local gauge-invariant states exists to eliminatethe need for the imposition of non-Abelian Gauss’slaws (that are more complicated to implement inhigher dimensions) at the level of basis states, andthat the Abelian Gauss’s law is still implementedin only one dimension, thanks to a point-splittingprocedure developed in Ref. [100, 119]. As a result,this framework appears to be the most appealingamong all the formulations considered for the pur-pose of Hamiltonian simulation of the SU(2) LGT.The generalization of such an efficient formulationto the SU(3) LGT coupled to matter is, therefore,an important next step in approaching the ultimategoal of Hamiltonian simulation of QCD. (cid:66) Hamiltonian-truncation methods, common in a va-riety of problems from condensed-matter physicsto nuclear physics, have been well developed overdecades and have a direct connection to the con-cept of Renormalization Group. In particular,Wegner [120] and Glazek and Wilson [121] inde-pendently studied the concept of renormalizationwithin a Hamiltonian approach. They establishedpowerful methods such as similarity renormaliza-tion group to continuously flow the transformedHamiltonian to a banded-diagonal form such thatthe high-energy and low-energy modes in a giventheory are maximally decoupled. This procedureleads to smaller truncation errors resulting fromconsidering only a partial sector of the Hilbertspace in generating and processing the Hamiltonianmatrix. More concretely, such approaches put thesystem in the asymptotic scaling regime such thatthe limit of Λ → ∞ can be taken using asymp-totic formulae, see e.g., Ref. [117]. In other words,a Hamiltonian truncation approach can be usedto derive the RG equations for the parameters ofthe theory, promoting the truncated bare Hamilto-nian to an effective renormalized Hamiltonian. Adedicated study of renormalized Hamiltonian gaugetheories which extends upon previous work is a ne-cessity as one moves toward using the Hamiltonianapproach in the simulations of gauge theories. Inthis first study, however, we have only presentedempirical observations for the scale dependence ofthe spectrum and dynamics in the SU(2) gauge the-ory in 1+1 D in small lattices to illuminate sev-eral seemingly general features. In particular, theexponential dependence of observables on the ex-citation of gauge fields is established for energiesand given dynamical expectation values, but a suf- ficiently large cutoff is needed to enter this scalingregime for high-energy states, as well as the long-time-evolved states. Approaches in constructing arenormalized theory in the Hamiltonian formula-tion will be studied in future work to shed light onthe interesting empirical features observed. In thiscontext, the formal analysis of digitization effectsin bose-fermi systems [75, 76] and in scalar fieldtheories [77] will potentially be relevant. (cid:66) Despite the steady progress in advanced simulationalgorithms such as tensor networks, the enormouscost of Hamiltonian simulation, even with the mostefficient formulations identified, suggests that thequantum simulation of LGTs of interest may bethe ultimate path to reaching accurate and large-scale dynamical simulations. It is, therefore, im-portant to extend the analysis of this work to ver-ify the efficiency of the LSH formulation comparedto other formulations in the context of quantumsimulation, and to find the most efficient quantumalgorithms for simulating it, similar to studies con-ducted to date for various quantum field theories,such as scalar field theories [24, 77] and the lat-tice Schwinger model [53]. The insights obtainedfrom the thorough comparative study of this workare the key to embarking on such investigations ina quantum-computing setting. Analog quantum-simulation protocols within the LSH formulationare being developed, see e.g., Ref. [122], and simi-lar developments are planned for digital protocols.In particular, explicitly solving Gauss’s laws in adigital quantum computation and projecting outthe physical Hilbert space of the non-Abelian SU(2)LGTs in any dimension has only been achieved us-ing the LSH formalism [40]. Presumably, a mixtureof various formulations, such as purely bosonic for-mulations or partially fermionic formulation withthe LSH basis, could provide more computationalbenefits in this context, but further investigationsneed to be conducted to reach accurate conclusions. ACKNOWLEDGMENTS We are grateful to Rudranil Basu for valuable discus-sions and to Jesse Stryker for valuable comments onthe manuscript. 