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arXiv:2404.12158

Protecting gauge symmetries in the the dynamics of SU(3) lattice gauge theories

Mathew, Emil, Indrakshi Raychowdhury

Published Apr 18, 2024• 0 citations
arXivHTMLPDF

Abstract

Quantum simulation of the dynamics of a lattice gauge theory demands imposing on-site constraints. Ideally, the dynamics remain confined within the physical Hilbert space, where all the states satisfy those constraints. For non-Abelian gauge theories, implementing local constraints is non-trivial, as is keeping the dynamics confined in the physical Hilbert space, considering the erroneous quantum devices. SU(3) gauge group, albeit crucial for describing the strong interaction of nature, is notorious for studying via Hamiltonian simulation. This work presents a couple of symmetry protection protocols for simulating the exact dynamics of SU(3) gauge theory in $1+1$ dimension. The first protocol doesn't require imposing any local symmetry but relies on protecting global symmetries, which are Abelian with a preferred choice of framework, namely the loop-string-hadron framework. Generalization to a higher dimension is possible, however, the protection scheme needs to be local for that case but is still Abelian and thus advantageous. The symmetry protection schemes presented here are important steps towards quantum simulating the full theory of quantum chromodynamics.

DOI
10.48550/arxiv.2404.12158

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Quantum simulation of the dynamics of a lattice gauge theory demands imposing on-site constraints. Ideally, the dynamics remain confined within the physical Hilbert space, where all the states satisfy those constraints. For non-Abelian gauge theories, implementing local constraints is non-trivial, as is keeping the dynamics confined in the physical Hilbert space, considering the erroneous quantum devices. SU(3) gauge group, albeit crucial for describing the strong interaction of nature, is notorious for studying via Hamiltonian simulation. This work presents a couple of symmetry protection protocols for simulating the exact dynamics of SU(3) gauge theory in 1+11+11+1 dimension. The first protocol doesn't require imposing any local symmetry but relies on protecting global symmetries, which are Abelian with a preferred choice of framework, namely the loop-string-hadron framework. Generalization to a higher dimension is possible, however, the protection scheme needs to be local for that case but is still Abelian and thus advantageous. The symmetry protection schemes presented here are important steps towards quantum simulating the full theory of quantum chromodynamics.
Abstract Quantum simulation of a lattice gauge theory demands imposing on-site constraints. Ideally, the dynamics remain confined within the physical Hilbert space, where all the states satisfy those constraints. For a non-Abelian gauge theory, implementing these local Gauss’ law constraints is non-trivial. The presence of noise in current quantum devices further complicates efforts to confine the theory to the physical Hilbert space. The SU(3) gauge theory, describing the strong interaction of nature contains 8 mutually non-commuting local constraints. An efficient Hamiltonian simulation for the same should preserve all of these simultaneously - which stands as a notoriously difficult task. In this work, we explore two symmetry protection protocols for simulating SU(3) gauge theory in 1+1 dimensions. The first protocol does not require the imposition of any local symmetry but relies on protecting global symmetries, which are Abelian within the preferred choice of framework, namely the loop-string-hadron framework. The second protocol employs a protection scheme that is local, Abelian, and generalizable to higher dimensions. The symmetry protection schemes presented here are important steps towards quantum simulating the full theory of Quantum Chromodynamics.
Efficient quantum simulation protocols for any quantum theories demand efficient protection protocols for its underlying symmetries. This task is nontrivial for gauge theories as it involves local symmetry/invariance. For non-Abelian gauge theories, protecting all the symmetries generated by a set of mutually noncommuting generators, is particularly difficult. In this paper, a global symmetry-protection protocol is proposed. Using the novel loop-string-hadron formalism of non-Abelian lattice gauge theory, we numerically demonstrate that all of the local symmetries get protected even for large time by this global symmetry protection scheme. With suitable protection strength, the dynamics of a (1+11+11+1)-dimensional SU(2) lattice gauge theory remains confined in the physical Hilbert space of the theory even in presence of explicit local symmetry violating terms in the Hamiltonian that may occur in both analog and digital simulation schemes as an error. The whole scheme holds for SU(3) gauge theory as well.
Within the aim of understanding quantum chromodynamics through simulation, an increasingly studied approach is that of quantum computation and simulation. Challenges exist in encoding the minimal and physical degrees of freedom for a non-Abelian gauge theory and maintaining physical or gauge-invariant dynamics in a simulation. In this work, the Loop-String-Hadron (LSH) formulation of the 1+1-dimensional SU(3) lattice gauge theory is used to define an efficient mapping of SU(3) invariant degrees of freedom onto qubits. It is shown that the required number of qubits in the LSH basis is significantly reduced compared to its IRREP basis counterpart. While the non-Abelian Gauss laws of the SU(3) theory are automatically satisfied by the usage of LSH variables, the remnant constraints on the consistency of the flux numbers still exist. During time evolution, the noise can accumulate and take the state out of the sector of the Hilbert space where the constraint is satisfied. With this motivation, an oracle algorithm is constructed to be applied to the qubits for checking the local constraint at a given link. Costs in terms of qubit and gate number, and circuit depth, are found.
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.
Simulation of quantum field theories and fundamental interactions are one of the most challenging tasks in modern particle physics. Classical computers generally fail to reproduce accurate results when it comes to strongly coupled theories such as QCD. Recent developments in quantum technologies open up the possibility of simulating such physical regimes by using quantum computers. In this paper, we study the quantum resource related to the simulability of a quantum theory, i.e. non-stabilizerness for Lattice Gauge Theory (LGT) with discrete symmetry gauge groups. We show that enforcing gauge constraints for Zl\mathbb{Z}_lZl​ LGTs has no cost in terms of this resource and discuss the relation between non-abelianity of the gauge group with the average non-stabilizerness of the gauge invariant Hilbert space.
Reformulating the Hamiltonian formulation of non-Abelian lattice gauge theories entirely in terms of gauge invariant loop-string-hadron degrees of freedom provides a set of advantages for simulating the theory on quantum hardware and in turn is expected to address a series of physics quests. The framework is manifestly (non-Abelian) gauge invariant, yet possesses a set of remnant Abelian gauge symmetries along with its global symmetry properties. In this talk, we describe all the symmetries of this framework and discuss the advantages/ challenges of the symmetry structure being present/preserved in a Hamiltonian simulation towards understanding real-time dynamics of non-Abelian gauge theories.