Simulation Complexity of Many-Body Localized Systems

TL;DR

This paper investigates the classical simulation complexity of many-body localized systems, showing efficient algorithms for short times and proving hardness for long times, revealing a transition in simulation difficulty.

Contribution

It introduces a quasipolynomial-time algorithm for simulating 1D MBL systems and proves that sampling becomes hard after exponential evolution time, highlighting a complexity transition.

Findings

Efficient quasipolynomial simulation for polynomial time evolution.

Sampling becomes computationally hard after exponential time.

Quantum circuit complexity for MBL systems is sublinear in time.

Abstract

We use complexity theory to rigorously investigate the difficulty of classically simulating evolution under many-body localized (MBL) Hamiltonians. Using the defining feature that MBL systems have a complete set of quasilocal integrals of motion (LIOMs), we demonstrate a transition in the classical complexity of simulating such systems as a function of evolution time. On one side, we construct a quasipolynomial-time tensor-network-inspired algorithm for strong simulation of 1D MBL systems (i.e., calculating the expectation value of arbitrary products of local observables) evolved for any time polynomial in the system size. On the other side, we prove that even weak simulation, i.e. sampling, becomes formally hard after an exponentially long evolution time, assuming widely believed conjectures in complexity theory. Finally, using the consequences of our classical simulation results, we also show that the quantum circuit complexity for MBL systems is sublinear in evolution time. This result is a counterpart to a recent proof that the complexity of random quantum circuits grows linearly in time.