Towards Complexity for Quantum Field Theory States
Shira Chapman, Michal P. Heller, Hugo Marrochio, Fernando Pastawski
TTowards Complexity for Quantum Field Theory States
Shira Chapman, ∗ Michal P. Heller, † Hugo Marrochio,
1, 3, ‡ and Fernando Pastawski
4, 2, § Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada Max Planck Institute for Gravitational Physics, Potsdam-Golm, D-14476, Germany Department of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada Dahlem Center for Complex Quantum Systems, Freie Universit¨at Berlin, Berlin, D-14195, Germany
We investigate notions of complexity of states in continuous quantum-many body systems. Wefocus on Gaussian states which include ground states of free quantum field theories and their approx-imations encountered in the context of the continuous version of Multiscale Entanglement Renor-malization Ansatz. Our proposal for quantifying state complexity is based on the Fubini-Studymetric. It leads to counting the number of applications of each gate (infinitesimal generator) in thetransformation, subject to a state-dependent metric. We minimize the defined complexity with re-spect to momentum preserving quadratic generators which form su (1 ,
1) algebras. On the manifoldof Gaussian states generated by these operations the Fubini-Study metric factorizes into hyper-bolic planes with minimal complexity circuits reducing to known geodesics. Despite working withquantum field theories far outside the regime where Einstein gravity duals exist, we find strikingsimilarities between our results and those of holographic complexity proposals.
1. Introduction.–
Applications of quantum informationconcepts to high energy physics and gravity have recentlyled to many far-reaching developments. In particular, ithas become apparent that special properties of entangle-ment in holographic [1] quantum field theories (QFT)sstates are crucial for the emergence of smooth higher-dimensional (bulk) geometries in the gauge-gravity du-ality [2]. Much of the progress in this direction wasachieved building on the holographic entanglement en-tropy proposal by Ryu and Takayanagi [3] which ge-ometrizes the von Neumann entropy of a reduced den-sity matrix of a QFT in a subregion in terms of thearea of codimension-2 bulk minimal surfaces anchoredat the boundary of this subregion (see e.g., Ref. [4] fora recent overview). However, Ryu-Takayanagi surfacesare often unable to access the whole holographic geom-etry [5–7]. This observation has led to significant inter-est in novel, from the point of view of quantum grav-ity, codimension-1 (volume) and codimension-0 (action)bulk quantities, whose behavior suggests conjecturing alink with the information theory notion of quantum statecomplexity [8–14]. In fact, a certain identification be-tween complexity and action was originally suggested byToffoli [15, 16] outside the context of holography.Quantum state complexity originates from the field ofquantum computations, which are usually modeled in afinite Hilbert space as the application of a sequence ofgates chosen from a discrete set. In this context, thecomplexity of a unitary U is roughly associated withthe minimal number of gates necessary to realize (or ap-proximate) U . Notable progress has been made in con-necting this notion to distances in Riemannian geome-tries derived from a set of generators [17]. The com-plexity of a target state | T (cid:105) is usually subordinated tounitary complexity by specifying a “simple” referencestate | R (cid:105) and minimizing the complexity of U subjectto | T (cid:105) = U | R (cid:105) [18, 19]; Our approach differs in defining state complexity directly.In the context of holography, the organization of dis-crete tensor networks (seen as a quantum circuit U ) hasbeen suggested to give a qualitative picture of how quan-tum states give rise to emergent geometries [20]. Thisheuristic analysis was applied to the multiscale entangle-ment renormalization ansatz (MERA) [21], employed tofind ground states of critical physical theories present-ing a tensor network structure reminiscent of an AdStime slice. This motivated proposing “complexity equalsvolume” (CV) [9] and “complexity equals action” (CA)[11, 12] as new entries in the holographic dictionary.However, in holography one naturally considers contin-uum setups, QFTs, and there are shortcomings of tra-ditional approaches to complexity when attempting toaddress field theory states. The aim of this letter is tobridge a pressing gap by exploring complexity-motivateddistance measures in QFTs.The main challenges in providing a workable definitionof complexity in the continuum are related to choosing:a) a reference state | R (cid:105) , b) a set of allowed gates (cor-respondingly infinitesimal generators), c) a measure forhow such gates contribute to the resulting distance func-tion and a procedure for how to minimize it, d) a wayto regulate ultraviolet (UV) divergences. Our proposedchoice for c) is to measure the path length by integratingthe Fubini-Study (FS) line element along a path from | R (cid:105) to | T (cid:105) associated to an allowed realization of U . Mini-mizing the path will amount to studying geodesics onthe manifold of quantum states induced by allowed gatesacting on the reference state. In this way, our approachderives complexity from the projective structure of theHilbert space in a universal way. In the FS prescription,directions which modify the state by an overall phasehave no effect on the complexity. Simultaneously withour work, Ref. [22] appeared which considers a differentapproach based on unitary complexity [17] (see [23]F for a r X i v : . [ h e p - t h ] N ov a comparison).While the FS prescription is quite general, our choicesfor a), b) and d) render the necessary calculationstractable. Some of these choices, are inspired by the con-tinuous MERA (cMERA) approach to free QFTs [24–26],which we briefly review in [23]A. Similarly to the statesin cMERA, our choices for the reference state | R (cid:105) andtarget state | T (cid:105) will be pure Gaussian states and allowedgenerators will be subsets of quadratic operators. Ourchoices include cMERA in the set of allowed circuits, let-ting us test its optimality. We perform our analysis inmomentum space and ignore frequencies above the UVcutoff Λ which equips us with a notion of approximation.Unlike in cMERA, Λ need not coincide with the referencestate characteristic scale M , defined below in Sec. 2, sincethe freedom of choosing the reference state is a part ofthe definition of complexity and is a priori independentfrom a notion of cutoff or regulator (this observation isdue to R. C. Myers).As a first step, we consider the two mode squeezing op-erator for each pair of opposite momenta ± (cid:126)k . We thenextend our analysis to include the full set of momentumpreserving quadratic generators which form su (1 ,
1) al-gebras. In this case the study of minimal complexityreduces to the study of geodesics on a product of hyper-bolic planes.While a full literature review is outside the scope of thisarticle, there is a substantial body of important recentdevelopments which include e.g., Refs. [27–35].
