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Digital lattice gauge theories
Erez Zohar, Alessandro Farace, Benni Reznik, J. Ignacio Cirac
TL;DR
Quantum simulation of lattice gauge theories must enforce local gauge symmetry on simulating systems that do not exhibit it explicitly. The paper gives a digital construction for lattice gauge theories with dynamical matter, using auxiliary degrees of freedom and concatenated interactions to implement gauge-matter couplings. The construction covers lattice gauge theories with dynamical matter for gauge groups that are compact or finite.
Problem
Quantum simulation of lattice gauge theories must enforce local gauge symmetry on simulating systems that do not exhibit it explicitly.
Source: §I · p. 2Method
Results
The construction covers lattice gauge theories with dynamical matter for gauge groups that are compact or finite.
Source: §I · p. 2Takeaways & Limitations
Each digitized component and its commutators are gauge invariant, so the digitized evolution respects the desired symmetry in principle.
Source: C. Limitations · p. 19Experimental imperfections can break symmetry and accumulate with the number of steps, so that number must balance implementation and digitization errors.
Source: C. Limitations · p. 19
Abstract
from arXiv · showhide
We propose a general scheme for a digital construction of lattice gauge theories with dynamical fermions. In this method, the four-body interactions arising in models with $2+1$ dimensions and higher, are obtained stroboscopically, through a sequence of two-body interactions with ancillary degrees of freedom. This yields stronger interactions than the ones obtained through pertubative methods, as typically done in previous proposals, and removes an important bottleneck in the road towards experimental realizations. The scheme applies to generic gauge theories with Lie or finite symmetry groups, both Abelian and non-Abelian. As a concrete example, we present the construction of a digital quantum simulator for a $\mathbb{Z}_{3}$ lattice gauge theory with dynamical fermionic matter in $2+1$ dimensions, using ultracold atoms in optical lattices, involving three atomic species, representing the matter, gauge and auxiliary degrees of freedom, that are separated in three different layers. By moving the ancilla atoms with a proper sequence of steps, we show how we can obtain the desired evolution in a clean, controlled way.
I. INTRODUCTION
Digital quantum simulation offers a route to studying lattice gauge theories while addressing the challenge of enforcing local gauge symmetry in a simulator. The paper develops a general construction and highlights a Z3 implementation with dynamical matter.
I. INTRODUCTION
- Quantum simulators are tailored systems that mimic target quantum theories, but lattice gauge simulation must enforce local gauge symmetry absent from the simulator itself.Digital simulation approximates the target dynamics through a precise sequence of short operations.Source: §I · p. 2
- Four-body plaquette interactions in 2+1 dimensions are difficult to implement and were obtained perturbatively in previous proposals, making them weak and experimentally challenging.The cited discussion notes that fourth-order processes are needed when constructing these terms from gauge-invariant building blocks.Source: §I · p. 2
- The proposed digital construction includes dynamical matter and applies to gauge groups that are compact or finite.It concatenates simpler two-body interactions to obtain required three- and four-body interactions while keeping each digitized step locally gauge invariant.Source: §I · p. 2
- The concrete example is a Z3 lattice gauge theory in 2+1 dimensions using ultracold atoms in a layered structure with movable ancillary atoms.The paper describes the Z3 case as containing ingredients beyond the Z2 implementation and readily extendable to higher N.Source: §I · pp. 2–3
1. Lattice gauge theories: the physical ingredients
- Matter fields reside on lattice vertices and gauge fields on links; the discussion restricts to two spatial dimensions while noting generalization to other dimensions.The paper uses staggered fermions, with particles on even vertices and antiparticles on odd vertices.Source: §1 · pp. 3–4
2. Lattice local gauge invariance
Local gauge transformations act on the degrees of freedom meeting at a vertex, with a transformation parameter that may vary by position. Gauge-invariant states and operators are defined by their behavior under these transformations.
