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arXiv:2301.12224 · Physical review. D/Physical review. D.

Hamiltonians and gauge-invariant Hilbert space for lattice Yang-Mills-like theories with finite gauge group

A. Mariani, Sunny Pradhan, Elisa Ercolessi

Published Jun 20, 2023• 20 citations
arXivHTMLPDF

Abstract

Motivated by quantum simulation, we consider lattice Hamiltonians for Yang-Mills gauge theories with finite gauge group, for example a finite subgroup of a compact Lie group. We show that the electric Hamiltonian admits an interpretation as a certain natural, nonunique Laplacian operator on the finite Abelian or non-Abelian group and derive some consequences from this fact. Independent of the chosen Hamiltonian, we provide a full explicit description of the physical, gauge-invariant Hilbert space for pure gauge theories and derive a simple formula to compute its dimension. We illustrate the use of the gauge-invariant basis to diagonalize a dihedral gauge theory on a small periodic lattice.

DOI
10.1103/physrevd.107.114513

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Motivated by quantum simulation, we consider lattice Hamiltonians for Yang-Mills gauge theories with finite gauge group, for example a finite subgroup of a compact Lie group. We show that the electric Hamiltonian admits an interpretation as a certain natural, nonunique Laplacian operator on the finite Abelian or non-Abelian group and derive some consequences from this fact. Independent of the chosen Hamiltonian, we provide a full explicit description of the physical, gauge-invariant Hilbert space for pure gauge theories and derive a simple formula to compute its dimension. We illustrate the use of the gauge-invariant basis to diagonalize a dihedral gauge theory on a small periodic lattice.
We consider the problem of the explicit description of the gauge-invariant subspace of pure lattice gauge theories in the Hamiltonian formulation, where the gauge group is either a compact Lie group or a finite group. The latter case is particularly interesting for quantum simulation. A basis of states where configurations are grouped according to their holonomies is shown to have several advantages over other descriptions. Using this basis, we compute some properties of interest for some non-Abelian finite groups on small lattices, and in particular we examine the question of whether a certain ansatz introduced long ago is a good approximation for the ground state. Published by the American Physical Society 2024
We construct the local Hamiltonian description of the Chern-Simons theory with discrete non-Abelian gauge group on a lattice. We show that the theory is fully determined by the phase factors associated with gauge transformations and classify all possible non-equivalent phase factors. We also construct the gauge invariant electric field operators that move fluxons around and create/anihilate them. We compute the resulting braiding properties of the fluxons. We apply our general results to the simplest class of non-Abelian groups, dihedral groups D_n.
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.
We examine the Kogut-Susskind formulation of lattice gauge theories under the light of fermionic and bosonic degrees of freedom that provide a description useful to the development of quantum simulators of gauge-invariant models. We consider both discrete and continuous gauge groups and adopt a realistic multicomponent Fock space for the definition of matter degrees of freedom. In particular, we express the Hamiltonian of the gauge theory and the Gauss law in terms of Fock operators. The gauge fields are described in two different bases based on either group elements or group representations. This formulation allows for a natural scheme to achieve a consistent truncation of the Hilbert space for continuous groups, and provides helpful tools to study the connections of gauge theories with topological quantum double and string-net models for discrete groups. Several examples, including the case of the discrete D3{D}_{3}D3​ gauge group, are presented.
We present an overview of a program to understand the low-energy physics of quantum Yang-Mills theory from a quantum-information perspective. Our setting is that of the Hamiltonian formulation of pure Yang-Mills theory in the temporal gauge on the lattice. Firstly, inspired by recent constructions for Z/2Z\mathbb{Z}/2\mathbb{Z}Z/2Z lattice gauge theory, in particular, Kitaev's toric code, we describe the gauge-invariant sector of Hilbert space by introducing a primitive quantum gate: the quantum parallel transporter. We then develop a non-Abelian generalization of Laplace interpolation to present an ansatz for the ground state of pure Yang-Mills theory which interpolates between the weak- and strong-coupling renormalization group fixed points. The resulting state acquires the structure of a tensor network, namely, a multiscale entanglement renormalization ansatz, and allows for the efficient computation of local observables and Wilson loops. Various refinements of the tensor network are discussed leading to several generalizations. Finally, the continuum limit of our ansatz as the lattice regulator is removed is then described. This paper is intended as an abstract for an ongoing program: there are still many open problems.
Quantum computers promise to simulate complex quantum field theories that are intractable for classical computers, potentially revealing new physics in strongly interacting systems. Current approaches for simulating Yang-Mills gauge theories face significant technical barriers due to complex group structures and complicated truncation schemes that vary drastically between different theories. Here we show that the orbifold lattice formulation provides a universal framework for quantum simulation of Yang-Mills theories with arbitrary gauge groups and dimensions. Our approach reduces all theories to the same simple Hamiltonian form, enabling implementation with standard quantum gates regardless of system complexity. We demonstrate explicit quantum circuits using only controlled-NOT and single-qubit operations, with concrete resource estimates for time evolution algorithms. This universal framework simplifies quantum simulation of gauge theories and enables systematic scaling to larger systems on fault-tolerant quantum computers.