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In this section, we briefly describe the QLM forthe SU(2) LGT using the ‘rishon’ representation in thesame setting as discussed in earlier sections, i.e., withstaggered fermionic fields on a one-dimensional spatiallattice, following the presentation of Ref. [92]. Like inall other formulations, the fermionic matter field is inthe fundamental representation of SU(2) and is definedat each lattice site. Gauge fields, defined in terms oflink variables, couple to the matter fields and constitutethe interaction part of the Hamiltonian. One importantdifference at this point is that the link operator is con-structed as a composite operator made up of ‘rishons’,which are fermions. The rishon and matter constituentson two staggered sites on the lattice are shown in Fig. 24.The mapping between the variables is: { ˆ E L/R ( x ) , ˆ U ( x ) , ψ ( x ) } → { ˆ c [ τ ] s ( x ) } , (A1)where s = ↑ , ↓ is the SU(2) component and τ = M, L, R ,where M refers to matter, and L ( R ) refers to rishons onthe left (right) side of the link. The canonical conjugatevariables, as well as the link variable and the matter field,are constructed asˆ E aL ( x ) = − ˆ c [ L ] † s ( x ) T as,s (cid:48) ˆ c [ L ] s (cid:48) ( x ) , (A2)ˆ E aR ( x ) = ˆ c [ R ] † s ( x + 1) T as,s (cid:48) ˆ c [ R ] s (cid:48) ( x + 1) , (A3)ˆ U s,s (cid:48) ( x ) = ˆ c [ L ] s ( x )ˆ c [ R ] † s (cid:48) ( x + 1) , (A4) ψ † s ( x ) = ˆ c [ M ] † s ( x ) , (A5)where T a = τ a , and τ a is the a th Pauli matrix, with a = 1 , , 3. These definitions yield the same commuta-tion relations between the electric fields and link variableas in Eqs. (5). However, unlike the KS theory, [ U s,s (cid:48) , U † t,t (cid:48) ]can be non-vanishing in the QLM formulation, and thelink operator has necessarily a finite-dimensional repre-sentation.6 · · · · · · staggered site x + 1 staggered site x c [ R ] " ( x ) c [ R ] ( x ) (A14) c [ L ] " ( x ) c [ L ] ( x ) (A15) c [ M ] " ( x ) c [ M ] ( x ) (A16) c [ R ] " ( x +1) c [ R ] ( x +1) (A17) c [ L ] " ( x +1) c [ L ] ( x +1) (A18) c [ M ] " ( x +1) c [ M ] ( x +1) (A19) E R ( x 1) (A20) E L ( x ) (A21) E R ( x ) (A22) E L ( x + 1) (A23) U ( x ) = c [ L ] " ( x ) c [ R ] †" ( x + 1) c [ L ] " ( x ) c [ R ] † ( x + 1) c [ L ] ( x ) c [ R ] †" ( x + 1) c [ L ] ( x ) c [ R ] † ( x + 1) ! (A24) Appendix B: Details [1] Hagen Kleinert, Gauge fields in condensed matter , Vol. 2(World Scientific, 1989).[2] Eduardo Fradkin, Field theories of condensed matterphysics (Cambridge University Press, 2013).[3] M. Tanabashi et al. (Particle Data Group), “Review ofParticle Physics,” Phys. Rev. D , 030001 (2018).[4] Paul Langacker, “The standard model and beyond,” in AIP Conference Proceedings , Vol. 150 (American Insti-tute of Physics, 1986) pp. 142–164.[5] David Tong, “Gauge theory,” (2018).[6] Kenneth G. Wilson, “Confinement of Quarks,” , 45–59(1974).[7] John B. Kogut, “An Introduction to Lattice Gauge The-ory and Spin Systems,” Rev. Mod. Phys. , 659 (1979).[8] Michael Creutz, Laurence Jacobs, and Claudio Rebbi,“Monte Carlo Computations in Lattice Gauge Theo- ries,” Phys. Rept. , 201–282 (1983).[9] Andrei Alexandru, Gokce Basar, Paulo F. Bedaque, So-han Vartak, and Neill C. Warrington, “Monte CarloStudy of Real Time Dynamics on the Lattice,” Phys.Rev. Lett. , 081602 (2016), arXiv:1605.08040 [hep-lat].[10] Xiang-Dong Ji, “Deeply virtual Compton scatter-ing,” Phys. Rev. D , 7114–7125 (1997), arXiv:hep-ph/9609381.[11] Ral A. Briceo, Zohreh Davoudi, Maxwell T. Hansen,Matthias R. Schindler, and Alessandro Baroni, “Long-range electroweak amplitudes of single hadrons from Eu-clidean finite-volume correlation functions,” Phys. Rev.D , 014509 (2020), arXiv:1911.04036 [hep-lat].[12] Xiangdong Ji, “Parton Physics on a Euclidean Lattice,”Phys. Rev. Lett. , 262002 (2013), arXiv:1305.1539 c [ R ] " ( x ) c [ R ] ( x ) (A14) c [ L ] " ( x ) c [ L ] ( x ) (A15) c [ M ] " ( x ) c [ M ] ( x ) (A16) c [ R ] " ( x +1) c [ R ] ( x +1) (A17) c [ L ] " ( x +1) c [ L ] ( x +1) (A18) c [ M ] " ( x +1) c [ M ] ( x +1) (A19) E R ( x 1) (A20) E L ( x ) (A21) E R ( x ) (A22) E L ( x + 1) (A23) U ( x ) = c [ L ] " ( x ) c [ R ] †" ( x + 1) c [ L ] " ( x ) c [ R ] † ( x + 1) c [ L ] ( x ) c [ R ] †" ( x + 1) c [ L ] ( x ) c [ R ] † ( x + 1) ! (A24) Appendix B: Details [1] Hagen Kleinert, Gauge fields in condensed matter , Vol. 2(World Scientific, 1989).[2] Eduardo Fradkin, Field theories of condensed matterphysics (Cambridge University Press, 2013).[3] M. Tanabashi et al. (Particle Data Group), “Review ofParticle Physics,” Phys. Rev. D , 030001 (2018).[4] Paul Langacker, “The standard model and beyond,” in AIP Conference Proceedings , Vol. 150 (American Insti-tute of Physics, 1986) pp. 142–164.[5] David Tong, “Gauge theory,” (2018).[6] Kenneth G. Wilson, “Confinement of Quarks,” , 45–59(1974).[7] John B. Kogut, “An Introduction to Lattice Gauge The-ory and Spin Systems,” Rev. Mod. Phys. , 659 (1979).[8] Michael Creutz, Laurence Jacobs, and Claudio Rebbi,“Monte Carlo Computations in Lattice Gauge Theo- ries,” Phys. Rept. , 201–282 (1983).[9] Andrei Alexandru, Gokce Basar, Paulo F. Bedaque, So-han Vartak, and Neill C. Warrington, “Monte CarloStudy of Real Time Dynamics on the Lattice,” Phys.Rev. Lett. , 081602 (2016), arXiv:1605.08040 [hep-lat].