2. Complexity from the Fubini-Study metric.–
We areinterested in considering unitary operators U arising fromiterating generators G ( s ) taken from some elementary setof Hermitian operators G . The allowed transformations U can then be represented as path ordered exponentials U ( σ ) = P e − i (cid:82) σsi G ( s ) ds . (1)Here, s parametrizes progress along a path, starting at s i and ending at s f and σ ∈ [ s i , s f ] is some intermediatevalue of s . The path-ordering P is required for non-commuting generators G ( s ). We seek a path achieving | T (cid:105) ≈ U ( s f ) | R (cid:105) , where ( ≈ ) indicates that states coincidefor momenta below a cutoff Λ. According to the FS lineelement (see e.g., [36]), ds F S ( σ ) = dσ (cid:114)(cid:12)(cid:12)(cid:12) ∂ σ | Ψ( σ ) (cid:105) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) (cid:104) Ψ( σ ) | ∂ σ | Ψ( σ ) (cid:105) (cid:12)(cid:12)(cid:12) , (2)the length of a path going via states | Ψ( σ ) (cid:105) is (cid:96) ( | Ψ( σ ) (cid:105) ) = (cid:90) s f s i ds F S ( σ ) . (3)For a path | Ψ( σ ) (cid:105) = U ( σ ) | R (cid:105) , with U ( σ ) given by Eq. (1),the line element of Eq. (3) can be re-expressed as ds F S ( σ ) = dσ (cid:113) (cid:104) G ( σ ) (cid:105) Ψ( σ ) − (cid:104) G ( σ ) (cid:105) σ ) , (4) and is independent of path reparametrizations.If the path | Ψ( σ ) (cid:105) is unrestricted, the unique unitarilyinvariant distance measure d R,T = arccos |(cid:104) R | T (cid:105)| ≤ π/ G ( s ), highly non-trivial notions of distance deservingthe name complexity may be obtained. Our proposal is todefine the complexity C as the minimal length accordingto Eq. (3) of a path from | Ψ( s i ) (cid:105) ≈ | R (cid:105) to | Ψ( s f ) (cid:105) ≈ | T (cid:105) driven by generators G ( s ) in GC ( | R (cid:105) , | T (cid:105) , G , Λ) = min G ( s ) (cid:96) ( | Ψ( σ ) (cid:105) ) . (5)The proposed complexity C inherits the properties of adistance function from the FS metric.
3. Gaussian states in free QFTs.–
We consider a theoryof free relativistic bosons in ( d + 1)-spacetime dimensionsdefined by the quadratic Hamiltonian H m = (cid:90) d d x : (cid:26) π + 12 ( ∂ (cid:126)x φ ) + 12 m φ (cid:27) : (6)with commutation relations [ φ ( (cid:126)x ) , π ( (cid:126)x (cid:48) )] = i δ d ( (cid:126)x − (cid:126)x (cid:48) ).This theory describes noninteracting particles createdand annihilated by operators a † (cid:126)k and a (cid:126)k obeying[ a (cid:126)k , a † (cid:126)k (cid:48) ] = δ d ( (cid:126)k − (cid:126)k (cid:48) ). These are related to the field andmomentum operators via ( ω k ≡ √ k + m ) φ ( (cid:126)k ) = 1 √ ω k (cid:16) a (cid:126)k + a †− (cid:126)k (cid:17) and π ( (cid:126)k ) = √ ω k √ i (cid:16) a (cid:126)k − a †− (cid:126)k (cid:17) (7) and diagonalize the Hamiltonian: H m = (cid:82) d d k ω k a † (cid:126)k a (cid:126)k .For m = 0 we obtain a free conformal field theory (CFT).A general translation invariant pure Gaussian state | S (cid:105) with momentum space correlation functions (cid:104) S | φ ( (cid:126)k ) φ ( (cid:126)k (cid:48) ) | S (cid:105) = 12 α k δ ( d ) (cid:16) (cid:126)k + (cid:126)k (cid:48) (cid:17) , (8)is specified by its nullifiers (annihilation operators): (cid:26)(cid:114) α k φ ( (cid:126)k ) + i √ α k π ( (cid:126)k ) (cid:27) | S (cid:105) = 0 . (9)The ground state | m (cid:105) of the free theory (6) is a pureGaussian state corresponding to α k = ω k . The groundstate | m (cid:105) is a product of vacuum states in momentumspace without particles according to the number opera-tors n (cid:126)k ≡ a † (cid:126)k a (cid:126)k . In momentum space, the only nontrivialcorrelations in | S (cid:105) are between (cid:126)k and ( − (cid:126)k ) modes. Inreal-space, the (cid:126)k -dependent factor on the RHS of Eq. (8)leads to spatial correlations (and entanglement).A natural choice for a reference state | R ( M ) (cid:105) is theGaussian state corresponding to | R ( M ) (cid:105) : α k = M . (10)Since here α k is independent of k this state is a prod-uct state with no spatial correlations, i.e., in real spacethe two point function of field operators takes the form (cid:104) R ( M ) | φ ( (cid:126)x ) φ ( (cid:126)x (cid:48) ) | R ( M ) (cid:105) = M δ d ( (cid:126)x − (cid:126)x (cid:48) ). Neverthelessin the basis associated with energy eigenstates of H m mo-mentum sectors (cid:126)k and − (cid:126)k are pairwise entangled accord-ing to (8). We will show that the reference state scale M is related to certain ambiguities encountered in thecontext of holographic complexity. The annihilation andcreation operators b (cid:126)k and b † (cid:126)k associated with the state | R ( M ) (cid:105) can be related to those of the vacuum state | m (cid:105) by the following Bogoliubov transformation b (cid:126)k = β + k a (cid:126)k + β − k a †− (cid:126)k ; b (cid:126)k | R ( M ) (cid:105) = 0; β + k = cosh 2 r k ; β − k = sinh 2 r k ; r k ≡ log (cid:114) Mω k . (11)As our target state, we consider the approximateground state (cid:12)(cid:12) m (Λ) (cid:11) characterized by the UV momentumcut-off Λ which corresponds to: (cid:12)(cid:12)(cid:12) m (Λ) (cid:69) : α k = (cid:26) ω k , k < Λ (QFT vacuum)
M, k ≥ Λ (product state) , (12)with correlation functions interpolating between those ofthe vacuum state | m (cid:105) and the reference state | R ( M ) (cid:105) asmomentum increases according to Eq. (8). This state isin fact identical to the real ground state | m (cid:105) up to thecut-off momentum. When M = ω Λ , this state is identicalto the one obtained by cMERA circuits [24, 25] (see e.g.Ref. [37]).The target states (12) can be reached from the refer-ence states (10) by a circuit constructed with two modesqueezing operators which entangle the (cid:126)k and − (cid:126)k modes, K ( (cid:126)k ) = φ ( (cid:126)k ) π ( − (cid:126)k ) + π ( (cid:126)k ) φ ( − (cid:126)k )= i (cid:16) a † (cid:126)k a †− (cid:126)k − a (cid:126)k a − (cid:126)k (cid:17) = i (cid:16) b † (cid:126)k b †− (cid:126)k − b (cid:126)k b − (cid:126)k (cid:17) . (13) This operator is the main building block in cMERA cir-cuits, and allows preparing the target state as follows (cid:12)(cid:12)(cid:12) m (Λ) (cid:69) = e − i (cid:82) k ≤ Λ d d k r k K ( (cid:126)k ) | R ( M ) (cid:105) , (14)which is the starting point for our complexity analysis.