2. Lattice local gauge invariance
- A local gauge transformation acts on the Hilbert spaces intersecting at a vertex and may use a different group element at each position.A gauge-invariant state is invariant under this transformation.Source: §2 · p. 5
- For compact Lie groups, the local invariance condition can be expressed using generators that satisfy the group algebra.The resulting condition is identified as a lattice Gauss law.Source: §2 · p. 5
- Gauge-invariant operators include traces of link operators along closed paths, with the plaquette as the simplest example.Products along a line with fermionic operators at the endpoints provide another type of gauge-invariant operator.Source: §2 · p. 5
1. The Electric Hamiltonian,
The section identifies the electric, magnetic, mass, and gauge-matter terms of the lattice Hamiltonian, then describes digital evolution by implementing its parts separately. Stators let auxiliary degrees of freedom mediate effective interactions on the physical system.
1. The Electric Hamiltonian,
- The electric and magnetic terms describe the gauge field and together form the Kogut-Susskind Hamiltonian for compact Lie groups.The electric term corresponds to electric-field energy in the SU(N) case; the magnetic term yields magnetic energy in the continuum limit for examples such as QED.Source: §1 · p. 6
- The total Hamiltonian also contains fermion mass and gauge-matter interaction terms.The mass term's alternating signs encode staggered fermions, while the paper's digital implementation treats the Hamiltonian's parts separately.Source: §1 · p. 6
B. The digitization
- Digital evolution approximates the total dynamics by applying short evolutions generated by separately implementable Hamiltonian terms.The implementation splits plaquette and gauge-matter terms into even/odd plaquettes and link classes before applying a Trotter sequence.Source: B. The digitization · p. 6
1. Stators
- A stator couples a physical system to an auxiliary degree of freedom so that evolution on the auxiliary can implement an effective Hamiltonian on the physical system.For plaquette interactions, the paper proposes connecting several physical degrees of freedom with a group element stator built through two-body interactions.Source: §1 · p. 7
- The stator procedure can return the auxiliary system to its initial state while the physical state undergoes the desired evolution.The paper uses the stator eigenoperator relation to obtain the magnetic plaquette interaction.Source: §1 · p. 7
2. The plaquette interactions
The magnetic plaquette evolution is built from plaquette stators and local operations on their controls. Because neighboring plaquettes share links, the construction processes even and odd plaquettes in separate passes.
2. The plaquette interactions
- For each plaquette, the scheme creates a stator, applies a control Hamiltonian, and undoes the stator to obtain the plaquette interaction.The control atoms are moved through link interactions in a sequence; the same controls can be moved from even to odd plaquettes for the second pass.Source: §2 · pp. 7–8
- The magnetic Hamiltonian's global evolution is exactly equivalent to combining the single-plaquette operations because the plaquette terms commute.Plaquettes can be evolved in parallel within a parity class, but shared links prevent creating all plaquette stators simultaneously.Source: §2 · pp. 7–8
- For non-Abelian groups, the order of the link interactions matters because it determines the order of group elements in the auxiliary state.For Abelian groups, this ordering does not affect the product.Source: §2 · p. 8
- The even-plaquette sequence yields the corresponding magnetic evolution while returning the controls to their initial state.An analogous operation on the odd plaquettes produces the other parity's evolution.Source: §2 · p. 8
3. The gauge-matter interactions
The gauge-matter interaction is obtained by turning free fermion tunneling into a charged interaction through a sequence of local two-body operations. The full time step combines these operations with local terms, respecting the noncommutation of its components.
3. The gauge-matter interactions
- The gauging transformation maps free fermionic tunneling into a charged interaction using a gauge transformation whose parameter is an operator.This realizes three-body gauge-matter interactions from two-body local interactions.Source: §3 · p. 9
- A link and its vertex interact before fermions tunnel along the link, producing the gauge-matter interaction through a sequence of operations.The link-vertex interaction and subsequent tunneling implement the desired interaction stroboscopically.Source: §3 · pp. 8–9
- Gauge-matter interactions are split into four sequential link sets— even and odd horizontal and vertical—because neighboring or differently oriented operations would involve unwanted gauge fields.All links within a set can be treated simultaneously, but the four sets are non-commuting terms.Source: §3 · p. 9
C. Examples - Abelian theories
The Abelian examples cover compact U(1) and finite ZN gauge groups. U(1) has infinite-dimensional local spaces, whereas ZN provides a finite-dimensional truncation suitable for experimental simulation.
