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arXiv:2410.14580 · Journal of High Energy Physics

Quantum computation of SU(2) lattice gauge theory with continuous variables

Victor Ale, Nora M. Bauer, Raghav G. Jha, Felix Ringer, George Siopsis

Published Jun 11, 2025• 9 citations
arXivHTMLPDF

Abstract

A bstract We present a quantum computational framework for pure SU(2) lattice gauge theory, using continuous variables instead of discrete qubits to represent the infinite-dimensional Hilbert space of the gauge fields. We consider a ladder as well as a two-dimensional grid of plaquettes, detailing the use of gauge fixing to reduce the degrees of freedom and simplify the Hamiltonian. We demonstrate how system dynamics, ground states, and energy gaps can be computed using the continuous-variable approach to quantum computing. Our results indicate that it is feasible to study non-Abelian gauge theories with continuous variables, providing new avenues for understanding the real-time dynamics of quantum field theories.

DOI
10.1007/jhep06(2025)084

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A bstract We present a quantum computational framework for pure SU(2) lattice gauge theory, using continuous variables instead of discrete qubits to represent the infinite-dimensional Hilbert space of the gauge fields. We consider a ladder as well as a two-dimensional grid of plaquettes, detailing the use of gauge fixing to reduce the degrees of freedom and simplify the Hamiltonian. We demonstrate how system dynamics, ground states, and energy gaps can be computed using the continuous-variable approach to quantum computing. Our results indicate that it is feasible to study non-Abelian gauge theories with continuous variables, providing new avenues for understanding the real-time dynamics of quantum field theories.
Quantum simulations of many-body systems offer novel methods for probing the dynamics of the Standard Model and its constituent gauge theories. Extracting low-energy predictions from such simulations rely on formulating systematically-improvable representations of lattice gauge theory Hamiltonians that are efficient at all values of the gauge coupling. One such candidate representation for SU(2) is the fully gauge-fixed Hamiltonian defined in the mixed basis. This work focuses on the quantum simulation of the smallest non-trivial system: two plaquettes with open boundary conditions. A mapping of the continuous gauge field degrees of freedom to qubit-based representations is developed. It is found that as few as three qubits per plaquette is sufficient to reach per-mille level precision on predictions for observables. Two distinct algorithms for implementing time evolution in the mixed basis are developed and analyzed in terms of quantum resource estimates. One algorithm has favorable scaling in circuit depth for large numbers of qubits, while the other is more practical when qubit count is limited. The latter algorithm is used in the measurement of a real-time observable on IBM's Heron superconducting quantum processor, ibm_fez. The quantum results match classical predictions at the percent-level. This work lays out a path forward for two- and three-dimensional simulations of larger systems, as well as demonstrating the viability of mixed-basis formulations for studying the properties of SU(2) gauge theories at all values of the gauge coupling.
The development of quantum technology and the advent of quantum computation have renewed interest in Hamiltonian formulations of lattice gauge theories. In contrast to Euclidean Monte Carlo calculations, Hamiltonian simulation requires a direct representation of the physical Hilbert space, an efficient treatment of enforcing Gauss' law as constraints, and a formulation of real-time dynamics in terms of degrees of freedom that can be encoded on quantum or quantum-inspired classical platforms. The starting point corresponds to finding a suitable basis for Hamiltonian lattice gauge theories. The existing literature focuses on several complementary approaches: electric basis, magnetic basis, gauge-fixed, quantum-link, qubit-regularised, finite-group, and orbifold formulations. This article primarily focuses on the prepotential and loop--string--hadron formulations, where non-Abelian Gauss laws are solved locally and the dynamics is written directly in terms of gauge-invariant loop, string and hadron excitations. The discussion covers the SU(2) and SU(3) constructions in one spatial dimension, the role of residual Abelian Gauss laws, and the extension to higher dimensions through point splitting. We conclude by highlighting its direct applications: eliminating redundant gauge degrees of freedom makes exact diagonalisation and tensor network calculations feasible for studying real-time dynamics, provides insights into thermalisation properties, enables mapping of physical degrees of freedom onto analog and digital quantum hardware, and enables quantum simulation of dynamics.
Quantum simulations of the dynamics of QCD have been limited by the complexities of mapping the continuous gauge fields onto quantum computers. By parametrizing the gauge invariant Hilbert space in terms of plaquette degrees of freedom, we show how the Hilbert space and interactions can be expanded in inverse powers of N_{c}. At leading order in this expansion, the Hamiltonian simplifies dramatically, both in the required size of the Hilbert space as well as the type of interactions involved. Adding a truncation of the resulting Hilbert space in terms of local energy states we give explicit constructions that allow simple representations of SU(3) gauge fields on qubits and qutrits. This formulation allows a simulation of the real time dynamics of a SU(3) lattice gauge theory on a 5×5 and 8×8 lattice on ibm_torino with a CNOT depth of 113.
A dynamical quantum simulation of SU(2) non-Abelian gauge field theory on a digital quantum computer is presented. This was enabled on current quantum hardware by introducing a mapping of the field onto a register of qubits that utilizes local gauge symmetry while preserving local constraints on the fields, reducing the dimensionality of the calculation. Controlled plaquette operators and gauge-variant completions in the unphysical part of the Hilbert space were designed and used to implement time evolution. The new techniques developed in this work generalize to quantum simulations of higher dimensional gauge field theories.
Simulating lattice gauge theories on quantum computers presents unique challenges that drive the development of novel theoretical frameworks. The orbifold lattice approach offers a scalable method for simulating SU(NNN) gauge theories in arbitrary dimensions. In this work, we present three improvements: (i) two new simplified Hamiltonians, (ii) an encoding of the SU(2) theory with smaller number of qubits, and (iii) a reduction in the requirement for large scalar masses to reach the Kogut-Susskind limit, achieved via the inclusion of an additional term in the Hamiltonian. These advancements significantly reduce circuit depth and qubit requirements for quantum simulations. We benchmarked these improvements using Monte Carlo simulations of SU(2) in (2+1) dimensions. Preliminary results demonstrate the effectiveness of these developments and further validate the use of noncompact variables as a promising framework for scalable quantum simulations of gauge theories.
Pure gauge theories in more than one spatial dimension are extremely interesting as they allow pure gauge observables such as glueballs and Wilson loops. Simulating the real-time dynamics of pure non-Abelian lattice gauge theories poses a significant computational challenge due to the large number of spurious gauge degrees of freedom. This work presents a brute-force (exact diagonalization) study of 2+1-dimensional SU(2) lattice gauge theory, leveraging the gauge-invariant Loop-String-Hadron (LSH) framework. Using only CPU-based computation, we have successfully demonstrated classical simulation on systems of up to 9 plaquettes and aim to push it further by leveraging combinations of CPUs and GPUs. This work provides crucial benchmarks for GPU implementations, near-term tensor network simulators, and quantum simulations, opening a new avenue for studying fundamental phenomena like thermalization in a non-Abelian gauge theory directly from the Hamiltonian perspective in higher dimensions.