4. Ground state complexity with a single generator perpair of momenta ± (cid:126)k .– We start by evaluating our pro-posed complexity under the assumption that we allow fora single generator per pair of momenta ± (cid:126)k which we taketo be K ( (cid:126)k ) of Eq. (13), i.e., G = Span { K ( (cid:126)k ) } , where Span is taken over the field of real numbers. These generatorscontinue to achieve minimal complexity within the larger su (1 ,
1) class considered in Sec. 5. We consider circuitsof the form (1) with G ( σ ) = (cid:90) k ≤ Λ d d k K ( (cid:126)k ) y (cid:126)k ( σ ) . (15) Since all the K ( (cid:126)k ) commute, the unitary U ( σ ) of (1) issimply specified by the integrated values Y (cid:126)k ( σ ) := (cid:90) σs i y (cid:126)k ( s ) ds ; Y (cid:126)k ( s f ) = r k , (16)where Y (cid:126)k ( s f ) was fixed to match Eq. (14). The commu-tation of generators allows the variance in the FS lineelement (4) to be evaluated at any state | Ψ( σ ) (cid:105) along thepath. Furthermore, the variance is additive with respectto the different K ( (cid:126)k ) contributions because only equalor opposite momenta can be correlated. The complexityminimization of Eq. (5) then reduces to C = min Y (cid:126)k ( σ ) (cid:90) s f s i dσ (cid:115) (cid:90) k ≤ Λ d d k (cid:0) ∂ σ Y (cid:126)k ( σ ) (cid:1) , (17)where Vol ≡ δ d (0) is the volume of the d -dimensionaltime slice. One recognizes a flat Euclidean geometrywith coordinates Y (cid:126)k ( σ ) continuously labeled by (cid:126)k . Toachieve minimal complexity the generators for the differ-ent momenta must act simultaneously with ratio dictatedby Eq. (16) (straight path). A particularly simple affineparametrization for the path is Y (cid:126)k ( σ ) = σ − s i s f − s i Y (cid:126)k ( s f ); y k ( σ ) = 1 s f − s i Y (cid:126)k ( s f ) . (18) As the corresponding cMERA circuit presents a σ depen-dent ratio, the complexity associated with it will generi-cally be larger (as shown in [23]A). Evaluating (17) with(18), the minimal complexity reads C (2) = (cid:115) (cid:90) k ≤ Λ d d k r k , (19)where the superscript (2) anticipates an interpretation ofEq. (19) as an L norm.Suppose on the other hand that G contains only indi-vidual generators K ( (cid:126)k ) and not their linear span. Thisis analogous to disallowing different elementary gates ina circuit to act simultaneously. Our path parameters inthis case consist of σ and (cid:126)k . The arguments leading toEq. (17) continue to hold except that now, the k inte-gral must be pulled out of the square root and an extra (cid:112) Vol / L norm (Man-hattan distance) complexity C (1) = Vol (cid:90) k ≤ Λ d d k | r k | . (20)More generally, and without reference to the FS metric,one can postulate L n norms as a measure of complexity C ( n ) = 2 n (cid:115) Vol2 (cid:90) k ≤ Λ d d k | r k | n . (21)The leading divergence in the complexity measures C ( n ) is proportional to C ( n ) ∼ Vol /n Λ d/n log( M/ Λ) , (22)when M and Λ are chosen independently and to C ( n ) ∼ Vol /n Λ d/n (23)when M = Λ. See [23]B for some additional details onevaluating the ground state complexities using the C ( n ) measures. The C (1) norm results carry resemblance tothose found in the context of holographic complexity aswe explain in Sec. 6.
5. Ground state complexity using su (1 , generators.– Here, we extend our minimization to a larger set of gen-erators G that transforms | R ( M ) (cid:105) into (cid:12)(cid:12) m (Λ) (cid:11) . Namely,we consider momentum preserving quadratic operators,which for each (cid:126)k are spanned by G = Span (cid:26) K , K ≡ K + + K − , K ≡ K + − K − i (cid:27) K + = b † (cid:126)k b †− (cid:126)k , K − = b (cid:126)k b − (cid:126)k , K = b † (cid:126)k b (cid:126)k + b − (cid:126)k b †− (cid:126)k . (24) These Hermitian operators form a larger (yet manage-able), algebraically closed extension of the generators K = − K of Eq. (13) used in cMERA circuits. Thealgebra formed is an infinite product of su (1 ,
1) subalge-bras of quadratic generators commuting with n (cid:126)k − n − (cid:126)k .The path in Eqs. (1), (15) and (18) is contained in thislarger set. We prove that it continues to be minimal anddetermine its complexity, although we emphasize thatthis does not follow automatically from the results of theprevious section. For instance, in [23]D we study an-other constant generator B ( (cid:126)k, M ) which belongs to theextended su (1 ,
1) subalgebras but does not lead to a min-imal length path. This generator has bounded norm anddrives constant period oscillations between the referencestate | R ( M ) (cid:105) and target state (cid:12)(cid:12) m (Λ) (cid:11) .We will see that the manifold of states generated byeach su (1 ,
1) is a hyperbolic plane, one for each pair of op-posite momenta. Minimal complexity paths correspondto geodesics in the resulting tensor product manifold. Atthe level of the state | Ψ( σ ) (cid:105) , the most general su (1 , | Ψ( σ ) (cid:105) = N ( σ ) e (cid:82) d d kγ + ( (cid:126)k,σ ) K + ( (cid:126)k ) | R ( M ) (cid:105) , (25)where N ( σ ) is a complex normalization and σ is thepath parameter from Eq. (1). This implies that thestate | Ψ( σ ) (cid:105) can be conveniently parametrized by a sin-gle complex parameter γ + ( σ ). The existence of spuriousparameters is a manifestation of the non-uniqueness ofthe unitary circuit U ( σ ) achieving a minimal complexitypath. Due to the non-commutative nature of the genera-tors (24), there is a non-trivial relationship between theircoefficients in the path ordered exponential in Eq. (1)and γ + ( (cid:126)k, σ ). Our reference state | R ( M ) (cid:105) corresponds to γ + ( (cid:126)k, s i ) = 0 while the target state (cid:12)(cid:12) m (Λ) (cid:11) correspondsto γ + ( (cid:126)k, s f ) = tanh(2 r k ). Evaluating the FS line element (2) along the path (25)leads to the following remarkably simple form ds F S ( σ ) = dσ (cid:118)(cid:117)(cid:117)(cid:116) Vol2 (cid:90) Λ d d (cid:126)k γ (cid:48) + ( (cid:126)k, σ ) γ (cid:48)∗ + ( (cid:126)k, σ )(1 − | γ + ( (cid:126)k, σ ) | ) , (26)(see [23]C for the derivation). This line element corre-sponds to a direct product of Poincar´e disks parametrizedby the complex coordinates γ + ( (cid:126)k ) = γ + ( − (cid:126)k ) ( | γ + ( (cid:126)k ) | < ± (cid:126)k (an example ofsuch a disk is illustrated in Fig. 1). The Poincar´edisk is the manifold naturally associated with the coset SU (1 , /U (1) (see e.g., [38–40]) and its structure ofgeodesics is well known. Given an affinely parametrizedgeodesic on a Riemannian product manifold such as (26),its natural projections are affinely parametrized geodesicswithin each factor manifold. The relative speeds of theseprojections are coupled and will, as in (18), be fixed bythe target state. FIG. 1. The Poincar´e disk, parametrized by real (horizon-tal) and imaginary (vertical) components of γ + . Examplesof geodesics appear as dashed lines. The two dots indicatethe reference state (the center) and the target state (on thereal axis). The geodesic connecting the two is the straightsolid line along the diameter, corresponding to the generator K ( (cid:126)k ). The solid semicircle is the non-geodesic path gener-ated by B ( (cid:126)k, M ) (see [23]D). The su (1 ,
1) algebra generatesisometries on the hyperbolic plane.