C. Examples - Abelian theories
- U(1) links and controls have infinite-dimensional local Hilbert spaces, making the theory infeasible for practical quantum simulation in this proposal.The finite-dimensional ZN truncation is presented as the next experimentally simulable example.Source: C. Examples - Abelian theories · pp. 9–10
1. U(1) (compact QED)
- For U(1), φ̂ is the vector potential and L the electric field; the states |m⟩ are electric-flux states.The operators are canonically conjugate, and L acts as a flux-raising operator.Source: §1 · p. 10
- The U(1) digital construction specifies control interactions for plaquette and fermion-gauge terms.The control system is initialized in a stated state and interacts with links to implement these operations.Source: §1 · p. 10
2. ZN
- ZN has finite-dimensional link spaces and, as N increases, converges to U(1), making it a truncation scheme for the continuous case.The ZN construction defines link operators and gauge transformations, then specifies control operations for plaquette and fermion interactions.Source: §2 · pp. 10–11
III. APPLICATION FOR DIGITAL QUANTUM SIMULATION - THE Z3 CASE
The paper applies its digital construction to a Z3 lattice gauge theory and describes a stator-based implementation using ultracold atoms. The design uses separate matter, gauge, and auxiliary species and moves the auxiliary atoms to implement interactions.
III. APPLICATION FOR DIGITAL QUANTUM SIMULATION - THE Z3 CASE
- Z3 is presented as a finite-group example for a feasible simulator and as a step toward compact Lie-group simulations using stators.The paper treats the example as a truncation of U(1).Source: §III · p. 11
- The proposed truncation uses the group-element basis because representation-basis truncation makes the U operators nonunitary and prevents group-element stators.ZN theories converge to U(1) as N approaches infinity.Source: §III · p. 11
1. The stator
- The Z3 stator construction includes SQ and P stators, with the P stator designated for gauge-matter interactions.The SQ stator satisfies an eigenoperator relation that the paper identifies as important to the construction.Source: §1 · pp. 11–12
1. The atomic ingredients
- The simulator assigns fermionic atoms to lattice vertices, three-level bosonic link atoms to gauge fields, and three-level auxiliary atoms to every second plaquette.The auxiliary atoms can move between layers to interact with both matter and link atoms.Source: §III · p. 11, §1 · p. 12
2. The optical lattices
- The optical-lattice design places auxiliary atoms at even-plaquette centers and uses temporal shaping to move them to odd plaquettes or near matter and links.Link-atom minima lie on links, while matter minima coincide with simulated-lattice sites.Source: §2 · p. 13
3. The interactions: atomic collisions
The proposed stator interactions rely on controlled collisions between link and auxiliary atoms. The paper describes energy constraints and spatially selective overlap as ways to obtain the desired unitary and reverse the stator operation.
3. The interactions: atomic collisions
- Ultracold-atom collisions are treated as s-wave scattering, with kinetic energy small relative to centrifugal barriers and hyperfine splitting.These conditions reduce transitions to other hyperfine levels for atoms initially in their ground states.Source: §3 · p. 13
- For link and auxiliary atoms with F = 1, collisions conserve total angular momentum and its z projection, while energy constraints can conserve each atom’s angular momentum.The scattering potential is expressed through projectors onto total-angular-momentum subspaces.Source: §3 · pp. 13–14
- Unequal Zeeman energy splittings penalize unwanted changes in mF, leaving processes that conserve both atoms’ internal levels energetically allowed.The proposal uses this constraint to make the collision unitary diagonal in the mF basis.Source: §3 · p. 14
- Spatially separating internal-level components makes only the mF = m̃F = 0 components overlap, allowing the desired interaction phase; a local spin flip implements the inverse stator operation.The tuned collision sequence yields U′† up to a global phase.Source: §3 · p. 14
4. Magnetic fields
The implementation uses magnetic-field control and a layered optical-lattice setup to realize the Z3 Hamiltonian’s electric and mass terms. The required link-level energy shifts and fermion mass evolution are produced with species-selective control and lattice shaping.
