Digital quantum simulation of lattice gauge theories in three spatial dimensions

1. Introduction

• Monte Carlo lattice-gauge simulations use Euclidean spacetime, leaving real-time dynamics and non-equilibrium phenomena inaccessible.The introduction also notes the sign problem in finite-chemical-potential fermionic systems.Source: §1 · pp. 1–2
• The proposed 3+1-dimensional scheme realizes magnetic four-body plaquette interactions through exact sequences of two-body interactions, avoiding fourth-order perturbation theory.The auxiliary system mediates the interactions; the resulting plaquette interactions are stronger than in analogue approaches.Source: §1 · p. 3
• The paper extends the digital proposal to three spatial dimensions, analyzes Trotter error bounds, and proposes an ultracold-atom implementation for the non-abelian group D3.The D3 proposal exploits the group's semidirect product structure.Source: §1 · p. 3

2. Hamiltonian formulation of lattice gauge theories

• Matter fields occupy lattice vertices while gauge fields reside on links, and local gauge transformations act on the fermionic and adjacent link degrees of freedom.The matter fields are fermionic Fock-space states; the link degrees of freedom are represented using group-element operators.Source: §2 · pp. 3–5
• A gauge-invariant empty state uses trivial-representation gauge fields, and other physical states can be generated by gauge-invariant operators.Staggered fermions distribute spinor components across neighboring sites, with occupied even sites representing particles and vacant odd sites representing antiparticles.Source: §2 · pp. 4–5
• The conventional Hamiltonian sums magnetic plaquette, electric, staggered fermion-mass, and gauge-invariant fermion-hopping terms.The non-interacting vacuum is the ground state of the electric and magnetic Hamiltonians.Source: §2 · pp. 5–6

3. Digital algorithm for the quantum simulation of lattice gauge theories in three dimensions

• The algorithm uses an entangled auxiliary degree of freedom to rewrite interactions involving three or more constituents as exact sequences of two-body interactions.This avoids perturbative limits on the plaquette-interaction energy scale and can access a bigger region of the phase diagram.Source: §3 · p. 7

3.1. Isometries

• An isometry maps the physical Hilbert space into the joint physical-auxiliary space, where a unitary acting on an initial auxiliary state creates the entanglement.The physical Hamiltonian's action is reproduced by a corresponding Hermitian operator acting on the auxiliary space.Source: §3.1 · p. 7
• Applying the auxiliary operator after the isometry yields the desired physical time evolution, after which the auxiliary system can be disentangled and returned to its initial state.The method requires stroboscopic evolution because the entangling operations are sequential.Source: §3 · p. 7, §3.1 · p. 8

3.2. The three-dimensional algorithm

• Auxiliary degrees of freedom in every second cube support sequential implementation of the three-dimensional lattice gauge theory Hamiltonian, with the desired evolution approximated by a Trotterized sequence.The gauge-matter sequence is split by direction and cube parity to avoid undesired interactions between adjacent links.Source: §3.2 · p. 8

3.2.1. Plaquette interactions

• Each four-gauge-field plaquette term is implemented by coupling its links to an auxiliary control, acting on the control, and undoing the isometry.The three plaquette terms are implemented for even cubes and then odd cubes; because the magnetic terms commute, this splitting gives the magnetic time evolution exactly.Source: §3.2.1 · pp. 9–11, Figure 3 · p. 11

3.2.2. Gauge-Matter interactions

• A gauge-field–fermion unitary acts as a fermionic transformation parameterized by the gauge field, allowing pure fermion tunneling to be mapped into gauge-matter interactions.The interactions are split across six steps for the six links connected to each fermion; an alternative isometry sequence mediates them through auxiliaries without interactions between physical degrees of freedom.Source: §3.2.2 · pp. 12–13, Figure 4 · p. 13

3.2.3. Other parts of the Hamiltonian

• Electric and matter evolution are implemented locally, and every evolution in the sequence is individually gauge-invariant, so digitization errors do not break gauge invariance.The magnetic terms are combined into one evolution because they commute.Source: §3.2.3 · p. 14

3.3. Error bounds for Trotterized time evolutions in lattice gauge theory

• The second-order formula has error that decreases faster with the number of time steps N, at the cost of a longer sequence.The bounds help choose N to balance digitization error against experimental errors that can accumulate and break gauge invariance.Source: §3.3 · p. 15

