Overcoming exponential scaling with system size in Trotter-Suzuki implementations of constrained Hamiltonians: 2+1 U(1) lattice gauge theories

TL;DR

This paper presents a method to reduce the quantum resource scaling from exponential to polynomial in simulating constrained Hamiltonians, exemplified by 2+1 U(1) lattice gauge theories, using a redefined operator basis.

Contribution

It introduces a basis redefinition technique that significantly improves the scaling of quantum simulations for constrained systems like lattice gauge theories.

Findings

Scaling is reduced from exponential to polynomial in system size.

Explicit construction of the basis change matrices is provided.

The method applies to systems with magnetic Gauss' law constraints.

Abstract

For many quantum systems of interest, the classical computational cost of simulating their time evolution scales exponentially in the system size. At the same time, quantum computers have been shown to allow for simulations of some of these systems using resources that scale polynomially with the system size. Given the potential for using quantum computers for simulations that are not feasible using classical devices, it is paramount that one studies the scaling of quantum algorithms carefully. This work identifies a term in the Hamiltonian of a class of constrained systems that naively requires quantum resources that scale exponentially in the system size. An important example is a compact U(1) gauge theory on lattices with periodic boundary conditions. Imposing the magnetic Gauss' law a priori introduces a constraint into that Hamiltonian that naively results in an exponentially deep circuit. A method is then developed that reduces this scaling to polynomial in the system size, using a redefinition of the operator basis. An explicit construction of the matrices defining the change of operator basis, as well as the scaling of the associated computational cost, is given.