[10] Xiang-Dong Ji, “Deeply virtual Compton scatter-ing,” Phys. Rev. D , 7114–7125 (1997), arXiv:hep-ph/9609381.[11] Ral A. Briceo, Zohreh Davoudi, Maxwell T. Hansen,Matthias R. Schindler, and Alessandro Baroni, “Long-range electroweak amplitudes of single hadrons from Eu-clidean finite-volume correlation functions,” Phys. Rev.D , 014509 (2020), arXiv:1911.04036 [hep-lat].[12] Xiangdong Ji, “Parton Physics on a Euclidean Lattice,”Phys. Rev. Lett. , 262002 (2013), arXiv:1305.1539 c [ R ] " ( x ) c [ R ] ( x ) (A14) c [ L ] " ( x ) c [ L ] ( x ) (A15) c [ M ] " ( x ) c [ M ] ( x ) (A16) c [ R ] " ( x +1) c [ R ] ( x +1) (A17) c [ L ] " ( x +1) c [ L ] ( x +1) (A18) c [ M ] " ( x +1) c [ M ] ( x +1) (A19) E R ( x 1) (A20) E L ( x ) (A21) E R ( x ) (A22) E L ( x + 1) (A23) U ( x ) = c [ L ] " ( x ) c [ R ] †" ( x + 1) c [ L ] " ( x ) c [ R ] † ( x + 1) c [ L ] ( x ) c [ R ] †" ( x + 1) c [ L ] ( x ) c [ R ] † ( x + 1) ! (A24) Appendix B: Details [1] Hagen Kleinert, Gauge fields in condensed matter , Vol. 2(World Scientific, 1989).[2] Eduardo Fradkin, Field theories of condensed matterphysics (Cambridge University Press, 2013).[3] M. Tanabashi et al. (Particle Data Group), “Review ofParticle Physics,” Phys. Rev. D , 030001 (2018).[4] Paul Langacker, “The standard model and beyond,” in AIP Conference Proceedings , Vol. 150 (American Insti-tute of Physics, 1986) pp. 142–164.[5] David Tong, “Gauge theory,” (2018).[6] Kenneth G. Wilson, “Confinement of Quarks,” , 45–59(1974).[7] John B. Kogut, “An Introduction to Lattice Gauge The-ory and Spin Systems,” Rev. Mod. Phys. , 659 (1979).[8] Michael Creutz, Laurence Jacobs, and Claudio Rebbi,“Monte Carlo Computations in Lattice Gauge Theo- ries,” Phys. Rept. , 201–282 (1983).[9] Andrei Alexandru, Gokce Basar, Paulo F. Bedaque, So-han Vartak, and Neill C. Warrington, “Monte CarloStudy of Real Time Dynamics on the Lattice,” Phys.Rev. Lett. , 081602 (2016), arXiv:1605.08040 [hep-lat].[10] Xiang-Dong Ji, “Deeply virtual Compton scatter-ing,” Phys. Rev. D , 7114–7125 (1997), arXiv:hep-ph/9609381.[11] Ral A. Briceo, Zohreh Davoudi, Maxwell T. Hansen,Matthias R. Schindler, and Alessandro Baroni, “Long-range electroweak amplitudes of single hadrons from Eu-clidean finite-volume correlation functions,” Phys. Rev.D , 014509 (2020), arXiv:1911.04036 [hep-lat].[12] Xiangdong Ji, “Parton Physics on a Euclidean Lattice,”Phys. Rev. Lett. , 262002 (2013), arXiv:1305.1539 c [ R ] " ( x ) c [ R ] ( x ) (A14) c [ L ] " ( x ) c [ L ] ( x ) (A15) c [ M ] " ( x ) c [ M ] ( x ) (A16) c [ R ] " ( x +1) c [ R ] ( x +1) (A17) c [ L ] " ( x +1) c [ L ] ( x +1) (A18) c [ M ] " ( x +1) c [ M ] ( x +1) (A19) E R ( x 1) (A20) E L ( x ) (A21) E R ( x ) (A22) E L ( x + 1) (A23) U ( x ) = c [ L ] " ( x ) c [ R ] †" ( x + 1) c [ L ] " ( x ) c [ R ] † ( x + 1) c [ L ] ( x ) c [ R ] †" ( x + 1) c [ L ] ( x ) c [ R ] † ( x + 1) ! (A24) Appendix B: Details [1] Hagen Kleinert, Gauge fields in condensed matter , Vol. 2(World Scientific, 1989).[2] Eduardo Fradkin, Field theories of condensed matterphysics (Cambridge University Press, 2013).[3] M. Tanabashi et al. (Particle Data Group), “Review ofParticle Physics,” Phys. Rev. D , 030001 (2018).[4] Paul Langacker, “The standard model and beyond,” in AIP Conference Proceedings , Vol. 150 (American Insti-tute of Physics, 1986) pp. 142–164.[5] David Tong, “Gauge theory,” (2018).[6] Kenneth G. Wilson, “Confinement of Quarks,” , 45–59(1974).[7] John B. Kogut, “An Introduction to Lattice Gauge The-ory and Spin Systems,” Rev. Mod. Phys. , 659 (1979).[8] Michael Creutz, Laurence Jacobs, and Claudio Rebbi,“Monte Carlo Computations in Lattice Gauge Theo- ries,” Phys. Rept. , 201–282 (1983).[9] Andrei Alexandru, Gokce Basar, Paulo F. Bedaque, So-han Vartak, and Neill C. Warrington, “Monte CarloStudy of Real Time Dynamics on the Lattice,” Phys.Rev. Lett. , 081602 (2016), arXiv:1605.08040 [hep-lat].[10] Xiang-Dong Ji, “Deeply virtual Compton scatter-ing,” Phys. Rev. D , 7114–7125 (1997), arXiv:hep-ph/9609381.[11] Ral A. Briceo, Zohreh Davoudi, Maxwell T. Hansen,Matthias R. Schindler, and Alessandro Baroni, “Long-range electroweak amplitudes of single hadrons from Eu-clidean finite-volume correlation functions,” Phys. Rev.D , 014509 (2020), arXiv:1911.04036 [hep-lat].[12] Xiangdong Ji, “Parton Physics on a Euclidean Lattice,”Phys. Rev. Lett. , 262002 (2013), arXiv:1305.1539 c [ R ] " ( x ) c [ R ] ( x ) (A14) c [ L ] " ( x ) c [ L ] ( x ) (A15) c [ M ] " ( x ) c [ M ] ( x ) (A16) c [ R ] " ( x +1) c [ R ] ( x +1) (A17) c [ L ] " ( x +1) c [ L ] ( x +1) (A18) c [ M ] " ( x +1) c [ M ] ( x +1) (A19) E R ( x 1) (A20) E L ( x ) (A21) E R ( x ) (A22) E L ( x + 1) (A23) U ( x ) = c [ L ] " ( x ) c [ R ] †" ( x + 1) c [ L ] " ( x ) c [ R ] † ( x + 1) c [ L ] ( x ) c [ R ] †" ( x + 1) c [ L ] ( x ) c [ R ] † ( x + 1) ! (A24) Appendix B: Details [1] Hagen Kleinert, Gauge fields in condensed matter , Vol. 2(World Scientific, 1989).