Operator backflow and the classical simulation of quantum transport

Abstract

Tensor product states have proved extremely powerful for simulating the area-law entangled states of many-body systems, such as the ground states of gapped Hamiltonians in one dimension. The applicability of such methods to the dynamics of many-body systems is less clear: The memory required grows exponentially in time in most cases, quickly becoming unmanageable. New methods reduce the memory required by selectively discarding/dissipating parts of the many-body wave function which are expected to have little effect on the hydrodynamic observables typically of interest: For example, some methods discard fine-grained correlations associated with $n$-point functions, with $n$ exceeding some cutoff ${\ensuremath{\ell}}_{*}$. In this paper, we present a theory for the sizes of backflow corrections, i.e., systematic errors due to discarding this fine-grained information. In particular, we focus on their effect on transport coefficients. Our results suggest that backflow corrections are exponentially suppressed in the size of the cutoff ${\ensuremath{\ell}}_{*}$. Moreover, the backflow errors themselves have a hydrodynamical expansion, which we elucidate. We test our predictions against numerical simulations run on random unitary circuits and ergodic spin chains. These results lead to the conjecture that transport coefficients in ergodic diffusive systems can be captured to a given precision $\ensuremath{\epsilon}$ with an amount of memory scaling as $exp[\mathcal{O}(log{(\ensuremath{\epsilon})}^{2})]$, significantly better than the naive estimate of memory $exp[\mathcal{O}(\mathrm{poly}({\ensuremath{\epsilon}}^{\ensuremath{-}1}))]$ required by more brute-force methods.