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Efficient Finite-Resource Formulation of Non-Abelian Lattice Gauge Theories beyond One Dimension

Pierpaolo Fontana1,2,*, Marc Miranda-Riaza1, and Alessio Celi1,†

  • *Contact author: Pierpaolo.Fontana@uab.cat
  • †Contact author: Alessio.Celi@uab.cat
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Phys. Rev. X 15, 031065 – Published 9 September, 2025

DOI: https://doi.org/10.1103/k9p6-c649

Abstract

Non-Abelian gauge theories provide the most accurate description of fundamental interactions, showing remarkable agreement with experimental data in cosmology and particle physics. Highly precise predictions can be made using standard techniques, both in the continuum and in the lattice frameworks. However, classical methods have limitations, particularly when attempting to extrapolate the continuum limit from the study of lattice gauge theories. Complementing classical computations or combining them with quantum computational methods, to improve the predictions toward the continuum limit with current quantum resources, is a formidable open challenge. In this paper, we propose a resource-efficient method to compute the running of the coupling in non-Abelian gauge theories beyond one spatial dimension. We first represent the Hamiltonian on periodic lattices in terms of loop variables and conjugate loop electric fields, exploiting the Gauss law to retain the gauge-independent ones. Then, we identify a local basis for small and large loops variationally to minimize the truncation error while computing the running of the coupling on small tori. Our method enables computations at arbitrary values of the bare coupling and lattice spacing with current quantum computers, simulators, and tensor-network calculations, in regimes otherwise inaccessible.

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Popular Summary

Non-Abelian gauge theories are fundamental to explaining how particles interact, especially the phenomenon of confinement, where charged particles cannot be separated due to extremely strong interactions. These theories are typically studied using lattice models and Monte Carlo simulations on classical computers, where predictions toward the continuum limit are obtained by progressively reducing the lattice spacing. However, some problems remain out of reach for classical methods. In this study, we develop a resource-efficient formulation of non-Abelian lattice gauge theories that works across all interaction strengths and can be implemented on quantum computers to enhance predictions.

The challenge in simulating these theories on quantum computers lies in representing gauge fields, which require an infinitely large vector on each lattice link. Current quantum machines, however, can handle only a limited number of components. We solve this by first reducing the number of necessary vectors using gauge symmetry and then rotating them based on interaction strength to approximate the original state with fewer components. We fine-tune this rotation using variational techniques, making calculations that were previously too demanding now possible with limited quantum resources.

Our approach lays the groundwork for using quantum computers to explore gauge theories in regimes that are difficult or impossible for classical simulations alone. Looking ahead, this technique could be extended to even more complex systems and used in future quantum devices, including those based on multilevel versions of qubits known as qudits.

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Phys. Rev. X 15, 031065– Published 9 September, 2025
  • Received 17 September 2024
  • Revised 28 April 2025
  • Accepted 24 July 2025
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