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arXiv:1805.02871 · Physical Review A

Self-averaging of random quantum dynamics

Marcin Łobejko, Jerzy Dajka, J. Łuczka

Published Aug 7, 2018• 7 citations
arXivHTMLPDF

Abstract

The stochastic dynamics of a quantum system driven by $N$ statistically independent random sudden quenches in a fixed time interval is studied. We reveal that with increasing $N$ the system approaches a deterministic limit, indicating self-averaging with respect to its temporal unitary evolution. This phenomenon is quantified by the variance of the unitary matrix governing the time evolution of a finite-dimensional quantum system which, according to an asymptotic analysis, decreases at least as $1/N$. For a special class of protocols (when the averaged Hamiltonian commutes at different times), we prove that for finite $N$ the distance (according to the Frobenius norm) between the averaged evolution unitary operator generated by the Hamiltonian $H$ and the unitary evolution operator generated by the averaged Hamiltonian $\ensuremath{\langle}H\ensuremath{\rangle}$ scales as $1/N$. Numerical simulations enlarge this result to a broader class of noncommuting protocols.

DOI
10.1103/physreva.98.022111

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The stochastic dynamics of a quantum system driven by NNN statistically independent random sudden quenches in a fixed time interval is studied. We reveal that with increasing NNN the system approaches a deterministic limit, indicating self-averaging with respect to its temporal unitary evolution. This phenomenon is quantified by the variance of the unitary matrix governing the time evolution of a finite-dimensional quantum system which, according to an asymptotic analysis, decreases at least as 1/N1/N1/N. For a special class of protocols (when the averaged Hamiltonian commutes at different times), we prove that for finite NNN the distance (according to the Frobenius norm) between the averaged evolution unitary operator generated by the Hamiltonian HHH and the unitary evolution operator generated by the averaged Hamiltonian \ensuremath⟨H\ensuremath⟩\ensuremath{\langle}H\ensuremath{\rangle}\ensuremath⟨H\ensuremath⟩ scales as 1/N1/N1/N. Numerical simulations enlarge this result to a broader class of noncommuting protocols.
The nature of randomness and complexity growth in systems governed by unitary dynamics is a fundamental question in quantum many-body physics. This problem has motivated the study of models such as local random circuits and their convergence to Haar-random unitaries in the long-time limit. However, these models do not correspond to any family of physical time-independent Hamiltonians. In this work, we address this gap by studying the indistinguishability of time-independent Hamiltonian dynamics from truly random unitaries. On one hand, we establish a no-go result showing that for any ensemble of constant-local Hamiltonians and any evolution times, the resulting time-evolution unitary can be efficiently distinguished from Haar-random and fails to form a 222-design or a pseudorandom unitary (PRU). On the other hand, we prove that this limitation can be overcome by increasing the locality slightly: there exist ensembles of random polylog-local Hamiltonians in one-dimension such that under constant evolution time, the resulting time-evolution unitary is indistinguishable from Haar-random, i.e. it forms both a unitary kkk-design and a PRU. Moreover, these Hamiltonians can be efficiently simulated under standard cryptographic assumptions.
Abstract A quantization of a Hamiltonian system is an ambiguous procedure. Accordingly, we introduce the notion of random quantization, related random variables with values in the set of self-adjoint operators, and random processes with values in the group of unitary operators. The procedures for the averaging of random unitary groups and averaging of random self-adjoint operators are defined. The generalized weak convergence of a sequence of measures and the corresponding generalized convergence in distribution of a sequence of random variables are introduced. The generalized convergence in distribution for some sequences of compositions of random mappings is obtained. In the case of a sequence of compositions of shifts by independent random vectors of Euclidean space, the obtained convergence coincides with the statement of the central limit theorem for a sum of independent random vectors. The results are applied to the dynamics of quantum systems arising in random quantization of a Hamiltonian system.
Despite its long history, a canonical formulation of quantum ergodicity that applies to general classes of quantum dynamics, including driven systems, has not been fully established. Here we introduce and study a notion of quantum ergodicity for closed systems with time-dependent Hamiltonians, defined as statistical randomness exhibited in their longtime dynamics. Concretely, we consider the temporal ensemble of quantum states (time-evolution operators) generated by the evolution, and investigate the conditions necessary for them to be statistically indistinguishable from uniformly random states (operators) in the Hilbert space (space of unitaries). We find that the number of driving frequencies underlying the Hamiltonian needs to be sufficiently large for this to occur. Conversely, we show that statistical —indistinguishability up to some large but finite moment—can already be achieved by a quantum system driven with a single frequency, i.e., a Floquet system, as long as the driving period is sufficiently long. Our work relates the complexity of a time-dependent Hamiltonian and that of the resulting quantum dynamics, and offers a fresh perspective to the established topics of quantum ergodicity and chaos from the lens of quantum information. Published by the American Physical Society 2024
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Three Hamiltonians are Sufficient for Unitary kkk-Design in Temporal Ensemble
Unitary kkk-designs are central to quantum information and quantum many-body physics, as they provide efficient proxies for Haar-random dynamics. We study how chaotic Hamiltonian evolution can generate unitary kkk-designs using the frame potential (FP). Unlike standard approaches based on independently sampled Hamiltonians or fine-tuned evolution times, we consider a quenched temporal ensemble in which the Hamiltonians are sampled once and then held fixed, while randomness enters only through the evolution times. We compare a two-step protocol (2SP), with evolution under two fixed Hamiltonians, to a three-step protocol (3SP) with one additional quench, with three evolution times sampled independently. Analytically and numerically, we show that the 2SP fails to realize general unitary kkk-designs, whereas the 3SP realizes them for arbitrary kkk. The random phases in the 3SP impose stronger index-matching constraints in the FP, eliminating the independent permutation degrees of freedom that still remain in the 2SP. Even with imperfect time averaging, the 3SP reaches a given accuracy within a parametrically shorter time window. These results hold universally across different symmetry classes and persist in both all-to-all and local models with random interactions.
We introduce a scheme for the quantum simulation of many-body decoherence based on the unitary evolution of a stochastic Hamiltonian. Modulating the strength of the interactions with stochastic processes, we show that the noise-averaged density matrix simulates an effectively open dynamics governed by kkk-body Lindblad operators. Markovian dynamics can be accessed with white-noise fluctuations; non-Markovian dynamics requires colored noise. The time scale governing the fidelity decay under many-body decoherence is shown to scale as N−2kN^{-2k}N−2k with the system size NNN. Our proposal can be readily implemented on a variety of quantum platforms including optical lattices, superconducting circuits and trapped ions.