Limits of Short-Time Evolution of Local Hamiltonians

Quantum 6, 744 (2022).

https://doi.org/10.22331/q-2022-06-27-744

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 $concentrated$ and satisfy an $textit{isoperimetric inequality}$. To showcase explicit applications of our results, we study the $M$$small{AX}$$C$$small{UT}$ problem and conclude that quantum annealing needs at least a run-time that scales logarithmically in the problem size to beat classical algorithms on $M$$small{AX}$$C$$small{UT}$. To establish our results, we also prove a Lieb-Robinson bound that works for time-dependent Hamiltonians which might be of independent interest.