4. Magnetic fields
- A uniform field can lift ground-state degeneracy, but distinct Landé factors across three atomic species—or species-selective AC Stark shifts—are needed to avoid splitting all atoms alike.Source: §4 · p. 15
- A magnetic-field gradient separates atomic components vertically, allowing collisions to be tailored by their mF values and selective mF = 0 interactions.With sufficient separation, only the selected components are brought together to implement Vscat.Source: §4 · p. 15
1. Standard configuration of the lattice
- The standard lattice traps link and auxiliary atoms in deep, separated minima, while half-filled matter sites realize the Dirac state and complete the global singlet.The control atoms and gauge-field atoms are also prepared in their specified initial states.Source: §1 · p. 15
2. The electric Hamiltonian
- The electric term is implemented by shifting the link atoms’ mF = ±1 levels for a time proportional to λE ΔEel τ; its constant contribution is only a global phase.Source: §2 · p. 15
3. The mass Hamiltonian
- The mass term is realized by shaping the matter lattice to raise even-site energies by Meven, then evolving for a time proportional to τ.Hamiltonian pieces are applied in a frame rotating with respect to H0, using the rotating-wave approximation.Source: §3 · pp. 15–16, §1 · p. 15
4. The even plaquette interactions
Even-plaquette magnetic interactions are generated by creating a plaquette stator, applying a local control operation, and using the stator to transfer that operation to the links. Gauge-matter evolution on an even horizontal link is likewise obtained from a sequence of stator-mediated interactions and tunneling.
4. The even plaquette interactions
- Auxiliary atoms interact successively with the four links around each even plaquette to build its stator, after which a local control operation generates the even plaquette interaction.The stator is then undone; moving the auxiliary atoms to odd plaquettes repeats the process for odd plaquettes.Source: §4 · p. 16
5. Gauge-matter interaction on even horizontal links
- On an even horizontal link, the sequence UW UtU†W produces the desired HGM evolution, while the accompanying fermion-dependent phases may be ignored for the full lattice.The construction begins with a P-stator, couples the control to the fermion, allows link tunneling, and repeats the fermion interaction with a control-level flip.Source: §5 · p. 17
- The P-stator is undone after the link operation when the auxiliary and link atoms must be disentangled for the next step.Source: §5 · p. 17
6. The complete sequence
A single Z3 time step combines stator-mediated gauge-matter and plaquette operations on even and odd links, with the controls moved between plaquettes. The sequence is completed with the non-interacting mass and electric steps.
6. The complete sequence
- The even-link gauge-matter operations and even-plaquette stators are followed by a local control operation that produces the even plaquette interaction.The sequence starts with controls centered on even plaquettes and creates stators for even vertical links before treating the horizontal links.Source: §6 · p. 17
- Controls move to odd plaquettes in a separate vertical layer to avoid unwanted link–auxiliary interactions during transit.Source: §6 · p. 17
- The odd-link and odd-plaquette interactions repeat the corresponding even operations, with the plaquette stator completed, locally driven, and then undone.Source: §6 · pp. 17–18
- The sequence finishes by completing the odd plaquette operation and adding the non-interacting mass and electric steps; repeating W(τ) with τ = t/M yields the Trotterized evolution.The complete sequence is shown in Fig. 4.Source: §6 · pp. 18–19
C. Limitations
Digitization error decreases as the number of time slices grows, but experimental imperfections can break gauge symmetry and accumulate across steps. A second-order Trotter formula reduces the total operation burden and simulation time under the stated conditions.
C. Limitations
- Digitization errors scale as t2/M, but increasing M reduces them while lengthening the simulation; each sequence component and its commutators remain gauge invariant.Source: C. Limitations · p. 19
- Experimental implementation errors can break gauge symmetry and accumulate with M, so M must balance them against digitization error rather than increase arbitrarily.Fixed-duration gates, including VF, Uscat, and Vscat, merit particular care.Source: C. Limitations · p. 19
1. Trotterization
- For the first-order Trotterization, the component Hamiltonians do not all commute, producing 15 non-vanishing commutators; their norms are bounded using the largest Hamiltonian piece.The Hamiltonian-piece norms scale roughly as L2 in the stated two-dimensional system.Source: §1 · p. 19
2. Number of steps and simulation time
- Atomic collisions are typically the slowest experimental process, and a 2π/3 stator-creation rotation takes about 1 ms without Feshbach resonances.The total experimental time scales quadratically with simulated time.Source: §2 · p. 20
3. Second-order formula
- The second-order formula doubles operations per slice but can greatly reduce total operations for reasonable L and λmaxT, helping keep experimental errors at bay.It also reduces the actual simulation time.Source: §3 · pp. 20–21
4. Higher-order bounds
The section weighs higher-order Trotter formulas against their experimental demands, estimates error sources, and outlines feasible observations and the broader scope of the digital scheme.
