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 wavefunction 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 * . In this work, 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 * . 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 with an amount of memory scaling as exp î O(log( ) 2 ) ó , significantly better than the naive estimate of memory exp O(poly( −1 )) required by more brute-force methods.

Real-time correlators in chaotic quantum many-body systems

“…As discussed below, this picture of a simple bound phase connects with recent work in Refs. [17] and [16] that appeared after the results in this paper for the random circuit were obtained. The perturbation theory will simplify in any limit where the typical cluster size (ξ above) becomes small: for example in the Haar circuit, this happens when v approaches its maximal value, as can be seen in Eq.…”

“…VIII we discuss applications to more general systems, in particular systems without randomness. We also make connections with recent works on efficient numerical methods for computing correlators [16] and on dual unitary circuits [17]. Finally in Sec.…”

“…At first sight it looks similar to the propagator for Schrodinger evolution of a particle, but a key difference is that K ∆t (x − x) is not unitary. 16 This is because we have effectively projected the full unitary dynamics of the operator string to a restricted subspace of small strings. This leads to exponential decay.…”

“…The fact that the operator trajectories that dominate 2-point functions involve much smaller strings than the trajectories that dominate the OTOC has practical consequences for numerical simulations [16,62]. Refs.…”

Scrambling Dynamics and Out-of-Time Ordered Correlators in Quantum Many-Body Systems: a Tutorial

“…It is tempting to assume the dynamics of sufficiently high order correlation functions does not feedback to the dynamics of simple correlation functions and to truncate the infinite series of equations or simplify the higher order equations by approximation. These ideas have led to multiple new numerical algorithms [128][129][130][131][132]. There is still work to do to justify the assumptions and better understand the interplay between the scrambling dynamics and dynamics of local observables.…”