Lattice Gauge Theory for a Quantum Computer
LLattice Gauge Theory for a Quantum Computer
Richard C. Brower ∗ Boston University, Boston, MA 02215, USA email: [email protected]
David Berenstein
University of California Santa Barbara, CA 93106, USA email: [email protected]
Hiroki Kawai
Boston University, Boston, MA 02215, USA email: [email protected]
The quantum link [1] Hamiltonian was introduced two decades ago as an alternative to Wilson’sEuclidean lattice QCD with gauge fields represented by bi-linear fermion/anti-fermion opera-tors. When generalized this new microscopic representation of lattice field theories is referred as
D-theory [2]. Recast as a Hamiltonian in Minkowski space for real time evolution, D-theoryleads naturally to quantum Qubit algorithms. Here to explore digital quantum computing forgauge theories, the simplest example of U(1) compact QED on triangular lattice is defined andgauge invariant kernels for the Suzuki-Trotter expansions are expressed as Qubit circuits capa-ble of being tested on the IBM-Q and other existing Noisy Intermediate Scale Quantum (NISQ)hardware. This is a modest step in exploring the quantum complexity of D-theory to guide futureapplications to high energy physics and condensed matter quantum field theories. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] F e b uantum Computing LGT Richard C. Brower
1. Introduction
Quantum Chromodynamics (QCD) remains a singularly difficult computational problem. Inspite of the success of Wilson’s Euclidean lattice QCD, there remains the notorious sign problem making predictions of real-time dynamics impossible, as well as severely limiting the determinationof parton distribution functions and thermodynamics at non-zero chemical potentials needed byHEP experiments. On quantum computers, these sign problems are solved in principle. One definesa Hamiltonian operator for unitary evolution in continuous time. Two discretizations are requiredfor any finite Qubit algorithm: i) Placing the Kogut Susskind Hamiltonian [3] on a finite volumelattice and replacing Euclidean time ( it ) by Minkowski time ( t ) to guarantee unitary real timequantum evolution. ii.) The less familiar step of replacing field variables by a finite set of Qubitsper lattice cell. The quantum link representation of Brower, Chandrasekar and Wiese [1] or
D-theory accomplishes this second, field space quantization .Since the introduction of D-Theory in the context of classical computing algorithms, it hascontinued to be developed for quantum computing of growing number of interesting theories re-viewed by Uwe Wiese [4]. This talk will briefly present D-theory for QCD and then consider asimple prototype example of U ( ) gauge theory on a 2d triangular lattice, which is capable oftesting small kernel consistent with the hardware constraints in the NISQ era.
2. Quantum link D-theory for QCD
In going from the Euclidean D-theory to the Minkowski Hamiltonian, what is required, roughlyspeaking, is swapping the extra-dimension fermion with a gauge-fixed temporal realtime axis. Theresult is a wide range of spin and gauge field theories represented in the terminology of Bravyi-Kitaev [5] as
Fermionic Quantum Computing algorithms. For example, for QCD the Hamiltonianis ˆ H = g ∑ x , µ Tr [ ˆ E L ( x , µ ) + ˆ E R ( x , µ )] − g ∑ x , µ , ν Tr [ ˆ U µν ( x )] + Quarks (2.1)The key to the construction is a faithful preservation of the symplectic gauge algebra by introducingon each link ( (cid:104) x , x + µ (cid:105) ) operators,ˆ E L ( x , µ ) = a i † (cid:104) x , x + µ (cid:105) a j (cid:104) x , x + µ (cid:105) , ˆ E R ( x , µ ) = b i † (cid:104) x , x + µ (cid:105) b j (cid:104) x , x + µ (cid:105) , ˆ U i j † ( x , µ ) = ˆ a i † (cid:104) x , x + µ (cid:105) b j (cid:104) x , x + µ (cid:105) , U i j ( x , µ ) = b i † (cid:104) x , x + µ (cid:105) a j (cid:104) x , x + µ (cid:105) , (2.2)expressed as bilinear of 6 color triplet and 6 anti-triplet fermionic operators with i , j = , ,
3. Thefull algebra of E (cid:48) s and U (cid:48) s are generators of U ( ) . For QCD the unitary evolution, exp [ ± i ˆ H ] ,in the Hilbert space was referred to as the QCD Abacus [6] illustrated in Fig. 1. The magneticterm acts on 3 color fermionic bits like beads on wires, changing the color flux on each link.The only deformation of the algebra is that on each link (cid:104) x , x + µ (cid:105) , the forward ( ˆ U ( x , µ ) ) andbackward ( ˆ U † ( x , µ ) ) gauge matrices no longer commute ( [ ˆ U , ˆ U † ] (cid:54) =
0) and they are paired withindependent left E L ( x , µ ) and right gauge E R ( x , µ ) . These no longer obey the identity E R = U † E L U .However since all link fields on two different links still commute, this algebraic deformation is anirrelevant UV cut-off effect typical of all lattice field theories. Extended Wilson paths at long1 uantum Computing LGT Richard C. Brower
Figure 1: QCD quantum link dynamics: On the left, the Hamiltonian induces a hopping of linkfermion to rotate various colors around a plaquette. On the right, by adding an extra determinantterm the gauge group is broken from U(3) to SU(3) by translating a baryon-like object on the link.distance will have only O ( a ) commutators. D-theory conjectures [2] that this novel microstructureat the cut-off under proper implementation can provide a fundamental alternative to the WilsonKogut-Suskind lattice gauge theories in the same universality class in the continuum limit. Theprecise conditions required to support this conjecture of course pose challenging theoretical andcomputational problems.