The geodesic connecting | R ( M ) (cid:105) and | m (Λ) (cid:105) follows theradial direction on the Poincar´e disc which correspondsto the affinely parametrized path γ + ( (cid:126)k, σ ) = tanh(2 r k σ ), σ ∈ [0 , K ( (cid:126)k ) of Sec. 4. Therefore thepath in Eqs. (15), (18) leads to minimal complexity evenwithin the larger class of su (1 ,
1) generators.
6. Comparison with Holographic Complexity Proposals.–
There are two proposals for the gravity dual of complex-ity in terms of maximal codimension-1 volumes (CV [9])or on-shell actions of the Wheeler-DeWitt patch boundedby null hypersurfaces (CA [11, 12]) in the dual bulkspacetime. The structure of the vacuum UV divergencesof holographic complexity can be characterized by a UVregularization scheme [13, 14, 41] with the cut-off dis-tance from the AdS boundary in Fefferman-Graham co-ordinates identified as δ ∼ / Λ. Eq. (22) for C (1) indi-cates a leading divergence of Vol Λ d (cid:12)(cid:12) log M Λ (cid:12)(cid:12) (with M andΛ independent), which resembles the result of the CAproposal. In the holographic CA calculation, the leadinglogarithmic divergence is due to the codimension-2 jointaction contributions associated with the intersection be-tween the null and timelike hypersurfaces that bound theregulated Wheeler-DeWitt patch near the AdS boundary[42]. These contributions depend on the parametriza-tion of null normals ([42] suggested working in an affineparametrization) and their overall rescaling. The lattergives rise to an extra freedom represented in [13] by afree parameter ˜ α inside the logarithm. In our calculationthe same type of ambiguity is related to the choice of thereference state scale M and we can identify M ∼ ˜ α/L AdS where L AdS is the AdS scale. When M = ω Λ , the lead-ing divergence becomes proportional to Vol Λ d , which isin agreement with the CV results [13] (or with the CAresults when including a counter term which renders theaction reparametrization invariant, see [41]). It is inter-esting that despite considering QFTs without semiclas-sical gravity duals (having small central charge and nointeractions), the C (1) norm exhibits close similarity tothe holographic calculations of leading UV divergences.
7. Summary and Outlook.–
We proposed a definitionof state complexity in QFTs, independent from a no-tion of unitary complexity. This measure is derived fromthe FS metric by restricting to directions, in the spaceof states, generated by exponentiating allowed genera-tors G , on which our measure crucially depends. Weidentified unitary paths that map simple Gaussian ref-erence states | R ( M ) (cid:105) with no spatial correlations to ap-proximate ground states of free QFTs, generated within su (1 ,
1) subalgebras of momentum preserving quadraticgenerators and singled out the paths corresponding tominimal complexity according to our measure. Re-markably, for some instances, the evaluated complexitypresents qualitative agreement with holographic results.We could verify using our methods that cMERA cir-cuits are optimal in the C (1) norm when interpreting therenormalization scale u of cMERA as the circuit param-eter σ . In contrast, the C (2) norm allows for lower FScomplexity than that achieved by cMERA circuits by re-organizing the circuit in such a way that all the differentmomentum gates are active at every step along the circuit(see [23]A for details). The C (1) norm results show closeresemblance to the holographic results which suggests itis a better predictor of circuit complexity.We worked in momentum space and restricted the gen-erators to be quadratic. In position space our generatorsare bi-local which suggests an analogy to the 2-qubit op-erations of traditional quantum circuits. However our gates are spread in position space and it would be inter-esting to explore the implications of working with localgates. Future directions include evaluating the complex-ity for fermionic systems and studying the time evolutionof thermofield double states. Finally it would be interest-ing to understand what universal data can be extractedfrom complexity, whether complexity in QFTs can serveas an order parameter, and if it plays a role in the contextof RG-flows. ACKNOWLEDGMENTS
We would like to express first our special gratitude toR. C. Myers for numerous illuminating discussions andsuggestions, that helped shape the ideas and results ofthis paper and for sharing with us his preliminary re-sults with R. Jefferson on defining complexity in QFTsusing Nielsen’s approach. S. C. would like to express per-sonal gratitude for multiple discussions during Strings2017 which motivated the development of the ideas ofSec. 5. We are also particularly thankful to R. Jeffer-son for many illuminating discussions and for providingcomments on the supplementary material comparing thetwo works. We are also grateful to J. Eisert for pro-viding numerous comments on this work. In addition,we would like to thank G. Verdon-Akzam, J. de Boer,P. Caputa, M. Fleury, A. Franco, K. Hashimoto, Q. Hu,R. Janik, J. Jottar, T. Osborne, G. Policastro, K. Re-jzner, D. Sarkar, V. Scholz, M. Spalinski, T. Takayanagi,K. Temme, J. Teschner, G. Vidal, F. Verstraete andP. Witaszczyk for valuable comments and discussions.Research at Perimeter Institute is supported by the Gov-ernment of Canada through Industry Canada and by theProvince of Ontario through the Ministry of Research &Innovation. S.C. acknowledges additional support froman Israeli Women in Science Fellowship from the IsraeliCouncil of Higher Education. The research of M.P.H. issupported by the Alexander von Humboldt Foundationand the Federal Ministry for Education and Researchthrough the Sofja Kovalevskaja Award. M.P.H is alsograteful to Perimeter Institute, ETH and the Universityof Amsterdam for stimulating hospitality during the com-pletion of this project and to Kyoto University where thiswork was presented for the first time during the work-shop
Quantum Gravity, String Theory and Holography in April 2017. F.P. would also like to acknowledge thesupport of the Alexander von Humboldt Foundation. ∗ [email protected] † [email protected]; aei.mpg.de/GQFI;On leave from: National Centre for Nuclear Research,00-681 Warsaw, Poland. ‡ [email protected] [email protected][1] J. M. Maldacena, Int. J. Theor. Phys. , 1113 (1999),[Adv. Theor. Math. Phys.2,231(1998)], arXiv:hep-th/9711200 [hep-th].[2] M. Van Raamsdonk, Gen. Rel. Grav. , 2323 (2010),[Int. J. Mod. Phys.D19,2429(2010)], arXiv:1005.3035[hep-th].[3] S. Ryu and T. Takayanagi, Phys. Rev. Lett. , 181602(2006), arXiv:hep-th/0603001 [hep-th].[4] M. Rangamani and T. Takayanagi, (2016),arXiv:1609.01287 [hep-th].[5] V. Balasubramanian, B. D. Chowdhury, B. Czech, andJ. de Boer, JHEP , 048 (2015), arXiv:1406.5859 [hep-th].[6] L. Susskind, Fortsch. Phys. , 49 (2016),arXiv:1411.0690 [hep-th].[7] B. Freivogel, R. A. Jefferson, L. Kabir, B. Mosk, and I.-S.Yang, Phys. Rev. D91 , 086013 (2015), arXiv:1412.5175[hep-th].[8] L. Susskind, Fortsch. Phys. , 24 (2016),arXiv:1403.5695 [hep-th].[9] D. Stanford and L. Susskind, Phys. Rev. D90 , 126007(2014), arXiv:1406.2678 [hep-th].[10] M. Alishahiha, Phys. Rev.