[2] Eduardo Fradkin, Field theories of condensed matterphysics (Cambridge University Press, 2013).[3] M. Tanabashi et al. (Particle Data Group), “Review ofParticle Physics,” Phys. Rev. D , 030001 (2018).[4] Paul Langacker, “The standard model and beyond,” in AIP Conference Proceedings , Vol. 150 (American Insti-tute of Physics, 1986) pp. 142–164.[5] David Tong, “Gauge theory,” (2018).[6] Kenneth G. Wilson, “Confinement of Quarks,” , 45–59(1974).[7] John B. Kogut, “An Introduction to Lattice Gauge The-ory and Spin Systems,” Rev. Mod. Phys. , 659 (1979).[8] Michael Creutz, Laurence Jacobs, and Claudio Rebbi,“Monte Carlo Computations in Lattice Gauge Theo- ries,” Phys. Rept. , 201–282 (1983).[9] Andrei Alexandru, Gokce Basar, Paulo F. Bedaque, So-han Vartak, and Neill C. Warrington, “Monte CarloStudy of Real Time Dynamics on the Lattice,” Phys.Rev. Lett. , 081602 (2016), arXiv:1605.08040 [hep-lat].[10] Xiang-Dong Ji, “Deeply virtual Compton scatter-ing,” Phys. Rev. D , 7114–7125 (1997), arXiv:hep-ph/9609381.[11] Ral A. Briceo, Zohreh Davoudi, Maxwell T. Hansen,Matthias R. Schindler, and Alessandro Baroni, “Long-range electroweak amplitudes of single hadrons from Eu-clidean finite-volume correlation functions,” Phys. Rev.D , 014509 (2020), arXiv:1911.04036 [hep-lat].[12] Xiangdong Ji, “Parton Physics on a Euclidean Lattice,”Phys. Rev. Lett. , 262002 (2013), arXiv:1305.1539 c [ R ] " ( x ) c [ R ] ( x ) (A14) c [ L ] " ( x ) c [ L ] ( x ) (A15) c [ M ] " ( x ) c [ M ] ( x ) (A16) c [ R ] " ( x +1) c [ R ] ( x +1) (A17) c [ L ] " ( x +1) c [ L ] ( x +1) (A18) c [ M ] " ( x +1) c [ M ] ( x +1) (A19) E R ( x 1) (A20) E L ( x ) (A21) E R ( x ) (A22) E L ( x + 1) (A23) U ( x ) = c [ L ] " ( x ) c [ R ] †" ( x + 1) c [ L ] " ( x ) c [ R ] † ( x + 1) c [ L ] ( x ) c [ R ] †" ( x + 1) c [ L ] ( x ) c [ R ] † ( x + 1) ! (A24) Appendix B: Details [1] Hagen Kleinert, Gauge fields in condensed matter , Vol. 2(World Scientific, 1989).[2] Eduardo Fradkin, Field theories of condensed matterphysics (Cambridge University Press, 2013).[3] M. Tanabashi et al. (Particle Data Group), “Review ofParticle Physics,” Phys. Rev. D , 030001 (2018).[4] Paul Langacker, “The standard model and beyond,” in AIP Conference Proceedings , Vol. 150 (American Insti-tute of Physics, 1986) pp. 142–164.[5] David Tong, “Gauge theory,” (2018).[6] Kenneth G. Wilson, “Confinement of Quarks,” , 45–59(1974).[7] John B. Kogut, “An Introduction to Lattice Gauge The-ory and Spin Systems,” Rev. Mod. Phys. , 659 (1979).[8] Michael Creutz, Laurence Jacobs, and Claudio Rebbi,“Monte Carlo Computations in Lattice Gauge Theo- ries,” Phys. Rept. , 201–282 (1983).[9] Andrei Alexandru, Gokce Basar, Paulo F. Bedaque, So-han Vartak, and Neill C. Warrington, “Monte CarloStudy of Real Time Dynamics on the Lattice,” Phys.Rev. Lett. , 081602 (2016), arXiv:1605.08040 [hep-lat].[10] Xiang-Dong Ji, “Deeply virtual Compton scatter-ing,” Phys. Rev. D , 7114–7125 (1997), arXiv:hep-ph/9609381.[11] Ral A. Briceo, Zohreh Davoudi, Maxwell T. Hansen,Matthias R. Schindler, and Alessandro Baroni, “Long-range electroweak amplitudes of single hadrons from Eu-clidean finite-volume correlation functions,” Phys. Rev.D , 014509 (2020), arXiv:1911.04036 [hep-lat].[12] Xiangdong Ji, “Parton Physics on a Euclidean Lattice,”Phys. Rev. Lett. , 262002 (2013), arXiv:1305.1539 c [ R ] " ( x ) c [ R ] ( x ) (A14) c [ L ] " ( x ) c [ L ] ( x ) (A15) c [ M ] " ( x ) c [ M ] ( x ) (A16) c [ R ] " ( x +1) c [ R ] ( x +1) (A17) c [ L ] " ( x +1) c [ L ] ( x +1) (A18) c [ M ] " ( x +1) c [ M ] ( x +1) (A19) E R ( x 1) (A20) E L ( x ) (A21) E R ( x ) (A22) E L ( x + 1) (A23) U ( x ) = c [ L ] " ( x ) c [ R ] †" ( x + 1) c [ L ] " ( x ) c [ R ] † ( x + 1) c [ L ] ( x ) c [ R ] †" ( x + 1) c [ L ] ( x ) c [ R ] † ( x + 1) ! (A24) Appendix B: Details [1] Hagen Kleinert, Gauge fields in condensed matter , Vol. 2(World Scientific, 1989).[2] Eduardo Fradkin, Field theories of condensed matterphysics (Cambridge University Press, 2013).[3] M. Tanabashi et al. (Particle Data Group), “Review ofParticle Physics,” Phys. Rev. D , 030001 (2018).[4] Paul Langacker, “The standard model and beyond,” in AIP Conference Proceedings , Vol. 150 (American Insti-tute of Physics, 1986) pp. 142–164.[5] David Tong, “Gauge theory,” (2018).[6] Kenneth G. Wilson, “Confinement of Quarks,” , 45–59(1974).[7] John B. Kogut, “An Introduction to Lattice Gauge The-ory and Spin Systems,” Rev. Mod. Phys. , 659 (1979).[8] Michael Creutz, Laurence Jacobs, and Claudio Rebbi,“Monte Carlo Computations in Lattice Gauge Theo- ries,” Phys. Rept. , 201–282 (1983).[9] Andrei Alexandru, Gokce Basar, Paulo F. Bedaque, So-han Vartak, and Neill C. Warrington, “Monte CarloStudy of Real Time Dynamics on the Lattice,” Phys.Rev. Lett. , 081602 (2016), arXiv:1605.08040 [hep-lat].[10] Xiang-Dong Ji, “Deeply virtual Compton scatter-ing,” Phys. Rev. D , 7114–7125 (1997), arXiv:hep-ph/9609381.