On entropy growth and the hardness of simulating time evolution

TL;DR

This paper proves that simulating quantum time evolution with classical methods is inherently hard due to entropy growth, even in symmetric systems, confirming the exponential resource requirement for accurate classical simulation.

Contribution

It provides a rigorous proof that local Hamiltonians cause linear entropy growth, establishing fundamental limits on classical simulation efficiency.

Findings

Local Hamiltonians induce linear entropy increase in translational invariant systems.

Exponential resources are necessary for classical simulation of quantum evolution.

Symmetry does not simplify the classical simulation complexity.

Abstract

The simulation of quantum systems is a task for which quantum computers are believed to give an exponential speedup as compared to classical ones. While ground states of one-dimensional systems can be efficiently approximated using Matrix Product States (MPS), their time evolution can encode quantum computations, so that simulating the latter should be hard classically. However, one might believe that for systems with high enough symmetry, and thus insufficient parameters to encode a quantum computation, efficient classical simulation is possible. We discuss supporting evidence to the contrary: We provide a rigorous proof of the observation that a time independent local Hamiltonian can yield a linear increase of the entropy when acting on a product state in a translational invariant framework. This criterion has to be met by any classical simulation method, which in particular implies that every global approximation of the evolution requires exponential resources for any MPS based method.