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1) algebras. Section D considers alower bounded constant generator which induces con-stant period oscillations between the reference and targetstates and evaluate the corresponding FS length. SectionE presents a simplified derivation of the hyperbolic planemetric for a single pair of modes. Section F explores sim-ilarities and differences between this work and the workby Jefferson and Myers.[24] J. Haegeman, T. J. Osborne, H. Verschelde, andF. Verstraete, Phys. Rev. Lett. , 100402 (2013),arXiv:1102.5524 [hep-th].[25] M. Nozaki, S. Ryu, and T. Takayanagi, JHEP , 193(2012), arXiv:1208.3469 [hep-th].[26] A. Mollabashi, M. Nozaki, S. Ryu, and T. Takayanagi,JHEP , 098 (2014), arXiv:1311.6095 [hep-th].[27] M. Miyaji and T. Takayanagi, (2015),10.1093/ptep/ptv089, arXiv:1503.03542.[28] J. Molina-Vilaplana, Journal of High Energy Physics , 2 (2015).[29] P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi, andK. Watanabe, (2017), arXiv:1703.00456 [hep-th].[30] P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi, andK. Watanabe, (2017), arXiv:1706.07056 [hep-th].[31] B. Czech, (2017), arXiv:1706.00965.[32] A. R. Brown and L. Susskind, (2017), arXiv:1701.01107[hep-th].[33] K. Hashimoto, N. Iizuka, and S. Sugishita, (2017),arXiv:1707.03840 [hep-th].[34] D. A. Roberts and B. Yoshida, Journal of High EnergyPhysics , 121 (2017).[35] W. Chemissany and T. J. Osborne, JHEP , 055 (2016),arXiv:1605.07768 [hep-th].[36] I. Bengtsson and K. Zyczkowski, Geometry of Quan-tum States: An Introduction to Quantum Entanglement (Cambridge University Press, 2007).[37] Q. Hu and G. Vidal, (2017), arXiv:1703.04798 [quant-ph].[38] A. M. Perelomov, Commun. Math. Phys. , 222 (1972).[39] J. P. Provost and G. Vallee, Commun. Math. Phys. ,289 (1980).[40] A. M. Perelomov, Generalized coherent states and theirapplications (1986).[41] A. Reynolds and S. F. Ross, Class. Quant. Grav. ,105004 (2017), arXiv:1612.05439 [hep-th].[42] L. Lehner, R. C. Myers, E. Poisson, and R. D. Sorkin,Phys. Rev. D94 , 084046 (2016), arXiv:1609.00207 [hep-th].[43] A. B. Klimov and S. M. Chumakov,
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Supplemental Material
A. cMERA circuit length according to the Fubini-Study metric
Here, we evaluate the cMERA circuit length according to the proposed Fubini-Study metric. We demonstrate thatin the C (2) norm, the cMERA circuit is longer (i.e., more complex) than the minimal circuit described in the maintext whereas in the C (1) norm its circuit length coincides with the corresponding minimal complexity. We reviewbelow the needed ingredients of cMERA and refer the reader to Ref. [24–26] for extra details.cMERA is a unitary map taking the Gaussian reference state | R ( M ) (cid:105) defined by Eqs. (9) and (10), which is aproduct state with no spatial correlations, to the approximate ground state (cid:12)(cid:12) m (Λ) (cid:11) given by Eqs. (9) and (12). Onecan view | R ( M ) (cid:105) as the ground state of an ultra-local Hamiltonian H M = (cid:90) d d x : (cid:8) π + M φ (cid:9) / , (27)where the ( ∂ (cid:126)x φ ) term is omitted and the mass M is kept arbitrary. Despite being a product state in real space, | R ( M ) (cid:105) contains pairwise-entanglement in momentum space between momentum sectors (cid:126)k and − (cid:126)k .The cMERA circuit alters correlations between the (cid:126)k and − (cid:126)k modes from those corresponding to a constant andset by M to the ones governed by α k of Eq. (12) in a scale (i.e., u ) dependent manner as follows (cid:12)(cid:12)(cid:12) m (Λ) (cid:69) = P e − i (cid:82) −∞ du (cid:82) k ≤ Λ eu d d k K ( (cid:126)k ) χ ( u ) | R ( M ) (cid:105) . (28) The two mode squeezing operator K ( (cid:126)k ) defined in Eq. (13) (dis)entangles the (cid:126)k and − (cid:126)k modes along the circuit.Energy minimization with respect to the free Hamiltonian H m as well as continuity of the transformation at the cutoffscale implies χ ( u ) = 12 e u e u + m / Λ and M = ω Λ ≡ (cid:112) Λ + m . (29)Comparing with Eq. (1) we see that the renormalization-group scale parameter u running from the infrared at u = −∞ to the ultraviolet at u = 0 plays the role of σ ∈ [ s i , s f ] and (cid:82) k ≤ Λ e u d d (cid:126)kK ( (cid:126)k ) χ ( u ) / G ( s ).We also note that for every value of u , the action of the circuit on momenta larger than Λ e u is suppressed. Since theoperators K ( (cid:126)k ) all commute, we may integrate over u in Eq. (28) to obtain (cid:12)(cid:12)(cid:12) m (Λ) (cid:69) = e − i (cid:82) k ≤ Λ d d k K ( (cid:126)k ) log (cid:113) ω Λ ωk | R ( ω Λ ) (cid:105) . (30)The above expression is the cMERA realization of Eq. (14).Evaluating the Fubini-Study metric distance according to Eq. (3) for the cMERA circuit yields (cid:96) (2)cMERA = (cid:90) −∞ du χ ( u ) (cid:115) Vol2 (cid:90) k< Λ e u d d k. (31)We can evaluate this expression analytically and obtainΓ (cid:0) d + 1 (cid:1) ( (cid:96) (2)cMERA ) π d/ Vol Λ d = Λ F (cid:16) , d +44 ; d +84 ; − Λ m (cid:17) d + 4) m , (32)where for the massless theory, the expression simplifiesΓ (cid:0) d + 1 (cid:1) π d/ Vol Λ d (cid:16) (cid:96) (2)cMERA (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) m =0 = 14 d . (33)This is equivalent to the complexity C (2) of Eq. (19). Comparing to the relevant entry in Eq. (35) (see [23]B below)we conclude that the path generated by cMERA is a √ K ( (cid:126)k ), studiedin the main text. Fig. 2 highlights the differences between these two circuits.If we now disallow the different elementary generators in the cMERA circuit to act simultaneously and considerthe equivalent of the C (1) norm, we end up with (cid:96) (1)cMERA = Vol2 (cid:90) −∞ du χ ( u ) (cid:90) k< Λ e u d d k, (34)which yields precisely the same circuit length as the one obtained in Eq. (20) (see also Eq. (35) in [23]B below). Thisis due to the fact that the C (1) norm is invariant under independent reparametrizations of the circuits associated withthe different pairs of momenta. B. Properties of the ground state complexities C ( n ) Here we analyze the properties of our complexity proposals (19)-(21). In particular we focus on the structureof divergences. As we saw in the main text, the C (1) norm results carry similarities to the results found using theholographic complexity proposals.The leading divergence in the complexity measures C ( n ) is proportional to Vol /n Λ d/n when M = Λ, and toVol /n Λ d/n log( M/ Λ) when M and Λ are independent. The structure of subleading divergences depends on theinterplay between m , M and Λ and we will analyze it in more detail for the n = 1 and n = 2 cases. For free CFTs( m = 0) we obtain the following exact expressions Γ (cid:0) d + 1 (cid:1) π d Λ d Vol C (1) CFT = (cid:12)(cid:12)(cid:12)(cid:12) log (cid:113) M Λ (cid:12)(cid:12)(cid:12)(cid:12) + M d d Λ d − d , M < Λ (cid:12)(cid:12)(cid:12)(cid:12) log (cid:113) M Λ (cid:12)(cid:12)(cid:12)(cid:12) + d , M ≥ ΛΓ (cid:0) d + 1 (cid:1) π d Λ d Vol (cid:16) C (2) CFT (cid:17) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log (cid:114) M Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + log (cid:113) M Λ d + 18 d . (35) If we denote M = e γ M Λ, we see that the first terms in Eq. (35) are indifferent to the sign of γ M . The remainder issmaller for a given value of | γ M | when γ M is negative and this leads to a smaller complexity. To understand this, notethat for M <
