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A resource efficient approach for quantum and classical simulations of gauge theories in particle physics
Jan F. Haase, Luca Dellantonio, Alessio Celi, Danny Paulson, Angus Kan, Karl Jansen, Christine A. Muschik
TL;DR
Hamiltonian simulations of continuous-gauge-group LGTs require truncation, but existing schemes become costly at weak coupling and obstruct the continuum limit. The paper combines Hilbert-space truncation with coupling-dependent Z2L+1 regularization, demonstrating a resource-efficient route toward continuum physics in 2+1D QED. Its scope includes a benchmark study and a limitation for current analog implementations of two-dimensional magnetic interactions.
Problem
Continuous gauge groups produce infinite-dimensional gauge degrees of freedom, while existing truncations require rapidly increasing resources at small bare coupling and hinder the continuum limit.
Method
The method combines state truncation with a gauge-group discretization using finite Z2L+1 groups whose order is scaled with the bare coupling.
Results
The framework supports simulations at all values of g and arbitrarily small lattice spacing a, providing a perspective for accessing non-perturbative physics near the continuum limit.
Takeaways & Limitations
The framework provides a route toward continuum-limit simulations on quantum and classical platforms, including physically meaningful regimes precluded to quantum Monte Carlo.
Takeaways & Limitations
Experimental realization of two-dimensional theories with magnetic terms is currently out of reach for the described analog bosonic approach.
Abstract
from arXiv · showhide
Gauge theories establish the standard model of particle physics, and lattice gauge theory (LGT) calculations employing Markov Chain Monte Carlo (MCMC) methods have been pivotal in our understanding of fundamental interactions. The present limitations of MCMC techniques may be overcome by Hamiltonian-based simulations on classical or quantum devices, which further provide the potential to address questions that lay beyond the capabilities of the current approaches. However, for continuous gauge groups, Hamiltonian-based formulations involve infinite-dimensional gauge degrees of freedom that can solely be handled by truncation. Current truncation schemes require dramatically increasing computational resources at small values of the bare couplings, where magnetic field effects become important. Such limitation precludes one from `taking the continuous limit' while working with finite resources. To overcome this limitation, we provide a resource-efficient protocol to simulate LGTs with continuous gauge groups in the Hamiltonian formulation. Our new method allows for calculations at arbitrary values of the bare coupling and lattice spacing. The approach consists of the combination of a Hilbert space truncation with a regularization of the gauge group, which permits an efficient description of the magnetically-dominated regime. We focus here on Abelian gauge theories and use $2+1$ dimensional quantum electrodynamics as a benchmark example to demonstrate this efficient framework to achieve the continuum limit in LGTs. This possibility is a key requirement to make quantitative predictions at the field theory level and offers the long-term perspective to utilise quantum simulations to compute physically meaningful quantities in regimes that are precluded to quantum Monte Carlo.
1 Introduction
LGT provides a nonperturbative framework, but MCMC faces sign-problem limitations and Hamiltonian simulations face resource challenges for continuous gauge groups. The paper introduces a resource-efficient truncation and gauge-group regularization, benchmarked on 2+1D QED, to approach the continuum limit.
- MCMC methods can be ineffective because of the sign problem, while classical Hamiltonian simulations remain mostly restricted to one spatial dimension.
- Quantum hardware offers an alternative route to higher-dimensional QFT simulations, but current devices require new approaches to make this extension practical.
- Continuous gauge groups give Hamiltonian formulations infinite-dimensional local gauge degrees of freedom, so simulations require truncation that conflicts with the continuum limit.
- The approach is presented as a resource-efficient solution for Hamiltonian LGT simulations and benchmarked using 2+1-dimensional QED.
- Electric-basis truncations preserve gauge symmetry but require dramatically more states as coupling decreases and magnetic contributions become important.
- The proposed method combines state truncation with coupling-dependent gauge-group discretization, connecting weak, strong, and intermediate coupling regimes.
2 Minimal encoding of LGTs with continuous gauge groups
The paper formulates continuous-gauge-group LGTs in Hamiltonian form while enforcing gauge invariance and removing redundant degrees of freedom. It develops the construction for lattice QED, including electric and magnetic operators, matter coupling, and physical-state constraints.
- Continuous Abelian gauge theories define electric and magnetic fields through the vector potential, with gauge invariance arising under local phase transformations.
- Gauss’ law relates the divergence of the electric field to charge density and identifies the physical states of the lattice theory.
- Lattice charges occupy sites, while electric fields and Wilson operators act on links and generate plaquette interactions.
- In staggered QED, fermions on the square lattice represent particles and antiparticles, while kinetic hopping changes the connecting electric-field string.
- The Hamiltonian’s electric and magnetic terms appear asymmetric because time is continuous, but spatial continuum limiting restores the corresponding duality.
