Limits of Short-Time Evolution of Local Hamiltonians

TL;DR

This paper proves limitations on short-time evolutions of local Hamiltonians, showing they produce concentrated measurement distributions and require logarithmic runtime for quantum annealing to outperform classical algorithms on MaxCut.

Contribution

It establishes new bounds on short-time local Hamiltonian evolutions, including a Lieb-Robinson bound for time-dependent cases, and applies these to quantum optimization.

Findings

Short-time evolutions are concentrated and satisfy isoperimetric inequalities.

Quantum annealing needs at least logarithmic runtime to outperform classical MaxCut algorithms.

A new Lieb-Robinson bound for time-dependent Hamiltonians was proven.

Abstract

Evolutions of local Hamiltonians in short times are expected to remain local and thus limited. In this paper, we validate this intuition by proving some limitations on short-time evolutions of local time-dependent Hamiltonians. We show that the distribution of the measurement output of short-time (at most logarithmic) evolutions of local Hamiltonians are \emph{concentrated} and satisfy an \emph{isoperimetric inequality}. To showcase explicit applications of our results, we study the \textsc{MaxCut} problem and conclude that quantum annealing needs at least a run-time that scales logarithmically in the problem size to beat classical algorithms on \textsc{MaxCut}. To establish our results, we also prove a Lieb-Robinson bound that works for time-dependent Hamiltonians which might be of independent interest.