Λ some (cid:126)k -modes already begin with the right correlations with their ( − (cid:126)k )-counterparts. So our minimalunitary transformation can act much more mildly in the vicinity of this locus in momentum space which leads to areduced complexity. For the CFT, we see that the subleading divergences except for Vol Λ d and a constant vanish.This is similar to what happens in the CA proposal when curvature invariants over the relevant time slice vanish.One can also consider the structure of divergences when the reference state is the CFT ( m = 0) vacuum and thetarget state is the vacuum of a free massive QFT. This means that instead of considering | R ( M ) (cid:105) as our referencestate, we consider the vacuum of a free CFT | (cid:105) ( m = 0). The transformation between | (cid:105) and (cid:12)(cid:12) m (Λ) (cid:11) takes the form (cid:12)(cid:12)(cid:12) m (Λ) (cid:69) = e − i (cid:82) k ≤ Λ d d k ¯ r k K ( (cid:126)k ) | (cid:105) ; ¯ r k ≡ log (cid:115) | (cid:126)k | ω k . (36)For the physically interesting case Λ (cid:29) m , we see that the parameter of the transformation, ¯ r k , at large momentaapproaches 0 in accord with the physical intuition that mass becomes irrelevant in the UV. However, this does notmean that the complexity of the transformation does not diverge with the cut-off due to the growth of the numberof momentum modes in the UV. Adapting the n = 1 instance of our complexity proposal given by Eq. (20) to thepresent case, i.e., replacing the r k ≡ log (cid:112) M/ω k by ¯ r k ≡ log (cid:113) | (cid:126)k | /ω k , we obtain that the complexity is now finitefor d = 1, diverges logarithmically with the cut-off, ( m Vol) log Λ m , for d = 2 and in higher number of dimensionsbehaves as ( m d Vol) (cid:0) Λ m (cid:1) d − , for large values of the cutoff. This behavior is subleading with respect to Eq. (35) validfor starting with the unentangled reference state. What we thus see is that taking the ground state of another QFT asa reference state significantly lowers the complexity of the transformation needed to obtain (cid:12)(cid:12) m (Λ) (cid:11) , cf. Eq. (35), sincesome of the correlations have already been built, but still leads to UV-divergent results in space dimensions d > C. su (1 , manifold – metric and geodesics Here, we provide the detailed derivation of the results presented in the main text for the ground state complexityusing su (1 ,
1) generators. In particular we study the general form of the manifold of states generated by the elementsof the su (1 ,
1) algebras of Eq. (24), and the metric and geodesics induced on this manifold by the FS metric.A path in the manifold of states will correspond to a path ordered exponential of the various su (1 ,
1) elements. Forevery point in the path such an exponent can always be regrouped as | Ψ( σ ) (cid:105) = U ( σ ) | R ( M ) (cid:105) ; U ( σ ) ≡ e (cid:82) Λ d d k g ( (cid:126)k,σ ) , (37)where g ( (cid:126)k, σ ) = α + ( (cid:126)k, σ ) K + ( (cid:126)k )+ α − ( (cid:126)k, σ ) K − ( (cid:126)k ) + ω ( (cid:126)k, σ ) K ( (cid:126)k ) , (38)and σ is the path parameter from Eq. (1). This is due to the fact that the su (1 ,
1) generators form a closed algebra. Thecoefficients α ± ( (cid:126)k, σ ) and ω ( (cid:126)k, σ ) are not simply related to the original coefficients of the various generators in the pathordered trajectory (1) because of the noncommutative nature of generators. Unitarity implies that α ∗ + ( σ ) = − α − ( σ )and ω ∗ ( σ ) = − ω ( σ ).The unitary (38) can then be decomposed as follows (see e.g., [43] Appendix 11.3.3): U ( σ ) = e (cid:82) Λ d d kγ + ( (cid:126)k,σ ) K + ( (cid:126)k ) × e (cid:82) Λ d d k log γ ( (cid:126)k,σ ) K ( (cid:126)k ) e (cid:82) Λ d d kγ − ( (cid:126)k,σ ) K − ( (cid:126)k ) , (39)where the mapping between the coefficients is γ ± = 2 α ± sinh Ξ2 Ξ cosh Ξ − ω sinh Ξ ,γ = (cid:16) cosh Ξ − ω
2Ξ sinh Ξ (cid:17) − , Ξ ≡ ω − α + α − . (40)Note that unitarity implies that | γ + | <
1, which fits nicely into the geometric picture describing the γ + manifold asa Poincar´e unit disk. We can now use the identities K − | R ( M ) (cid:105) = 0; K | R ( M ) (cid:105) = δ ( d ) (0)4 | R ( M ) (cid:105) . (41)to get rid of the information contained in the path parameters which only change our state by an overall phase. Thiscan be done by showing that the state in Eq. (39) can be recast as | Ψ( σ ) (cid:105) = N e (cid:82) Λ d d kγ + ( (cid:126)k,σ ) K + ( (cid:126)k ) | R ( M ) (cid:105) , N = e δ ( d )(0)4 (cid:82) Λ d d k log γ ( (cid:126)k,σ ) , (42)where N is a complex constant containing information about the overall normalization and phase. The identity | γ | = 1 − | γ + | allows to demonstrate that up to an unphysical overall phase, the state depends only on γ + .One can then use the following set of adjoint conjugation identities to evaluate the Fubini-Study line element alongthe path (42) e (cid:82) Λ d d k (cid:48) λ ( (cid:126)k (cid:48) ,σ ) K + ( (cid:126)k (cid:48) ) K ( (cid:126)k ) e − (cid:82) Λ d d k (cid:48) λ ( (cid:126)k (cid:48) ,σ ) K + ( (cid:126)k (cid:48) ) = K ( (cid:126)k ) − K + ( (cid:126)k ) λ ( (cid:126)k, σ ) ,e (cid:82) Λ d d k (cid:48) λ ( (cid:126)k (cid:48) ,σ ) K − ( (cid:126)k (cid:48) ) K + ( (cid:126)k ) e − (cid:82) Λ d d k (cid:48) λ ( (cid:126)k (cid:48) ,σ ) K − ( (cid:126)k (cid:48) ) = K + ( (cid:126)k ) + 2 λ ( (cid:126)k, σ ) K ( (cid:126)k ) + λ ( (cid:126)k, σ ) K − ( (cid:126)k ) . (43)This leads to a remarkably simple form, resulting in the following expression for the complexity C = min γ + ( (cid:126)k,σ ) (cid:90) s f s i dσ (cid:118)(cid:117)(cid:117)(cid:116) Vol2 (cid:90) Λ d d (cid:126)k γ (cid:48) + ( (cid:126)k, σ ) γ (cid:48)∗ + ( (cid:126)k, σ )(1 − | γ + ( (cid:126)k, σ ) | ) , (44)where the prime denotes differentiation with respect to the path parameter σ . We identify the line element of amanifold which consists of a direct product of hyperbolic unit discs, one for each pair of momenta with γ + for thedifferent momenta playing the role of the complex coordinates on the discs. The Poincar´e unit disk is known to be themanifold associated with the coset SU (1 , /U (1) (see e.g., Refs. [38–40] and Ref. [28] in the context of cMERA). The su (1 ,