[11] Ral A. Briceo, Zohreh Davoudi, Maxwell T. Hansen,Matthias R. Schindler, and Alessandro Baroni, “Long-range electroweak amplitudes of single hadrons from Eu-clidean finite-volume correlation functions,” Phys. Rev.D , 014509 (2020), arXiv:1911.04036 [hep-lat].[12] Xiangdong Ji, “Parton Physics on a Euclidean Lattice,”Phys. Rev. Lett. , 262002 (2013), arXiv:1305.1539 c [ R ] " ( x ) c [ R ] ( x ) (A14) c [ L ] " ( x ) c [ L ] ( x ) (A15) c [ M ] " ( x ) c [ M ] ( x ) (A16) c [ R ] " ( x +1) c [ R ] ( x +1) (A17) c [ L ] " ( x +1) c [ L ] ( x +1) (A18) c [ M ] " ( x +1) c [ M ] ( x +1) (A19) E R ( x 1) (A20) E L ( x ) (A21) E R ( x ) (A22) E L ( x + 1) (A23) U ( x ) = c [ L ] " ( x ) c [ R ] †" ( x + 1) c [ L ] " ( x ) c [ R ] † ( x + 1) c [ L ] ( x ) c [ R ] †" ( x + 1) c [ L ] ( x ) c [ R ] † ( x + 1) ! (A24) Appendix B: Details [1] Hagen Kleinert, Gauge fields in condensed matter , Vol. 2(World Scientific, 1989).[2] Eduardo Fradkin, Field theories of condensed matterphysics (Cambridge University Press, 2013).[3] M. Tanabashi et al. (Particle Data Group), “Review ofParticle Physics,” Phys. Rev. D , 030001 (2018).[4] Paul Langacker, “The standard model and beyond,” in AIP Conference Proceedings , Vol. 150 (American Insti-tute of Physics, 1986) pp. 142–164.[5] David Tong, “Gauge theory,” (2018).[6] Kenneth G. Wilson, “Confinement of Quarks,” , 45–59(1974).[7] John B. Kogut, “An Introduction to Lattice Gauge The-ory and Spin Systems,” Rev. Mod. Phys. , 659 (1979).[8] Michael Creutz, Laurence Jacobs, and Claudio Rebbi,“Monte Carlo Computations in Lattice Gauge Theo- ries,” Phys. Rept. , 201–282 (1983).[9] Andrei Alexandru, Gokce Basar, Paulo F. Bedaque, So-han Vartak, and Neill C. Warrington, “Monte CarloStudy of Real Time Dynamics on the Lattice,” Phys.Rev. Lett. , 081602 (2016), arXiv:1605.08040 [hep-lat].[10] Xiang-Dong Ji, “Deeply virtual Compton scatter-ing,” Phys. Rev. D , 7114–7125 (1997), arXiv:hep-ph/9609381.[11] Ral A. Briceo, Zohreh Davoudi, Maxwell T. Hansen,Matthias R. Schindler, and Alessandro Baroni, “Long-range electroweak amplitudes of single hadrons from Eu-clidean finite-volume correlation functions,” Phys. Rev.D , 014509 (2020), arXiv:1911.04036 [hep-lat].[12] Xiangdong Ji, “Parton Physics on a Euclidean Lattice,”Phys. Rev. Lett. , 262002 (2013), arXiv:1305.1539 c [ R ] " ( x ) c [ R ] ( x ) (A14) c [ L ] " ( x ) c [ L ] ( x ) (A15) c [ M ] " ( x ) c [ M ] ( x ) (A16) c [ R ] " ( x +1) c [ R ] ( x +1) (A17) c [ L ] " ( x +1) c [ L ] ( x +1) (A18) c [ M ] " ( x +1) c [ M ] ( x +1) (A19) E R ( x 1) (A20) E L ( x ) (A21) E R ( x ) (A22) E L ( x + 1) (A23) U ( x ) = c [ L ] " ( x ) c [ R ] †" ( x + 1) c [ L ] " ( x ) c [ R ] † ( x + 1) c [ L ] ( x ) c [ R ] †" ( x + 1) c [ L ] ( x ) c [ R ] † ( x + 1) ! (A24) Appendix B: Details [1] Hagen Kleinert, Gauge fields in condensed matter , Vol. 2(World Scientific, 1989).[2] Eduardo Fradkin, Field theories of condensed matterphysics (Cambridge University Press, 2013).[3] M. Tanabashi et al. (Particle Data Group), “Review ofParticle Physics,” Phys. Rev. D , 030001 (2018).[4] Paul Langacker, “The standard model and beyond,” in AIP Conference Proceedings , Vol. 150 (American Insti-tute of Physics, 1986) pp. 142–164.[5] David Tong, “Gauge theory,” (2018).[6] Kenneth G. Wilson, “Confinement of Quarks,” , 45–59(1974).[7] John B. Kogut, “An Introduction to Lattice Gauge The-ory and Spin Systems,” Rev. Mod. Phys. , 659 (1979).[8] Michael Creutz, Laurence Jacobs, and Claudio Rebbi,“Monte Carlo Computations in Lattice Gauge Theo- ries,” Phys. Rept. , 201–282 (1983).[9] Andrei Alexandru, Gokce Basar, Paulo F. Bedaque, So-han Vartak, and Neill C. Warrington, “Monte CarloStudy of Real Time Dynamics on the Lattice,” Phys.Rev. Lett. , 081602 (2016), arXiv:1605.08040 [hep-lat].[10] Xiang-Dong Ji, “Deeply virtual Compton scatter-ing,” Phys. Rev. D , 7114–7125 (1997), arXiv:hep-ph/9609381.[11] Ral A. Briceo, Zohreh Davoudi, Maxwell T. Hansen,Matthias R. Schindler, and Alessandro Baroni, “Long-range electroweak amplitudes of single hadrons from Eu-clidean finite-volume correlation functions,” Phys. Rev.D , 014509 (2020), arXiv:1911.04036 [hep-lat].[12] Xiangdong Ji, “Parton Physics on a Euclidean Lattice,”Phys. Rev. Lett. , 262002 (2013), arXiv:1305.1539 c [ R ] " ( x ) c [ R ] ( x ) (A14) c [ L ] " ( x ) c [ L ] ( x ) (A15) c [ M ] " ( x ) c [ M ] ( x ) (A16) c [ R ] " ( x +1) c [ R ] ( x +1) (A17) c [ L ] " ( x +1) c [ L ] ( x +1) (A18) c [ M ] " ( x +1) c [ M ] ( x +1) (A19) E R ( x 1) (A20) E L ( x ) (A21) E R ( x ) (A22) E L ( x + 1) (A23) U ( x ) = c [ L ] " ( x ) c [ R ] †" ( x + 1) c [ L ] " ( x ) c [ R ] † ( x + 1) c [ L ] ( x ) c [ R ] †" ( x + 1) c [ L ] ( x ) c [ R ] † ( x + 1) ! (A24) Appendix B: Details [1] Hagen Kleinert, Gauge fields in condensed matter , Vol. 2(World Scientific, 1989).