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arXiv:2610.23681

Impact of Irrep Truncations on Quantum Simulations of QCD

Patrick Draper, Drishti Gupta

hep-lathep-phquant-phSubmitted Oct 16, 2026• Announced Oct 19, 2026• 0 citations
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Abstract

Quantum simulations of lattice gauge theories with continuous gauge groups require a choice of basis (how states of the gauge theory map to the qubit computational basis) and truncation scheme (which of the infinitely many gauge theory basis states should be omitted in the map to the finite dimensional qubit Hilbert space.) The most common choices are adapted to an electric basis for the gauge theory, where the basis states for each link are associated with irreducible representations of the gauge group. Electric bases and truncations are generally advantageous for preserving gauge symmetries and simulating the strong coupling regime, but their effectiveness in the weak coupling regimes of physical interest are less well understood. In this work, we compute relative probabilities $p(R')/p(R)$ for finding a link in the irreps $R,R'$ in the ground state of SU(3) lattice gauge theory using Euclidean Monte Carlo methods and weak-coupling lattice perturbation theory. The observables that compute these relative probabilities are constructed by introducing seams into an anisotropic lattice, and by a similar construction we are also able to measure relative probabilities of different site singlets (ways that electric flux braids together into a gauge singlet) at each vertex. With these methods we can quantify, for example, the importance of higher electric flux states in the weak coupling ground state. These results directly inform what truncations are needed to accurately capture non-abelian gauge dynamics near the continuum limit.

Comments
38 pages, 7 figures, 2 tables

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Quantum simulations of lattice gauge theories with continuous gauge groups require a choice of basis (how states of the gauge theory map to the qubit computational basis) and truncation scheme (which of the infinitely many gauge theory basis states should be omitted in the map to the finite dimensional qubit Hilbert space.) The most common choices are adapted to an electric basis for the gauge theory, where the basis states for each link are associated with irreducible representations of the gauge group. Electric bases and truncations are generally advantageous for preserving gauge symmetries and simulating the strong coupling regime, but their effectiveness in the weak coupling regimes of physical interest are less well understood. In this work, we compute relative probabilities p(R′)/p(R)p(R')/p(R)p(R′)/p(R) for finding a link in the irreps R,R′R,R'R,R′ in the ground state of SU(3) lattice gauge theory using Euclidean Monte Carlo methods and weak-coupling lattice perturbation theory. The observables that compute these relative probabilities are constructed by introducing seams into an anisotropic lattice, and by a similar construction we are also able to measure relative probabilities of different site singlets (ways that electric flux braids together into a gauge singlet) at each vertex. With these methods we can quantify, for example, the importance of higher electric flux states in the weak coupling ground state. These results directly inform what truncations are needed to accurately capture non-abelian gauge dynamics near the continuum limit.
We study methods for efficient preparation of approximate ground states of S U ( 3 ) lattice gauge theory on quantum hardware. Working in a variant of the electric basis, we introduce a refinement of the irrep truncation based on the energy density of site singlets, which provides a finer gradation of simulation complexity. Using strong-coupling perturbation theory as a guide, we develop simple ansatz circuits for ground state preparation and test them via classical simulation on small lattices, including the 2 × 2 plaquette lattice in d = 2 and the cube in d = 3 . We contrast state fidelities and resource requirements of variational methods against adiabatic state preparation and introduce a method that hybridizes the two approaches. Finally, we report on the public release of —a package of tools for building S U ( 3 ) circuits and processing measurements—and , a package for efficiently computing S U ( N ) Clebsch–Gordan coefficients.
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.
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+12+12+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.
Due to rapidly improving quantum computing hardware, Hamiltonian simulations of relativistic lattice field theories have seen a resurgence of attention. This computational tool requires turning the formally infinite-dimensional Hilbert space of the full theory into a finite-dimensional one. For gauge theories, a widely-used basis for the Hilbert space relies on the representations induced by the underlying gauge group, with a truncation that keeps only a set of the lowest dimensional representations. This works well at large bare gauge coupling, but becomes less efficient at small coupling, which is required for the continuum limit of the lattice theory. In this work, we develop a new basis suitable for the simulation of an SU(2) lattice gauge theory in the maximal tree gauge. In particular, we show how to perform a Hamiltonian truncation so that the eigenvalues of both the magnetic and electric gauge-fixed Hamiltonian are mostly preserved, which allows for this basis to be used at all values of the coupling. Little prior knowledge is assumed, so this may also be used as an introduction to the subject of Hamiltonian formulations of lattice gauge theories.
Deforming the Trail: Baseline Quantum Circuitry for SU(2)k\text{SU(2)}_kSU(2)k​ Lattice Gauge Theory
Quantifying quantum resources for simulating the fundamental forces of Nature is sensitive to the mapping of gauge fields onto finite quantum computational architectures. When locally truncating lattice gauge theories in the irreducible representation basis, it has been proposed to further deform the theory via quantum groups. The purpose of this deformation is (1) to provide an infinite tower of finite-dimensional (d=k+1d = k+1d=k+1) groups systematically approximating the infinite-dimensional gauge links and (2) to restore the physical unitarity of a plaquette operator diagonalization procedure analytically derived from the field continuum by recontracting vertex pairs. For the SU(2)k_kk​ Yang-Mills pure-gauge theory, we provide a constructive strategy of gauge-variant completions to extend this unitarity to the entire computational Hilbert space, leading to well-defined time evolution unitaries as targets for optimized circuit synthesis. Leveraging basic circuit decompositions and symmetries of the diagonalized plaquette operator, we report resource upper-bounds on the generalized-controlled-X two-qudit gates for arbitrary local truncation ddd, reducing estimates and scaling relative to the non-deformed theory by three polynomial powers from O(d8)O(d^8)O(d8) to O(d5)O(d^5)O(d5). Examining the stronger q-deformed gauge constraint, which softens the total flux at vertices, we show that the physical Hilbert space dimension of the deformed plaquette operator scales equivalently to its non-deformed counterpart with a constant factor 0.2563(5)0.2563(5)0.2563(5). Thus, despite affecting interactions at all scales as exemplified by the observed flux hierarchy inversion symmetry, q-deformation continues to pass scrutiny as a reliable truncation offering advantages in quantum circuit synthesis.
Due to rapidly improving quantum computing hardware, Hamiltonian simulations of relativistic lattice field theories have seen a resurgence of attention. This computational tool requires turning the formally infinite-dimensional Hilbert space of the full theory into a finite-dimensional one. For gauge theories, a widely used basis for the Hilbert space relies on the representations induced by the underlying gauge group, with a truncation that keeps only a set of the lowest dimensional representations. This works well at large bare gauge coupling, but becomes less efficient at small coupling, which is required for the continuum limit of the lattice theory. In this work, we develop a new basis suitable for the simulation of an SU(2) lattice gauge theory in the maximal tree gauge. In particular, we show how to perform a Hamiltonian truncation so that the eigenvalues of both the magnetic and electric gauge-fixed Hamiltonian are mostly preserved, which allows for this basis to be used at all values of the coupling. Little prior knowledge is assumed, so this may also be used as an introduction to the subject of Hamiltonian formulations of lattice gauge theories. Published by the American Physical Society 2024