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Operator backflow and the classical simulation of quantum transport

Research output: Contribution to journalArticlepeer-review

Curt Von Keyserlingk, Frank Pollmann, Tibor Rakovszky

Original languageEnglish
Article number245101
JournalPhysical Review B
Issue number24
Accepted/In press3 May 2022
Published15 Jun 2022

Bibliographical note

Funding Information: We thank Luca Delacretaz, Christopher White, Daniel Parker, and Gabriele Pinna for useful discussions. C.v.K. is supported by UKRI Future Leaders Fellowship No. MR/T040947/1. The computations described in this paper were performed in part using the University of Birmingham's BlueBEAR HPC service. F.P. acknowledges support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant Agreement No. 771537) and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy No. EXC-2111-390814868 and No. TRR 80. T.R. is supported by the Stanford Q-Farm Bloch Postdoctoral Fellowship in Quantum Science and Engineering. T.R. acknowledges the hospitality of the Aspen Center for Physics, supported by National Science Foundation Grant No. PHY-1607611 and the Kavli Institute for Physics, supported by the National Science Foundation under Grant No. NSF PHY-1748958. Publisher Copyright: © 2022 American Physical Society.

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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 ℓ∗. 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 ℓ∗. 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 ϵ with an amount of memory scaling as exp[O(log(ϵ)2)], significantly better than the naive estimate of memory exp[O(poly(ϵ-1))] required by more brute-force methods.

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