3. Quantum Links for U(1) gauge Theory
To test D-theory operation on existing Quantum Computing hardware, we now consider thesimplest non-trivial confining gauge theory: the 2+1 Abelian U(1) theory on a triangular lattice.The compact U ( ) group manifold itself is a circle S , and the gauge algebra is realized as E L = − E R = − i ∂ θ , U = exp ( i θ ) . (3.1)The single link compact manifold, L ( S ) , is infinite dimensional but it is a discrete set of Fouriermodes in the flux basis, | q (cid:105) , enumerated by integer or half integer wave number q . In this Hilbertspace we have that the action of E L , U is given by E L | q (cid:105) = q | q (cid:105) , U | q (cid:105) = | q + (cid:105) . It is easy tocheck that [ E L , U ] = U and that this representation is irreducible. When we truncate the chargebasis, U and U † needs to annihilate the largest and smallest charge states respectively.D-theory replaces this manifold by the simplest truncation with a fermion bilinear representa-tion, ˆ U = b † a , on a two Qubit state. However since Fermion number, ˆ N = a † a + b † b , is conservedon each link, we may project onto the two states with ˆ N = a † a + b † b = U = σ + , ˆ U † = σ − , ˆ E L = − ˆ E R = σ z , (3.2)in a single Qubit space. Note in this Abelian example both left and right gauge generators at theends of each link are the same up to a sign. Relative to the full U ( ) manifold, now the flux isrestricted to ± / [ E L , U ] = U , [ E L , U † ] = − U † , with E = L z and U ∝ L + , so that the electric field squared is not just a constant like inthe spin 1 / uantum Computing LGT Richard C. Brower angular momentum. This way we write the E , U in terms of sums of spin 1 /
4. Hamiltonian for 2 +1 U ( ) gauge theory In D-theory higher flux as we approach the continuum is built up by coherent states eitherin an extra dimension and/or at larger spatial distances . The dynamics of this coherence is anexample of the typical physical mechanism requirement at a second order phase point in order totake the continuum limit. If achieved, universality requires a similar phase boundary familiar in theclassical example of the Ising vs the φ Hamiltonian dynamics.We consider the 2 + 1 d U ( ) gauge theory Hamiltonian on a 2d spatial lattice. Beginningwith a single plaquette with ± / The Hamiltonian for a single triangular plaquette stacked up in an extra dimension enumeratedby s with periodic boundary conditions isˆ H = ∑ s (cid:20) g ∑ j = ( σ zj , s + σ zj , s + ) + α g ∑ j = ( σ + j , s σ − j , s + + σ − j , s σ + j , s + ) − g ( σ + , s σ + , s σ + , s + σ − , s σ − , s σ − , s ) (cid:21) . (4.1)The lattice structure of this spin-basis Hamiltonian is as in Fig. 2 with triangle links, each of whichcan be represented as one Qubit spatial link, stacked in the direction of the extra dimension. The (a) Electric (b) XY Coupling (c) Plaquette Figure 2: Visualization of each term of the Hamiltonian of the single triangle plaquette model withtwo layers.first term corresponds to the electric term ˆ E of (2.1), and third term to the plaquette term of (2.1)represents the magnetic ˆ B part of the energy. The positive and negative flux is created by σ + and σ − respectively. Notice that on a single link ˆ E is constant, so we need at least one nearestneighbor to separate states based on differences of total electric flux.3 uantum Computing LGT Richard C. Brower
The second term can be rewritten as a sum of XX and YY interactions: σ + j , s σ − j , s + + σ − j , s σ + j , s + = ( σ xj , s σ xj , s + + σ yj , s σ yj , s + ) . It is an anti-ferromagnetic nearest neighbor XY spin 1/2 chain for eachlink. Together with the first term, it composes a 1D spin-1/2 anisotropic XXZ model. There are nogauge variables on links into the extra direction so that the gauge theory still has only the proper2d spatial gauge rotations at each site. For example for a single (cid:52) ( , , ) plaquette model thegenerators are given by E i j = ∑ s ( σ zi , s − σ zj , s ) (4.2)where ( i , j ) = ( , ) , ( , ) , ( , ) . Given the periodic conditions and the commutation relations [ σ zj , σ ± i ] = ± δ i j σ ± , we may easily see that all three gauge operators satisfy [ E i j , ˆ H P ] =