[2] Eduardo Fradkin, Field theories of condensed matterphysics (Cambridge University Press, 2013).[3] M. Tanabashi et al. (Particle Data Group), “Review ofParticle Physics,” Phys. Rev. D , 030001 (2018).[4] Paul Langacker, “The standard model and beyond,” in AIP Conference Proceedings , Vol. 150 (American Insti-tute of Physics, 1986) pp. 142–164.[5] David Tong, “Gauge theory,” (2018).[6] Kenneth G. Wilson, “Confinement of Quarks,” , 45–59(1974).[7] John B. Kogut, “An Introduction to Lattice Gauge The-ory and Spin Systems,” Rev. Mod. Phys. , 659 (1979).[8] Michael Creutz, Laurence Jacobs, and Claudio Rebbi,“Monte Carlo Computations in Lattice Gauge Theo- ries,” Phys. Rept. , 201–282 (1983).[9] Andrei Alexandru, Gokce Basar, Paulo F. Bedaque, So-han Vartak, and Neill C. Warrington, “Monte CarloStudy of Real Time Dynamics on the Lattice,” Phys.Rev. Lett. , 081602 (2016), arXiv:1605.08040 [hep-lat].[10] Xiang-Dong Ji, “Deeply virtual Compton scatter-ing,” Phys. Rev. D , 7114–7125 (1997), arXiv:hep-ph/9609381.[11] Ral A. Briceo, Zohreh Davoudi, Maxwell T. Hansen,Matthias R. Schindler, and Alessandro Baroni, “Long-range electroweak amplitudes of single hadrons from Eu-clidean finite-volume correlation functions,” Phys. Rev.D , 014509 (2020), arXiv:1911.04036 [hep-lat].[12] Xiangdong Ji, “Parton Physics on a Euclidean Lattice,”Phys. Rev. Lett. , 262002 (2013), arXiv:1305.1539 c [ R ] " ( x ) c [ R ] ( x ) (A14) c [ L ] " ( x ) c [ L ] ( x ) (A15) c [ M ] " ( x ) c [ M ] ( x ) (A16) c [ R ] " ( x +1) c [ R ] ( x +1) (A17) c [ L ] " ( x +1) c [ L ] ( x +1) (A18) c [ M ] " ( x +1) c [ M ] ( x +1) (A19) E R ( x 1) (A20) E L ( x ) (A21) E R ( x ) (A22) E L ( x + 1) (A23) U ( x ) = c [ L ] " ( x ) c [ R ] †" ( x + 1) c [ L ] " ( x ) c [ R ] † ( x + 1) c [ L ] ( x ) c [ R ] †" ( x + 1) c [ L ] ( x ) c [ R ] † ( x + 1) ! (A24) Appendix B: Details [1] Hagen Kleinert, Gauge fields in condensed matter , Vol. 2(World Scientific, 1989).[2] Eduardo Fradkin, Field theories of condensed matterphysics (Cambridge University Press, 2013).[3] M. Tanabashi et al. (Particle Data Group), “Review ofParticle Physics,” Phys. Rev. D , 030001 (2018).[4] Paul Langacker, “The standard model and beyond,” in AIP Conference Proceedings , Vol. 150 (American Insti-tute of Physics, 1986) pp. 142–164.[5] David Tong, “Gauge theory,” (2018).[6] Kenneth G. Wilson, “Confinement of Quarks,” , 45–59(1974).[7] John B. Kogut, “An Introduction to Lattice Gauge The-ory and Spin Systems,” Rev. Mod. Phys. , 659 (1979).[8] Michael Creutz, Laurence Jacobs, and Claudio Rebbi,“Monte Carlo Computations in Lattice Gauge Theo- ries,” Phys. Rept. , 201–282 (1983).[9] Andrei Alexandru, Gokce Basar, Paulo F. Bedaque, So-han Vartak, and Neill C. Warrington, “Monte CarloStudy of Real Time Dynamics on the Lattice,” Phys.Rev. Lett. , 081602 (2016), arXiv:1605.08040 [hep-lat].[10] Xiang-Dong Ji, “Deeply virtual Compton scatter-ing,” Phys. Rev. D , 7114–7125 (1997), arXiv:hep-ph/9609381.[11] Ral A. Briceo, Zohreh Davoudi, Maxwell T. Hansen,Matthias R. Schindler, and Alessandro Baroni, “Long-range electroweak amplitudes of single hadrons from Eu-clidean finite-volume correlation functions,” Phys. Rev.D , 014509 (2020), arXiv:1911.04036 [hep-lat].[12] Xiangdong Ji, “Parton Physics on a Euclidean Lattice,”Phys. Rev. Lett. , 262002 (2013), arXiv:1305.1539 FIG. 24. Rishon constituents at two adjacent staggered sites x and x +1 in a one-dimensional lattice within the QLM formulationof SU(2) LGT. This construction re-expresses the different parts of theSU(2) LGT Hamiltonian in terms of rishon operators as H (QLM) I = t (cid:88) x,s,s (cid:48) (cid:104) ˆ c [ M ] † s ( x ) ˆ U s,s (cid:48) ( x )ˆ c [ M ] s (cid:48) ( x + 1) + h . c . (cid:105) = t (cid:88) x,s,s (cid:48) (cid:104) ˆ c [ M ] † s ( x )ˆ c [ L ] s ( x )ˆ c [ R ] † s (cid:48) ( x + 1)ˆ c [ M ] s (cid:48) ( x + 1)+ h . c . (cid:105) , (A6) H (QLM) E = g (cid:88) x (cid:104) ˆ E R ( x − 1) + ˆ E L ( x ) (cid:105) ≡ g (cid:88) x (cid:34)(cid:16) ˆ n [ R ] ↑ ( x − 1) + ˆ n [ R ] ↓ ( x − − n [ R ] ↑ ( x − n [ R ] ↓ ( x − (cid:17) + (cid:16) ˆ n [ L ] ↑ ( x ) + ˆ n [ L ] ↓ ( x ) − n [ L ] ↑ ( x )ˆ n [ L ] ↓ ( x ) (cid:17) (cid:35) , (A7) H (QLM) M = m (cid:88) x ( − x (cid:104) ˆ n [ M ] ↑ ( x ) + ˆ n [ M ] ↓ ( x ) (cid:105) , (A8)where ˆ n [ τ ] s ( x ) ≡ ˆ c [ τ ] † s ( x ) ˆ c [ τ ] s ( x ) is the fermionic occupa-tion number at site x for the s = ↑ , ↓ components of the τ = L, R, M type fermion. The Hamiltonian, being thesum of the three parts given in Eqs. (A7), (A8), and (A6),conserves the total number of fermions at each site, i.e., (cid:88) s = ↑ , ↓ (cid:104) n [ M ] s ( x ) + n [ L ] s ( x ) + n [ R ] s ( x ) (cid:105) = const ., (A9)implying that the local symmetry is instead SU (2) ⊗ U (1). Hence, in order to recover the SU(2) gauge sym-metry of interest, one needs to add the following U(1)symmetry-breaking term to the Hamiltonian, H (QLM)break = (cid:15) (cid:88) x (cid:104) det