3.3.1. First order formula

• The first-order error bound is finite for finite gauge groups or suitably truncated infinite gauge-field Hilbert spaces.In these settings, max_j |f(j)| is finite, yielding sensible bounds.Source: §3.3.1 · p. 16

3.3.2. Second order formula

• For a cubic lattice with L sites per side, the general second-order bound gives N scaling as L^d/2 t^3, while intensive observables are expected to scale better.Wilson loops need not be intensive, so they may require the more general estimates.Source: §3.3.2 · p. 16

4. Implementation of digital lattice gauge theories with ultracold atoms

• The proposed scheme covers lattice gauge theories with either compact or finite gauge groups.The paper proceeds from the general three-dimensional construction to concrete ultracold-atom examples.Source: §4 · pp. 16–17
• Compact gauge groups such as U(1) have infinite link Hilbert spaces, so truncation is required for feasible quantum simulation.Representation-basis truncation spoils the unitarity of U and prevents use of isometries.Source: §4 · p. 17
• Truncating U(1) in the group-element basis gives finite groups ZN, which converge to U(1) as N approaches infinity.Source: §4 · p. 17

4.1. Implementation of lattice gauge theories with gauge group ZN

• A ZN gauge field has N states, and finite abelian gauge groups approximate compact quantum electrodynamics.Source: §4.1 · p. 17
• Because ZN is abelian, its representations are one-dimensional, and the model uses a single fermionic species on the vertices.Source: §4.1 · p. 17
• The three-dimensional ultracold-atom implementation uses fermionic atoms for matter, a second atomic species for gauge fields, and auxiliary atoms at cube centers.The authors report that the resulting ZN Trotter-error bounds are more accurate than the original analysis in.Source: §4.1 · p. 18

4.2. Implementation of lattice gauge theories with a dihedral gauge group

• D_N is represented as Z_N ⋊ Z_2, with each link encoded in an N-dimensional rotation space and a two-dimensional reflection space; the implementation focuses on D_3.For the smallest faithful irreducible representation, matter uses two fermionic components, ψ1 and ψ2.Source: §4.2 · pp. 18–19, §4.2.1 · p. 19

4.2.1. Simulating system

• The D3 platform uses fermionic atoms for matter and encodes each link and control in a product of three- and two-level atomic systems.The proposed arrangement uses two matter levels, two atoms per link, and two independently addressed control atoms at the center of every second cube; the figure depicts the setup and staggered occupation.Source: §4.2.1 · pp. 19–20, Figure 5 · p. 21

4.2.2. Initial configuration and background Hamiltonian

• Auxiliary sites are initialized in the identity group-element state, and the desired D3 Hamiltonian is implemented on top of a background Hamiltonian in its interaction picture using the rotating-wave approximation.The gauge- and matter-field preparation depends on the physical initial state being simulated.Source: §4.2.2 · p. 22

4.2.3. The mass Hamiltonian

• The mass-term evolution is produced by smoothly raising the energy of even fermion-lattice minima by M_even and holding the resulting Hamiltonian for time M_even τ.This yields the desired unitary evolution up to an irrelevant global phase.Source: §4.2.3 · pp. 22–23

4.2.4. Creating the isometry

• The link isometry is built from separate three- and two-level atom interactions, using local basis changes and mF-preserving scattering; flipping auxiliary levels implements the conjugate sequence to undo it.The two interactions follow the procedures for Z3 and Z2 gauge theories, respectively.Source: §4.2.4 · pp. 23–24

4.2.5. Plaquette interactions

• Plaquette evolution uses the isometry to map the plaquette term to local control operations, then applies the control Hamiltonian and reverses the isometry.The procedure is repeated for the three plaquette terms on even cubes, then on odd cubes by moving the auxiliary atoms.Source: §4.2.4 · pp. 24–26

4.2.6. Gauge-matter interactions

• The gauge–matter term is decomposed using the product structure of U into two-body interactions mediated by control atoms, with interactions applied one link at a time.The isometry transfers link interactions to the auxiliary atoms, which then interact with the fermions; the sequence is described for links in the 1-direction on even cubes.Source: §4.2.6 · p. 26
• Gauge-matter interactions are obtained up to fermionic phases that correspond, on the full lattice, to an unphysical static vector potential with zero magnetic field.Repeating the sequence for the other links yields interactions in all three directions for even and odd links.Source: §4.2.6 · p. 28