- Unphysical states create exponential computational overhead, motivating elimination of redundant degrees of freedom by solving the local constraints.
3 Transformation into the magnetic representation
The method replaces U(1) with Z2L+1, applies a discrete Fourier transform to diagonalize magnetic contributions, and truncates the resulting magnetic basis. The parameters L and l jointly control discretization accuracy and retained Hilbert-space resources.
- Group discretization: The approach replaces the continuous U(1) gauge group with the discrete group Z2L+1 and uses a completely compact formulation.This replacement provides a discrete basis for vector-potential operators and supports the subsequent basis transformation.
- Hilbert-space truncation: The truncated electric Hilbert space retains states with r_j from −l through l, reducing the total dimension to (2l + 1)^3.This truncation is required because the rotators have discrete but infinite spectra, but it becomes restrictive in the weak-coupling regime.
- Basis transformation: The replacement rules reassemble finite Fourier series exactly because the discrete spectrum makes trigonometric coefficients periodic and equivalent.Only 2L coefficients remain after summing equivalent sine and cosine terms.
- Basis transformation: A discrete Fourier transform diagonalizes the magnetic contributions while sacrificing the electric part’s diagonal structure.The resulting magnetic representation permits efficient weak-coupling computations, with l acting as a cutoff for magnetic-field energy.
- Accuracy and resources: In the magnetic representation, L controls group-discretization accuracy, whereas l selects the retained states and is constrained by computational resources.The approximation first maps U(1) to Z2L+1 and then truncates from (2L + 1)^3 to (2l + 1)^3.
4 Performance and application of the new approach
The approach assesses convergence between electric and magnetic representations, selects truncation and resolution parameters across coupling regimes, and estimates the plaquette expectation value. The analysis shows how representation choice and regularization control computational resources and convergence toward the U(1) theory.
- Representation choice: The electric and magnetic representations are related by a Fourier transform, but each is naturally accurate in opposite coupling regimes.The electric representation is favored at strong coupling, while the magnetic representation is favored at weak coupling because the dominant Hamiltonian term is diagonal in the corresponding basis.
- Resource selection: For each bare coupling, perturbative estimates determine a minimal truncation l and resolution L that agree with the untruncated U(1) theory.The required truncation must cover the population tails; one estimate requires l^5 > g^8L^8, while decreasing g increases the needed resolution.
- Convergence analysis: The optimal magnetic resolution increases as L ∼ g^-1 when g decreases, reflecting the need for finer resolution around the localized state |0⟩.For fixed l, Lopt balances resolution near |0⟩ against representation of distribution tails; in broader settings, multiple optimal L values may occur.
- Convergence analysis: Fourier and sequence fidelities provide complementary convergence tests: Fourier fidelity is useful in intermediate coupling, while sequence fidelity tests extremal regimes.The Fourier fidelity quantifies agreement between truncated representations, whereas sequence fidelity identifies convergence toward the untruncated theory and can reveal freezing.
- Plaquette estimation: The plaquette expectation value benchmarks the method because it is related to the running of the coupling, while fixed group resolution can cause freezing and rising costs across coupling regimes.A resolution suitable for g ≪ 1 can require very large l near g ≈ 1, producing dramatically increasing computational costs.
- Plaquette estimation: At g^-2 = 10, both representations agree through the fourth decimal at l = 10, yielding ⟨□⟩ = 0.9572 ± 0.0001.The asymptotic values obtained by increasing l estimate the untruncated U(1) result, although convergence need not be monotonic.
5 Generalisations: Dynamical matter and arbitrary torus
The formulation extends to dynamical matter and arbitrary periodic tori while retaining transformations between electric and magnetic representations. Numerical tests with charges show intermediate-coupling features, but the harsh truncation limits Fourier fidelity.
- Dynamical matter: Including staggered fermions introduces no fundamental complication for the completely compact formulation and extends the simulations to matter.The mass Hamiltonian becomes representation-independent, while the magnetic field Hamiltonian remains unchanged and the kinetic term is modified.
- Dynamical matter: The dynamical-matter simulations use a harsh l = 2 truncation because the truncated Hilbert-space dimension scales as 24(2l + 1)^5.The estimate contains four charges, three rotators, and two strings, with L fixed to pure-gauge optimal values.
- Dynamical matter: For dynamical charges, the plaquette expectation value retains the pure-gauge asymptotic behavior but develops novel intermediate-coupling features, including a negative dip.The study uses l = 2 and m = κ = 10 for the displayed curves.
- Arbitrary torus and charges: The arbitrary-torus construction reduces a periodic lattice to two strings and NxNy − 1 rotators after removing redundant degrees of freedom.One rotator is explicitly fixed to zero, and the electric fields include charge-dependent corrections consistent with Gauss’ law.