ˆ U ( x, x + 1) + h . c . (cid:105) = (cid:15) (cid:88) x (cid:104) ˆ c [ L ] †↑ ( x )ˆ c [ L ] †↓ ( x )ˆ c [ R ] ↓ ( x + 1)ˆ c [ R ] ↑ ( x + 1)+h . c . (cid:105) . (A10) Hence, the total Hamiltonian of the SU(2) QLM in 1+1 Dis given by H (QLM) = H (QLM) I + H (QLM) M + H (QLM) E + H (QLM)break , (A11)with the Gauss’s law operator ˆ G a ( x ) = − ˆ E aL ( x ) + ˆ E aR ( x − 1) + c [ M ] † s ( x ) T as,s (cid:48) c [ M ] s (cid:48) ( x ) . (A12)It should be noted that the U(1) symmetry-breakingterm in Eq. (A10) is only non-vanishing if the total num-ber of fermionic rishons on each link is equal to two [123]: n [ L ] ↑ ( x ) + n [ L ] ↓ ( x ) + n [ R ] ↑ ( x + 1) + n [ R ] ↓ ( x + 1) = 2 . (A13)Since the Hamiltonian in Eq. (A11) conserves the rishonnumber on the link, this constraint is preserved.The QLM is only equivalent to the KS SU(2) gaugetheory in the continuum limit and through a dimensionalreduction [81, 82]. As a result, the spectrum and dynam-ics of the theory outlined here are not the same as those ofthe KS LGT obtained in this paper, even when the samelattice size and couplings are considered, a feature thatcan be verified by numerical computations. Nonetheless,it is interesting to ask whether the QLM is computation-ally efficient in the context of Hamiltonian simulation.While a thorough analysis of the computational cost ofconstructing the physical Hilbert space, generating theHamiltonian, and computing observables with the QLMare not analyzed in this paper, one important observa-tion can be made regarding the dimensionality of thephysical Hilbert space of the QLM. Figure 25 plots theratio of the dimension of the physical Hilbert space inthe QLM to that in the KS formulation (or equivalentlythe LSH formulation) when the cutoff is set to its sat-urating value with OBC. As is seen from the empiricalfit to the values for the first lowest N values, this ratiogrows exponentially with N , and so from a computa-tional standpoint, such a finite-dimensional representa-tion of the SU(2) LGT is still costly. Note that here, one To compare with Ref. [92], note that E L ≡ − J L and E R ≡ J R . log( R ) = 0 . 050 + 0 . N . N ▽ ▽ ▽ ▽ N l og ( R ) FIG. 25. The (logarithm of the) ratio of the dimension ofthe physical Hilbert space within the QLM formulation tothat within the KS (and LSH) formulation (the latter with asufficiently large cutoff such that the number of basis statessaturates to a fixed value), both with OBC, for several valuesof the lattice size, N . The empirical fit to the N dependenceof this quantity is shown in the figure. The numerical valuesassociated with this plot are listed in Supplemental Material. should also account for a comparable computational costto the LSH formulation in generating the Hilbert space:there are six types of fermions present locally, requiring2 configurations to be generated at each site. Addi-tionally, the Gauss’s law constraints and the fixed-rishonnumber per link must be imposed when constructing theHilbert space.It is worth noting that while in the LSH formulation,the infinite-dimensional bosons are present (and are cutoff at some finite value), with OBC their value can befixed given the string configurations. In higher dimen-sions, there may still be an advantage in working withthe finite-dimensional QLM since with other formula-tions, the Hilbert space grows with the cutoff on thelink quantum numbers as these can no longer be fullyfixed with boundary conditions. Nonetheless, the LSHformulation appears to be a competitive formulation ofthe SU(2) LGT, that not only is equivalent to the originalKS theory, but also its economical resource requirementscan bring it to the same footing as the QLM when itcomes to quantum-simulation proposals on analog anddigital simulators, an avenue that will be explored in theupcoming studies. Appendix B: Physical Hilbert-space dimensionality In this appendix, the numerical values of the dimensionof the physical Hilbert space in the KS SU(2) LGT as afunction of lattice sites N are listed in Tables III and IVfor PBC and OBC, respectively. Appendix C: Observables with open boundaryconditions This appendix contains the same plots as in Sec. VI forthe spectrum and dynamics of the KS SU(2) LGT but with OBC. These plots were removed from the main textfor brevity, and while the overall features and the generalconclusions remain the same as those for the PBC, theyare included in this appendix for completeness.8 N = 20 1 2 3 4 5 6 7 8 9 104 10 16 22 28 34 40 46 52 58 64 N = 40 1 2 3 4 5 6 7 8 9 1016 82 152 222 292 362 432 502 572 642 712 N = 60 1 2 3 4 5 6 7 8 9 1064 730 1,648 2,572 3,496 4,420 5,344 6,268 7,192 8,116 9,040 N = 80 1 2 3 4 5 6 7 8 9 10256 6,562 18,720 31,582 44,452 57,322 70,192 83,062 95,932 108,802 121,672 N = 100 1 2 3 4 5 6 7 8 9 101,024 59,050 216,256 399,502 584,248 769,004 953,760 1,138,516 1,323,272 1,508,028 1,692,784TABLE III. The dimension