4. Higher-order bounds
- Second-order Trotterization is a practical compromise: it improves error bounds over first order, while higher orders require more tunability and can slow the simulation.Higher-order sequences can reduce the number of time slices, but require different—and sometimes negative—tunneling times.Source: §4 · p. 21
5. Propagation of experimental errors
- Errors from gates with duration independent of the simulation timescale are more dangerous, while random errors scale as the square root of the number of repetitions.The authors recommend reducing the timescale or error magnitude of these fixed-duration gates.Source: §5 · p. 21
- The error bound is not tight because independently distributed gate errors can partially cancel across the sequence.The authors consider such cancellation probable when accounting for the sequence’s multiple blocks and gates.Source: §5 · p. 21
6. Experimental errors
- The proposal requires precise pulse timing and collision control, and elastic collisions require sufficiently different mF = ±1 level splittings across the three species.These controls help keep implemented gates close to their intended unitaries and avoid undesired spin flips.Source: §6 · p. 22
D. Possible observations
- A quench from half-filled fermions and zero-flux gauge atoms could test dynamical generation of gauge-invariant configurations without single-site addressing.The proposed setup estimates roughly 20 or 40 Trotter steps within a coherence time of about 1 s, depending on the sequence.Source: D. Possible observations · p. 22
- Static-charge string-breaking experiments require single-site addressing, while adiabatic preparation of the full-model ground state may require longer coherence times.The latter requirement depends on the model’s spectral gap and further refinements to the stroboscopic sequence.Source: D. Possible observations · p. 22
IV. SUMMARY
- The digital evolution is formulated for compact Lie or finite gauge groups, but compact Lie groups require truncation in the group-element basis.The stators use that basis, making conventional representation-basis truncations incompatible with the scheme.Source: §IV · p. 22
- The scheme implements a 2+1-dimensional Z3 lattice gauge theory with ultracold atoms, using layered species and stroboscopic two-body interactions to tailor three- and four-body terms.The layered structure allows species to be trapped, evolved, and measured separately, with interactions controlled through timing and overlap.Source: §IV · pp. 22–23
Appendix A: Optical lattices for Z3
Appendix A describes optical-lattice configurations for trapping and manipulating matter fermions alongside link and auxiliary atoms. Appendix B shows that the static phases arising from the physical realization have zero lattice curl and do not affect the physics.
1. Trapping the matter fermions
- Three pairs of counter-propagating out-of-plane laser waves can implement the matter-fermion potential, with confinement along z yielding the target potential.The equal-wavelength condition constrains the simulated lattice spacing.Source: §1 · p. 23
2. Shaping the lattice
- Tuning f(t), g(t), h(t), and φ(t) shapes the lattice; changes must be adiabatic to keep atoms in the ground state.Staggering uses h(t) to create an energy difference between even and odd minima; fermion tunneling is implemented in four directional steps.Source: §2 · pp. 23–24
- For both even and odd plaquettes, the physical realization produces static link phases with zero lattice curl, so they do not influence the physics.These phases may be removed or practically ignored.Source: Figure 5 · p. 24
3. Trapping the other atoms
- Link atoms require a static square lattice, while auxiliary atoms require a square lattice movable rigidly in three directions.The simulated spacing must be compatible with the wavelengths used for the matter, link, and auxiliary lattices.Source: §3 · p. 24
Appendix B: The unphysical phases of the Z3 simulation
- The residual phases in the Z3 simulation have zero lattice curl, so they can be gauged away or ignored.The appendix concludes that the phase-modified sequence acting on the physical state is equivalent to the intended sequence.Source: Appendix B · p. 25