0. Theinteger flux values for E i j at each site represent exact conserved Gauss’ law sectors.Now, we can expand this model to finite volume (areas in the two spatial dimensions) withmultiple triangles for very small systems with periodic boundary conditions giving a 2d torus. Forexample in Fig. 3 is a lattice of eight plaquettes triangulating a space for a 2 × ×
2. The triangle plaquette tiling forms a bipartite dual hexagonallattice in Fig. 3(right), where the Pauli terms of the Hamiltonian associated with one fixed colorcommute with each other. This leads naturally to a term Trotter expansion where one can parallelizethe evolution on a quantum computing algorithm with gauge invariant kernels on plaquettes. Theseare implemented next as quantum circuits for single 2 layer single plaquette model.Figure 3: Example of triangulations of two dimensional spaces for 2 × × In order to simulate the time evolution e − i ˆ Ht | ψ (cid:105) of the plaquette model on a digital quantumcomputer, the interactions between the links are represented as entanglements between the cor-responding Qubits. Each of the exponentials of ˆ E , coupling, and ˆ B terms can be representedas a quantum circuit as Fig. 4a, 4b, and 4c respectively. The coupling constant information andthe evolution time t in encoded in the rotation angle of the RZ and RX gates. The X X and YY terms commute with each other and together are gauge invariant. The other two are gauge invarianton their own. The separate non-commuting terms in ˆ H can be approximated via the Trotter de-composition exactly maintaining gauge invariance and unitarity. Introducing small time intervals4 uantum Computing LGT Richard C. Brower(a) Electric ( e − i θσ z ⊗ σ z ) (b) XY term ( e − i θ ( σ + ⊗ σ − + h . c . ) )(c) Plaquette (cid:52) (d) Overall step Figure 4: The quantum circuit representing each term of the Hamiltonian of the single plaquettemodel (a)-(c), and the circuit overview for the entire one single Trotter step (d). ∆ t = t / n , the Trotter evolution for n steps is e − i ˆ Ht | ψ (cid:105) = (cid:18) e − i ˆ H E ∆ t e − i α g ∑ j = ( σ + j , s σ − j , s + + σ − j , s σ + j , s + ) ∆ t e − i ˆ H B ∆ t (cid:19) n | ψ (cid:105) . (4.3)For the single plaquette model, one Trotter step is illustrated in Fig. 4d. Since different Qubit Paulimatrices commute with each other, the multiple terms for ˆ E , ˆ B , and the inter layer XY couplingmay be implemented in parallel on a quantum circuit. Fig. 5 demonstrates an example of the timeevolution for the single plaquette model Hamiltonian with Trotter decomposition with two differenttime intervals ∆ t = . ∆ t = . A ( t ) = (cid:104) φ | e − i ˆ Ht | φ (cid:105) for a single triangle model with g = .
75 and α = t = t =
10. The state | φ (cid:105) is initialized in the | + (cid:105) state for all Qubits. The bluedotted line represents the exact values of this inner product, the red and green points represent theTrotter result with a time interval ∆ t = . ∆ t = . uantum Computing LGT Richard C. Brower
5. Future Directions
We have discussed a possible implementation of the U ( ) gauge theory in 2 + d = d = d +
1. For the Abelian case, a plausibleroute to the coherent formation of large flux at physical scales is less well understood. Perhaps apromising avenue of investigation of this two state triangular lattice Abelian quantum link modelis to look at its duality to a spin model on a hexagonal lattice following arguments similar to theclassic duality of the 3d of Ising model and Wegnerâ ˘A ´Zs 3d Z gauge theories on a square lattice.Finally there are many interesting small quantum lattice prototypes. For example in additionto toroidal geometries with periodic boundary condition, one may consider spherical or de-Sitterlattices starting with the sequence of platonic solids. One can even consider small hyperbolic orAnti-de-Sitter lattices generated by the triangle group [7]. In conclusion even within the limitedcontext of U ( ) / Z quantum links in the 2 + Acknowledgements
We are grateful to Cameron V. Cogburn for fruitful discussions. This work was supported inpart by the U.S. Department of Energy (DOE) under Award No. DE-SC0019139.
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