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International Symposium on Lattice Field Theory (LATTICE)

A tensorial toolkit for quantum computing in lattice gauge theory

Yannick Meurice

Published May 29, 2019• 2 citations
Publisher

Abstract

In most lattice simulations, the variables of integration are compact and character expansion (for instance Fourier analysis for U(1) models) can be used to rewrite the partition function and average observables as discrete sums of contracted tensors. These reformulations have been used for RG blocking but they also naturally fit the needs of quantum computing. We discuss FAQ about tensorial reformulations: effects of truncations on symmetries, boundary conditions, Grassmann variables, and other recent aspects of quantum computing in the same context.

DOI
10.22323/1.334.0231

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In most lattice simulations, the variables of integration are compact and character expansion (for instance Fourier analysis for U(1) models) can be used to rewrite the partition function and average observables as discrete sums of contracted tensors. These reformulations have been used for RG blocking but they also naturally fit the needs of quantum computing. We discuss FAQ about tensorial reformulations: effects of truncations on symmetries, boundary conditions, Grassmann variables, and other recent aspects of quantum computing in the same context.
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We discuss recent progress in Tensor Lattice Field Theory and economical, symmetry preserving, truncations suitable for quantum computations/simulations. We focus on spin and gauge models with continuous Abelian symmetries such as the Abelian Higgs model and emphasize noise-robust implementations of Gauss’s law. We discuss recent progress concerning the comparison between field digitizations and character expansions, symmetry breaking in tensor language, wave-packet preparation and possible new implementations of Abelian models using Rydberg atoms.
One of the methods proposed in recent years for studying nonperturbative gauge theory physics is quantum simulation, where lattice gauge theories are mapped onto quantum devices which can be built in the laboratory, or quantum computers. While being very promising and already showing some experimental results, these methods still face several challenges related to the interface between the technological capabilities and the demands of the simulated models; in particular, one such challenge is the need to simulate infinitely dimensional local Hilbert spaces, describing the gauge fields on the links in the case of compact Lie gauge groups, requiring some truncations and approximations which are not completely understood or controllable in the general case. This work proposes a way to obtain arbitrarily large such local Hilbert spaces by using coarse graining of simple, low-dimensional qubit systems, made of components available on most quantum simulation platforms, and thus opening the way to new types of lattice gauge theory quantum simulations. Published by the American Physical Society 2024
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We have recently shown that the partition function of any classical spin model, including all discrete standard statistical models and all Abelian discrete lattice gauge theories, can be expressed as the specific instance of the partition function of a four dimensional Ising lattice gauge theory. This unifies models with apparently very different features (global vs. local symmetries, many body interactions, different dimensions, etc) in a single complete model. The proof of the result uses techniques from quantum information. In this talk we review these mappings and discuss recent developments, including quantum algorithms to approximate the partition function.