4.2.7. Electric Hamiltonian

• Local atomic operations and tailored two-body scattering implement the electric Hamiltonian up to local operations, and repeating the sequence N times gives a Trotter approximation.The sequence evolves the system for τ = T/N per repetition.Source: §4.2.7 · p. 29

4.2.8. Errors

• Trotter errors cause quantitative deviations but preserve gauge invariance, while experimental errors may break the symmetry and accumulate across steps.Statistical errors scale with the square root of N, compared with linear scaling for systematic errors; gates independent of simulated time are especially error-prone, so N must balance digitization and implementation errors.Source: §4.2.8 · p. 30, Table 1 · p. 31
• Implementation requires controlling decoherence, keeping atoms in the lowest Bloch band during motion, and accurately tailoring two-body scattering.The stated timescale conditions are tsim << tcoh, tmov >> 1/ω, and tmov << tcoh; scattering also requires control of wave-function overlap, collision timing, and magnetic-field gradients.Source: §4.2.8 · p. 31

5. Summary

• The scheme realizes 3+1-dimensional lattice gauge theories with dynamical fermions using only two-body interactions.An auxiliary system mediates the more complicated three- and four-body terms; each time evolution in the algorithm is individually gauge invariant.Source: §5 · pp. 31–32
• Magnetic plaquette interactions are obtained without fourth-order perturbation theory, yielding stronger interactions and access to wider phase-space regions than analogue approaches.Source: §5 · p. 31
• The paper studies intrinsic Trotter errors arising from the sequential time evolution, noting that bounds can provide more leeway for experimental errors.Source: §5 · p. 31
• The scheme applies to compact Lie groups or finite groups, and the paper proposes an ultracold-atom optical-lattice implementation for the non-abelian group D3.For compact Lie groups, gauge-field Hilbert spaces require truncation in the group-element basis rather than the representation basis.Source: §5 · pp. 31–32
• For D3, its semidirect-product structure permits gauge fields to be represented as a tensor product of a three-level and a two-level system.The authors identify precise control over atomic collisions, especially auxiliary-system entangling gates, as an experimental need.Source: §5 · p. 32

Appendix A. Details on DN lattice gauge theory

• DN describes rotations by 2π/N and reflections in a plane, with elements represented as (p, m).Source: Appendix A · p. 33
• For odd N ≥ 3, DN has the three irreducible representations shown in the appendix; even N is excluded as it has additional sign representations.Source: Appendix A · p. 33
• The electric Hamiltonian is transformed from the representation basis to group-element states because its representation-basis form is not feasible for the proposed simulation.The appendix specifies the trivial- and sign-representation traces used in the transformation.Source: Appendix A · p. 33
• In the conjugate basis, viewed as an angular-momentum basis, the expression simplifies under the constraint f_l = f_-l.The second term acts trivially on the gauge-field Hilbert space for reflections and can be viewed as the electric energy of ZN.Source: Appendix A · p. 34

Appendix B. Trotter errors

• Bounding Trotter error requires computing commutators and nested commutators among Hamiltonian components.The general lattice-gauge-theory calculation is only sketched because it is very lengthy.Source: Appendix B · p. 34

Appendix B.0.1. First order

• The first-order formula requires ordinary commutators, including those between gauge-matter interactions on different links.Unitary operators UW reduce the gauge-matter terms to pure fermionic tunneling terms for estimating this commutator.Source: Appendix B.0.1 · pp. 34–35
• The analysis also bounds commutators between matter and gauge-matter interactions, using UW to rewrite the expression.The tunneling-term operator norm is identified with d_U, the dimension of the representation U.Source: Appendix B.0.1 · p. 35
• The gauge-matter Hamiltonian is considered as a whole when commuting with the electric part, since each part fails to commute with HE.Source: Appendix B.0.1 · p. 35
• The magnetic-electric commutator estimate uses that each link belongs to 2(d −1) plaquettes.Source: Appendix B.0.1 · p. 35

• Nested commutators are evaluated as ordinary commutators, with no additional tricks, although the calculations are lengthy.Source: Appendix B.0.2 · p. 36
• For gauge-matter interactions, 2d separately implemented Hamiltonians determine how often their mutual commutator appears, yielding two natural-number partial sums in the second-order formula.Source: Appendix B.0.2 · p. 36
• The nested-commutator bounds are inserted into the total Trotter-error formulas to obtain the bounds reported in Sec. 3.Source: Appendix B.0.2 · p. 36