1) algebra whose generators are listed in Eq. (24) generates isometries on the disk, with K generating rotationsaround the origin and K and K generating pure translations along the imaginary and real axes respectively. Howevernote that the metric (26) couples the different speeds associated to the paths for different values of the momentum.As explained in the main text, affinely parametrized geodesic paths on the product space correspond to affinelyparametrized geodesics in each one of the spaces, where the relative speeds for the paths of the different momenta aredictated by the requirement that we reproduce the target state γ + = tanh(2 r k ) at the end of the path ( σ = s f ). Thegeodesics on the Poincar´e disk are well known and we identify the one connecting our reference state γ + = 0 to thetarget state as the solid line lying along the diameter in Fig. 1.The paths generated by K ( (cid:126)k ) in Eqs. (1), (15) and (18) and B ( (cid:126)k, M ) of Eq. (48) (see [23]D below) can be decomposedaccording to (37)-(40) as follows K ( (cid:126)k ) : α ± ( (cid:126)k, σ ) = ± r k σ, ω ( (cid:126)k, σ ) = 0 ,γ + ( (cid:126)k, σ ) = tanh(2 r k σ ) , (45)and B ( (cid:126)k, M ) : α ± ( (cid:126)k, σ ) = iπ r k ) σ ,ω ( (cid:126)k, σ ) = − iπ cosh(2 r k ) σ ,γ + ( (cid:126)k, σ ) = i sinh(2 r k ) sin( πσ )cos( πσ ) + i cosh(2 r k ) sin( σ π ) , (46)where here we have taken σ ∈ [0 , K ( (cid:126)k ) is an affinelyparametrized geodesic according to Eq. (44) while the one generated by B ( (cid:126)k, M ) is not a geodesic as it does notsatisfy the Euler Lagrange equations derived from Eq. (44).For completeness we also specify the path corresponding to the cMERA circuit (see [23]A). After integrating thecircuit (here it is possible since the generators commute) and decomposing according to Eq. (40) we obtain: γ + ( (cid:126)k, σ ) = tanh (cid:20)
14 log (cid:20) m + σ Λ k + m (cid:21)(cid:21) θ (cid:18) σ − | k | Λ (cid:19) , (47)where we have redefined the cMERA parameter σ ≡ e u . Fig. 2 is a comparison of the minimal path (45) and thecMERA path (47). It presents γ + as a function of (cid:126)k for different values of σ and demonstrates in this way the progressof the circuit. D. Another constant generator
What might be viewed as a deficiency of the transformation (14) adopted from cMERA is that its generators K ( (cid:126)k ),see Eq. (13), do not have a spectrum bounded from below. As such, they cannot be naturally interpreted as Hamil-tonians of some fiducial physical system. Here, we present an alternative constant generator B ( (cid:126)k, M ) which inducesconstant period oscillations between the ground state and the reference state and does admit an interpretation as alower bounded Hamiltonian and evaluate the associated path length.The approximate ground state | m (Λ) (cid:105) of Eq. (12) can also be reached, up to an overall phase, starting from thereference state | R ( M ) (cid:105) of Eq. (10) by repeatedly applying the operator B ( (cid:126)k, M ) = − r k )[ K + + K − ] + 4 cosh(2 r k ) K . (48) Fig. 1 contains a solid semicircle which illustrates the generated path. Indeed, using the relevant decompositionformulas Eqs. (37)-(40) (see e.g., [43] Appendix 11.3.3), one can establish that (cf. (14)) (cid:12)(cid:12)(cid:12) m (Λ) (cid:69) (cid:39) e − i π (cid:82) k ≤ Λ d d k B ( (cid:126)k, M ) | R ( M ) (cid:105) , (49) FIG. 2. Graphical description of the minimal circuit (45) (left) and the cMERA circuit (47) (right). The plots present γ + as afunction of | (cid:126)k | / Λ for different values of σ represented by the different colored contours. We see that while the cMERA circuitonly acts on momenta | (cid:126)k | / Λ < σ , the minimal circuit alters γ + for all the different momenta at every step of the circuit. Inthis plot we have chosen m/ Λ = 0 . where (cid:39) indicates that the states are equal up to an irrelevant global phase. Another way to obtain this transformationis to derive it using the properties of the Wigner distribution. It is interesting to note that in contrast to Eq. (14) thegenerators in Eq. (48) explicitly depend on both the reference and the target states. However, the number of timeseach of them is applied in Eq. (49) is fixed and equal to π/
4. Generalizing Eq. (21) to the present case, we obtain forthe different L n norms C ( n ) = π n (cid:115) Vol2 (cid:90) k ≤ Λ d d k (cid:12)(cid:12)(cid:12)(cid:112) M/ω k − (cid:112) ω k /M (cid:12)(cid:12)(cid:12) n , (50)which reads at leading order in Λ:2 n +1 Γ (cid:0) d + 1 (cid:1) (cid:0) C ( n ) (cid:1) n π d + n Λ d Vol= (cid:40) d d + n (cid:0) Λ M (cid:1) n/ + . . . , M (cid:28) Λ , d d − n (cid:0) M Λ (cid:1) n/ + . . . , M (cid:29) Λ , d > n. (51)One can clearly see that the leading divergences at large UV cut-off Λ got now altered from Λ dn to Λ dn ± dependingon the reference state scale M . Notice also that for M (cid:28) Λ ( M (cid:29) Λ), the number of gates K ( (cid:126)k ) needed for thetransformation from | R ( M ) (cid:105) to (cid:12)(cid:12) m (Λ) (cid:11) is smaller than the number of needed B ( M, (cid:126)k ) gates. This is in line with ourpredictions in the main text since B ( (cid:126)k, M ) deviates from the geodesic path generated by K ( (cid:126)k ).To make this statement more precise, let us study the length of the minimal path constructed with the generator B ( (cid:126)k, M ) included inside the larger manifold spanned by the su (1 ,
1) generators of (24) | Ψ( σ ) (cid:105) ≡ e − i σ π (cid:82) k ≤ Λ d d k B ( (cid:126)k, M ) | R ( M ) (cid:105) . (52)This path can be shown to be minimal if only B ( (cid:126)k, M ) gates are allowed. According to the decomposition in Eqs. (37)-(40) we can show that this path corresponds to (see [23]C) γ + ( (cid:126)k, σ ) = i sinh(2 r k ) sin( σ π )cos( π σ ) + i cosh(2 r k ) sin( σ π ) . (53)Checking the Euler Lagrange equations explicitly for this path one concludes that it is not a geodesic. The pathcorresponding to B ( (cid:126)k, M ) is represented in Fig. 1 by a solid semicircle. It would be interesting to explore if theoperators B ( M, (cid:126)k ) can lead to an alternative construction for a cMERA circuit.