- Arbitrary torus and charges: On a torus, background and dynamical charges are incorporated through electric-field corrections and string conventions, whose kinetic terms require a gate count linear in system size.The construction connects each charge to the origin through a specified path and permits static background charges through the Q̂_n operators.
- Scope: The electric and magnetic representations remain directly transformable, while detailed convergence analysis of the matter-induced effects is left for future work.The authors conclude that the method can be scaled to more complex systems, but do not analyze the novel effects and convergence in detail here.
6 Conclusions and outlook
The paper introduces a resource-efficient strategy for Hamiltonian simulations of continuous-group LGTs, combining truncation with coupling-dependent gauge-group discretisation. Applied to 2+1-dimensional QED, the framework supports simulations across coupling regimes and offers a route toward the continuum limit.
- Outlook: The framework is intended for both quantum and classical Hamiltonian simulations and may extend to higher dimensions and non-Abelian gauge theories.The proposed extensions are stated as possibilities rather than demonstrated results.
- Core strategy: The method combines finite Hilbert-space truncation with a Z2L+1 regularisation of U(1), scaling L with the bare coupling g.This keeps the number of simulated states constant for any g.
- Core strategy: At weak coupling the gauge fields are truncated in the magnetic representation, while strong-coupling truncation uses the electric representation.The representation is chosen according to the regime to reduce truncation error.
- Benchmark: The scheme was benchmarked using plaquette expectation values on a small periodic lattice, with and without dynamical matter.The benchmark estimated the accuracy of the regularised and truncated calculations.
- Continuum limit: Because the method works at all g and arbitrarily small lattice spacing a, it provides a perspective for accessing non-perturbative physics near the continuum limit.The paper identifies this as a route toward physically meaningful continuum simulations.
- Outlook: The approach addresses a central need in continuum-oriented LGT: extracting physically relevant results with finite computational resources.The paper links this goal to simulations of fundamental particle interactions and quantum-simulation platforms.
A Dimensions of the (2+1) dimensional QED Hamiltonian
This appendix establishes the dimensional conventions for the 2+1-dimensional QED Hamiltonian and describes how the continuum-limit scaling is defined.
- Dimensional conventions: Natural units c = ℏ = 1 are used, with dimensionless gauge-field and matter operators.The lattice spacing a has units [a] = mass^-1, while the Hamiltonian has dimension [H] = [energy].
- Continuum scaling: The continuum limit is defined through the scaling of the lattice spacing a toward zero.The appendix introduces this transition before analyzing the Hamiltonian terms separately.
Electric Hamiltonian
The electric Hamiltonian is dimensionally normalized by rescaling the electric-field operators and absorbing the remaining units into the coupling.
- Electric-energy dimensions: Each electric-energy summand has units of two-dimensional energy density, [energy]/[length]^2 = [energy]^3.This sets the dimensional requirement for the electric contribution.
- Rescaling: The rescaling a^3 J^2 = E^2 and definition g = g̃/√a make the electric-field operators dimensionless.The resulting coupling satisfies [g^2] = [energy].
Magnetic Hamiltonian
The magnetic Hamiltonian is normalized so that its continuum-limit contribution survives with the required scaling, while the gauge-potential dimensions remain consistent.
- Magnetic operators: Plaquette operators are dimensionless because they are constructed from dimensionless field creation operators.The product ag  must also be dimensionless for a valid series representation.
- Dimensional consistency: The gauge potential has dimension [Â] = [mass], ensuring that ag  is dimensionless.This fixes the units used in the magnetic expansion.
- Continuum limit: The denominator in the magnetic term ensures that the desired second-order contribution of P̂† + P̂ survives in the continuum limit.The relevant second-order term is proportional to a^4.
- Dimensional consistency: The resulting scaling is consistent with [g^2] = [mass], since [1/(g^2a^2)] = [mass] = [energy].This provides the dimensional check for the magnetic Hamiltonian.
Mass Hamiltonian
The section formulates the mass and kinetic terms alongside Gauss-law constraints in a rotator-and-string description of two-dimensional lattice gauge theory. Particle-pair creation is implemented through link-field updates, plaquette operators, and boundary strings that preserve gauge invariance.
- Mass Hamiltonian: The fermionic field is rescaled as ψ = φ/a, with M denoting the bare mass and K = 1/(2a) setting the kinetic scale.A further rescaling defines effective parameters m = M/α and κ = K/α.
- Mass Hamiltonian: Gauss’ law constrains the electric fields through charge operators at lattice vertices, leaving only three independent constraints because of charge conservation.The constrained electric Hamiltonian is obtained by eliminating selected electric-field operators.
- Mass Hamiltonian: Rotators and strings are represented as signed sums of link electric fields, analogous to Kirchhoff loop relations, and redundant degrees of freedom are removed.The resulting inverted relations express rotator and string operators through remaining electric fields and charges.