of the physical Hilbert space of the KS (and LSH) formulation with PBC, for N = 2 , , · · · , J max ) = 0 , , · · · , N = 20 1 24 9 10 N = 40 1 2 3 416 81 116 125 126 N = 60 1 2 3 4 5 664 729 1,352 1,625 1,702 1,715 1,716 N = 80 1 2 3 4 5 6 7 8256 6,561 15,760 21,250 23,494 24,157 24,292 24,309 24,310 N = 100 1 2 3 4 5 6 7 8 9 101,024 59,049 183,712 278,125 326,382 345,401 351,176 352,485 352,694 352,715 352,716TABLE IV. The dimension of the physical Hilbert space of the KS (and LSH) formulation with OBC, for N = 2 , , · · · , J max ) = 0 , , · · · , 10. For Λ > N , the dimension of the physical Hilbert space saturates to the value at Λ = N , andis hence not shown. Q q , 716 with OBC and for N = 10 and Λ ≥ 10. Thedimensionalities of the sectors with charge 2 N − Q are the same as those with charge Q ∈ [0 , N = 6, ν = 1, and various values of x and Λ. More precisely, the quantity plotted is E (cid:48) Nx , where E (cid:48) is the scaled energy corresponding to the scaled Hamiltonianin Eq. (89). The numerical values associated with these plots are provided in Supplemental Material. ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - Λ Δ E n / E n ▽ ◦ ◇ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - Λ Δ E n / E n ▽ ◦ ◇ x = 1 x = 25 x = 100 x = 400 E / E ⇤ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ - - - Λ Δ E n / E n ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ - - - Λ Δ E n / E n ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ - - - - Λ Δ E n / E n ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ - - - Λ Δ E n / E n ▽ ◦ ◇ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ - - - - - Λ Δ E n / E n FIG. 27. The quantity ∆ E (cid:48) E (cid:48) ≡ E (cid:48) (Λ) − E (cid:48) (Λ=8) E (cid:48) (Λ) as a functionof Λ for various values of x , and for the 1st, 21st, and 142ndstates in the spectrum of the KS Hamiltonian in the physicalHilbert space with OBC and with N = 6 and ν = 1. E (cid:48) (Λ)is the scaled energy corresponding to the scaled Hamiltonianin Eq. (89). The dashed lines denote the first Λ values atwhich the corresponding scaled energies become equal or lessthan 10% of their value at Λ = 8 (which are the Λ → ∞ values). When needed for presentational clarity, the pointsare artificially displaced along the horizontal axes by a smallamount. The numerical values associated with these plots areprovided in Supplemental Material. E / E p x ⇤ = 0⇤ = 1⇤ = 2 ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - Λ Δ E n / E n ▽ ◦ ◇ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ - Λ Δ E n / E n ▽ ◦ ◇ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ - - - Λ Δ E n / E n ▽ ◦ ◇ ▽▽▽▽ ▽ ▽ ▽ ▽ ▽ ▽ - - - - - x Δ E / E ◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ - - - - - x Δ E / E ◇◇◇◇ ◇ ◇ ◇ ◇ ◇ ◇ - - - - - x Δ E / E r ⇡ . r ⇡ . r ⇡ . FIG. 28. The quantity ∆ E (cid:48) E (cid:48) ≡ E (cid:48) (Λ) − E (cid:48) (Λ=8) E (cid:48) (Λ) as a functionof √ x for given values of Λ as denoted in the plots, and forthe 1st, 21st, and 142nd states in the spectrum of the KSHamiltonian in the physical Hilbert space with OBC and with N = 6 and ν = 1. E (cid:48) (Λ) is the scaled energy corresponding tothe scaled Hamiltonian in Eq. (89). The asymptotic ( x → ∞ )values of the quantity are obtained from the fits to data pointsin each case with an exponentially varying function of x , andare denoted in the plot. The colored regions denote the √ x values excluded from the fits. The numerical values associatedwith these plots are provided in Supplemental Material. ⇤ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ Λ Δ N p ▽ t = ◦ t = ◇ t = ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ Λ Δ N p ▽ t = ◦ t = ◇ t = ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ Λ Δ N p ▽ t = ◦ t = ◇ t = x = 1 x = 25 x = 100 x = 400 ◇ ◇ ◇ ◇ ◇ ◇ Λ Δ N p ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ Λ Δ N p ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ Λ Δ N p ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ Λ Δ N p ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ Λ Δ N p N ( t ) FIG. 29. The quantity N ( t ) defined in Eq. (90) as a functionof Λ for various values of x and t in the KS Hamiltonian inthe physical Hilbert space with N = 6 and ν = 1 with OBC. t is the absolute time as defined in the text in units of a .The exact values of the quantity are shown in dashed lines.Note that for Λ = 6, the full Hilbert space with this quantumnumber is achieved, nonetheless the quantity shown reachesthe exact values with Λ < p x ⇤ = 4 ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ Λ Δ N p ▽ t = ◦ t = ◇ t = N ( t ) ◇◇◇◇◇◇◇◇◇◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ x N r ⇡ . FIG. 30. The quantity N ( t ) defined in Eq. (90) as a functionof √ x for Λ = 4 and t = 100 in the KS Hamiltonian in thephysical Hilbert space with N = 6 and ν = 1 with OBC.The points are fit to N ∼ e − r √ xx