E. Fubini-Study metric derivation for a single su (1 , Here, we present an alternative, simpler, derivation of Eq. (44) for a single pair of momenta. For this purpose, inthe commutation relations of creation and annihilation operators, Dirac delta functions will be replaced by Kroneckerdeltas and integrals will be suppressed. We will restore them at the end of the calculation. We will consider states ofthe form | Ψ( σ ) (cid:105) = N e γ + K + | , (cid:105) , (54)where the state | , (cid:105) contains no particle excitation in the (cid:126)k and − (cid:126)k modes according to the annihilation operators b (cid:126)k . In fact the state | R ( M ) (cid:105) is a product of such states | , (cid:105) in the different momentum sectors. We can rewrite thestate | Ψ( σ ) (cid:105) up to an overall phase as follows: | Ψ( σ ) (cid:105) ≡ (cid:112) − | γ + | ∞ (cid:88) n =0 ( γ + ) n | n, n (cid:105) , (55)where we have fixed the constant by normalization and where | n, n (cid:105) is the normalized state containing n excitationswith momentum (cid:126)k and n with momentum − (cid:126)k . Small changes in γ + will result in the following change in the state | Ψ( σ ) (cid:105) | δ Ψ( σ ) (cid:105) = (cid:32) γ + δγ ∗ + + γ ∗ + δγ + (cid:112) − | γ + | ∞ (cid:88) n =0 ( γ + ) n + (cid:112) − | γ + | ∞ (cid:88) n =0 n ( γ + ) n − δγ + (cid:33) | n, n (cid:105) . (56)Evaluating the Fubini-Study line element ds F S = (cid:104) δψ | δψ (cid:105) − (cid:104) δψ | ψ (cid:105)(cid:104) ψ | δψ (cid:105) , (57)results in ds F S = | δγ + | (1 − | γ + | ) . (58)Restoring the continuum structure, including the momentum integrals and delta functions one reaches Eq. (44). Notethat Vol / F. Comparison with Ref. [22]
Ref. [22], which appeared simultaneously with this article has some overlap with our results. There, the authorsuse a lattice setup to study the complexity of the ground state of a free scalar field theory. An important differencebetween our approach and the one used in Ref. [22] is the nature of the distance metric minimized to obtain complexity.While in our approach, we minimize the Fubini-Study distance over states restricted to the manifold carved by allowedgenerators, in Ref. [22] the distance measure is over unitaries in the spirit of [17]. In addition, the set of gates weconsider here is different from the one in Ref. [22]. However, the two sets contain the gates K ( (cid:126)k ) of Eq. (13), whichturns out to be the one which in both approaches is used to construct the minimal length circuit. We compare belowthe various components of the construction, i.e., the gates, the distance function, the regularization method and thefinal result. Distance measures:
We use the Fubini-Study metric over states to measure the length of our path while Ref. [22]considers metrics in Finsler geometry over unitary circuits inspired by Nielsen’s approach, see Ref. [17]. Being basedon the projective nature of the Hilbert space, the FS metric does not account for generators acting trivially on thestate, namely generators which only modify the state up to an overall phase, while this is not in general the case fora unitary based approach. For the gates K ( (cid:126)k ) of Eq. (13) the two approaches give the same result up to an overallnumerical factor. It would be interesting however to explore to what extent they yield similar notions of complexitywhen studying complexity of pure Gaussian states with a larger set of gates. For instance it would be interesting tounderstand if the geometry remains hyperbolic when considering the full set of su (1 ,
1) generators using the approachof Ref. [22]. For this it is important to understand how to extend the Nielsen construction and assign a “weight” togates constructed from the different generators of su (1 , Generator sets:
Both works consider quadratic generators in order to build the allowed unitary transformationon the state. Gates constructed from these generators transform Gaussian states among themselves which allows torestrict the analysis to Gaussian states. Ref. [22] minimizes over all gates constructed by exponentiating bilineargenerators of the form φ ( x ) π ( y ). In fact, the authors do this on the lattice for generators G ab = x a p b + p b x a where a and b enumerate the lattice sites which form a GL ( N, IR) group structure for a one dimensional lattice with N sites.In the absence of penalty factors (Penalty factors correspond to modifying the cost function associated to a generatorin order to favor/penalize certain generators; this could for example be used to penalize long range generators withrespect to local ones), Ref. [22] finds that an optimal circuit admits a normal mode decomposition and thus requiresonly momentum preserving generators of the form φ ( k ) π ( − k ) + π ( k ) φ ( − k ) for its construction. On the lattice theseare given by G k = ˜ x k ˜ p − k + ˜ p k ˜ x − k , where ˜ x k ≡ √ N (cid:80) N − a =0 exp (cid:0) − πik aN (cid:1) x a . Inspired by cMERA, our starting pointwas to consider (a larger set of) momentum preserving generators. In this sense, our work is complementary to thatof Ref. [22]. We minimized over gates constructed from the following set of generators G ,k = φ ( (cid:126)k ) π ( − (cid:126)k ) + π ( (cid:126)k ) φ ( − (cid:126)k ); G ,k = φ ( (cid:126)k ) φ ( − (cid:126)k ); G ,k = π ( (cid:126)k ) π ( − (cid:126)k );which form su (1 ,
1) subalgebras of quadratic generators conserving momentum. These are bi-local in real space andcontain the generator G ,k (equivalently, K ( (cid:126)k ) of Eq. (13)) of cMERA circuits. In both works it was found that thepreferred motion is in the direction of G ,k which lies in the intersection of the two sets of generators. It would beinteresting to minimize the complexity using an algebra of generators which accounts for both our gates and those ofRef. [22] and check whether the minimal circuit is still generated by G ,k . Regulating divergences:
Our method of regulating divergences was that our circuit only reproduced the groundstate faithfully up to the cutoff momentum. This allowed us to obtain finite results. Ref. [22] uses a lattice regular-ization in order to obtain finite results and the regulator is set by the lattice spacing.
Final result:
The finite result is very similar, the main difference (except for an overall factor) is due to thedifferent regularization schemes. Recall that we obtained in Eq. (19) C (2) = 12 (cid:115) Vol2 (cid:90) k ≤ Λ d d k (cid:16) log ω k M (cid:17) . (59)This is to be compared to the result in Eqs. (4.32)-(4.33) of Ref. [22] which reads C (2) = 12 (cid:118)(cid:117)(cid:117)(cid:116) N − (cid:88) k i =0 (cid:18) log ˜ ω (cid:126)k ω (cid:19) , (60)with the normal mode frequencies given by ˜ ω (cid:126)k = m + 4 δ d − (cid:88) i =1 sin πk i N . (61)We see that δ the lattice spacing in Ref. [22] and Λ the momentum cutoff in our work play a similar role. We also seethat the reference state scale M is essentially the same as the reference state parameter ω in Ref. [22]. The leadingdivergence in C (1)(1)