- Mass Hamiltonian: Kinetic pair creation requires four construction rules that raise selected link fields while compensating induced changes on other links through rotators, charge strings, or plaquette operators.The rules cover horizontal and vertical links, including links crossing periodic boundaries.
- Mass Hamiltonian: For boundary pair creation, strings are required because equal and opposite arrow modifications can leave a link field unchanged while maintaining gauge invariance.The periodic-lattice construction is illustrated by the four panels of Figure 7.
C Diagonalisation of the magnetic gauge field Hamiltonian
The magnetic gauge-field Hamiltonian is diagonalised by exploiting the cyclant-matrix structure of the Z2L+1 lowering operators. A discrete Fourier transform provides the eigenbasis and extends componentwise across the plaquette subsystems.
- C Diagonalisation of the magnetic gauge field Hamiltonian: The magnetic Hamiltonian consists of lowering and raising operators that are cyclant matrices in the Z2L+1 group.These operators can therefore be diagonalised exactly before truncation.
- C Diagonalisation of the magnetic gauge field Hamiltonian: The lowering-operator spectrum is indexed by k = −L, ..., L, with eigenvectors obtained from the discrete Fourier transform.The same diagonalisation applies to the corresponding gauge-field operators.
- C Diagonalisation of the magnetic gauge field Hamiltonian: For multiple subsystems, the Fourier transform is the product of transforms over the separate N−1 spaces, with (P^γ)† = P^−γ.The construction uses vector labels r and γ for the subsystem bases.
D Asymptotic behaviour of the ground state expectation value of □
The appendix analyzes how truncation changes the ground-state expectation value of the plaquette observable in electric and magnetic representations. The continuous limit is recovered as the electric truncation grows, while magnetic truncation reshapes the state distribution and observable contributions.
- D Asymptotic behaviour of the ground state expectation value of □: In the magnetic representation, ⟨□(b)⟩ decreases monotonically with increasing l in the strong-coupling regime.This behavior is interpreted using a Riemann sum over discrete magnetic-Hamiltonian eigenvalues.
- D Asymptotic behaviour of the ground state expectation value of □: In the electric representation, the plaquette expectation approaches 1 at g = 0 when the Hamiltonian is untruncated.This follows from the maximum eigenvalue λmax = 2.
- D Asymptotic behaviour of the ground state expectation value of □: Increasing the electric truncation level l monotonically increases ⟨□(e)(g = 0)⟩, with the continuous limit reached as l →∞.The truncated U(1) representation approaches the true value from below for g ≪1.
- D Asymptotic behaviour of the ground state expectation value of □: For l < L, truncation shifts ground-state weight toward low-|r| states, enhancing positive Riemann-sum contributions while removing negative contributions from large |r|.The truncated distribution remains center-symmetric rather than uniform.
E Truncation effects in the strong coupling regime
This section isolates strong-coupling truncation effects in the magnetic representation by comparing the full cyclic Z2L+1 structure with versions that remove cyclic terms. The resulting ground-state distribution shifts toward lower |r| states, and cyclic terms substantially affect the distribution.
- E Truncation effects in the strong coupling regime: In the strong-coupling limit g →∞, the electric term dominates the magnetic representation, so the magnetic Hamiltonian can be neglected for the truncation analysis.The analysis uses an ansatz for the truncated electric-field Hamiltonian.
- E Truncation effects in the strong coupling regime: The truncation is analyzed by decomposing each untruncated plaquette operator into cyclic and noncyclic contributions, then collecting their effects in an auxiliary operator V.V is introduced specifically to study truncation effects.
- E Truncation effects in the strong coupling regime: For l = 7 and L = 8, truncation shifts ground-state population from high-|r| states toward lower-|r| states.The comparison is shown against amplitudes from the untruncated Hamiltonian.
- E Truncation effects in the strong coupling regime: Removing only the cyclic elements of the plaquette operators strongly changes the ground-state distribution because those elements produce uniform coefficients in the untruncated case.Figure 8 compares the full truncation with removal of only the cyclic contributions.
- E Truncation effects in the strong coupling regime: At g−2 = 100, the preferred L minimizes sequence infidelity by balancing available domain against resolution, although this minimum need not be global.The compromise is evaluated for l = 2, 3, ..., 10.
F Numerical determination of Lopt
Sequence infidelity reveals an optimum truncation parameter L_opt through a visible kink, although the kink is not always the global minimum.
- F Numerical determination of Lopt: For g^-2 = 100, sequence infidelity is plotted as a function of l and L for the pure gauge QED ground state.The fidelity calculation involves optimization over L.
- F Numerical determination of Lopt: A visible kink occurs at L = L_opt for every value of l.This kink identifies the optimized truncation scale in the plotted sequence infidelity.
- F Numerical determination of Lopt: The kink is not always a global minimum, notably for points with l = 2, indicating the freezing effect.