Simulation Complexity of Many-Body Localized Systems

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-networkinspired 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.

Computational advantage of quantum random sampling

“…Here, the idea is to vary a physical parameter of the system, in this case, the spacing between bosons in the initial state and consider the complexity as a function of time when evolving the system. In a similar vein, Ehrenberg et al (2022) study transitions in the complexity of sampling from the output distribution of many-body-localizing time evolution. In such approaches, the hope is to narrow down and better understand the physical mechanisms underlying sampling complexity.…”