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arXiv:2312.14617 · Physical Review Research

Arbitrary relaxation rate under non-Hermitian matrix iterations

Jaš Bensa

Published Jun 28, 2024• 2 citations
arXivHTMLPDF

Abstract

We study the exponential relaxation of observables propagated with a non-Hermitian transfer matrix, an example being out-of-time-ordered correlations (OTOC) in brick-wall (BW) random quantum circuits. Until a time that scales as the system size, the exponential decay of observables is not usually determined by the second largest eigenvalue of the transfer matrix, as one can naively expect, but it is, in general, slower—this slower decay rate was dubbed “phantom eigenvalue.” Generally, this slower decay is given by the largest value in the pseudospectrum of the transfer matrix; however, we show that the decay rate can be an arbitrary value between the second largest eigenvalue and the largest value in the pseudospectrum. This arbitrary decay can be observed, for example, in the propagation of OTOC in periodic boundary conditions BW circuits. To explore this phenomenon, we study a matrix iteration made from a simple tridiagonal Toeplitz matrix. This setting can be used to propagate OTOC in random circuits with open boundary conditions and to describe a one-dimensional biased random walk with dissipation at the edges. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.023331

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We study the exponential relaxation of observables propagated with a non-Hermitian transfer matrix, an example being out-of-time-ordered correlations (OTOC) in brick-wall (BW) random quantum circuits. Until a time that scales as the system size, the exponential decay of observables is not usually determined by the second largest eigenvalue of the transfer matrix, as one can naively expect, but it is, in general, slower—this slower decay rate was dubbed “phantom eigenvalue.” Generally, this slower decay is given by the largest value in the pseudospectrum of the transfer matrix; however, we show that the decay rate can be an arbitrary value between the second largest eigenvalue and the largest value in the pseudospectrum. This arbitrary decay can be observed, for example, in the propagation of OTOC in periodic boundary conditions BW circuits. To explore this phenomenon, we study a matrix iteration made from a simple tridiagonal Toeplitz matrix. This setting can be used to propagate OTOC in random circuits with open boundary conditions and to describe a one-dimensional biased random walk with dissipation at the edges. Published by the American Physical Society 2024
We study out-of-time-ordered correlation (OTOC) functions in various random quantum circuits and show that the average dynamics is governed by a Markovian propagator. This is then used to study relaxation of OTOC to its long-time average value in circuits with random single-qubit unitaries, finding that relaxation in general proceeds in two steps: In the first phase that lasts upto an extensively long time the relaxation rate is given by a phantom eigenvalue of a nonsymmetric propagator, whereas in the second phase the rate is determined by the true 2nd largest propagator eigenvalue. We also obtain exact OTOC dynamics on the lightcone and an expression for the average OTOC in finite random circuits with random two-qubit gates.
The out-of-time-ordered correlator (OTOC) is a popular probe for quantum information spreading and thermalization. In systems with local interactions, the OTOC defines a characteristic butterfly light cone that separates a regime unperturbed by chaos from one where time-evolved operators and the OTOC approach equilibrium. This relaxation has been shown to proceed in two stages. The first stage exhibits an extensive timescale and a decay rate known as the “phantom eigenvalue”, which is slower than the gap of the transfer matrix. In this paper, we investigate the two-stage relaxation of the OTOC towards its equilibrium value in various local quantum circuits. We apply a systematic framework based on an emergent statistical model, where the dynamics of two single-particle modes—a domain wall and a magnon—govern the decay rates. We show that a configuration with coexisting domain wall and magnon modes generates the phantom rate in the first stage, while competition between these two modes determines the second stage. We also examine this relaxation within the operator cluster picture. The magnon modes translate into a bound state of clusters, and the domain wall into a random operator, giving consistent rates. Finally, we extend our findings from random-in-time circuits to a broad class of Floquet models.
Abstract We want to understand how relaxation process from an initial non-generic state proceeds towards a long-time typical state reached under unitary quantum evolution. One would expect that after some initial correlation time relaxation will be a simple exponential decay with constant decay rate. We show that this is not necessarily the case. Studying various Floquet systems with fixed two-qubit gates, and focusing on purity and out-of-time-ordered correlation functions, we find that in many situations relaxation proceeds in two phases of exponential decay having different relaxation rates. Namely, in the thermodynamic limit the relaxation rate exhibits a change at a critical time proportional to system’s size. The initial thermodynamically relevant rate can be slower or faster than the asymptotic one, demonstrating that the recently discovered phantom relaxation, in which the decay is slower than predicted by a nonzero transfer matrix gap, is not limited to only random circuits.
We study random quantum circuits and their rate of producing bipartite entanglement, specifically with respect to the choice of 2-qubit gates and the order (protocol) in which these are applied. The problem is mapped to a Markovian process, and we prove that there are large spectral equivalence classes—different configurations have the same spectrum. Optimal gates and the protocol that generate entanglement with the fastest theoretically possible rate are identified. Relaxation towards the asymptotic thermal entanglement proceeds via a series of phase transitions in the local relaxation rate, which is a consequence of non-Hermiticity. In particular, non-Hermiticity can cause the rate to be either faster or, even more interestingly, slower than predicted by the matrix eigenvalue gap. This result is caused by expansion coefficients that grow exponentially with system size, resulting in a “phantom” eigenvalue and is due to nonorthogonality of non-Hermitian eigenvectors. We numerically demonstrate that the phenomenon also occurs in random circuits with nonoptimal generic gates, random U(4) gates, and also without spatial or temporal randomness, suggesting that it could be of wide importance in other non-Hermitian settings, including correlations.
We study random quantum circuits and their rate of producing bipartite entanglement, specifically with respect to the choice of 2-qubit gates and the order (protocol) in which these are applied.The problem is mapped to a Markovian process, and we prove that there are large spectral equivalence classes-different configurations have the same spectrum.Optimal gates and the protocol that generate entanglement with the fastest theoretically possible rate are identified.Relaxation towards the asymptotic thermal entanglement proceeds via a series of phase transitions in the local relaxation rate, which is a consequence of non-Hermiticity.In particular, non-Hermiticity can cause the rate to be either faster or, even more interestingly, slower than predicted by the matrix eigenvalue gap.This result is caused by expansion coefficients that grow exponentially with system size, resulting in a "phantom" eigenvalue and is due to nonorthogonality of non-Hermitian eigenvectors.We numerically demonstrate that the phenomenon also occurs in random circuits with nonoptimal generic gates, random U(4) gates, and also without spatial or temporal randomness, suggesting that it could be of wide importance in other non-Hermitian settings, including correlations.
We study purity decay---a measure of bipartite entanglement---in a chain of nnn qubits under the action of various geometries of nearest-neighbor random two-site unitary gates. We use a Markov chain description of average purity evolution, using further reduction to obtain a transfer matrix of only polynomial dimension in nnn. In most circuits, an exception being the brick-wall configuration, purity decays to its asymptotic value in two stages: the initial thermodynamically relevant decay persisting up to extensive times is $\ensuremath{\sim}{\ensuremath{\lambda}}_{\mathrm{eff}}^{t}$, with ${\ensuremath{\lambda}}_{\mathrm{eff}}$ not necessarily being in the spectrum of the transfer matrix, while the ultimate asymptotic decay is given by the second largest eigenvalue ${\ensuremath{\lambda}}_{2}$ of the transfer matrix. The effective rate ${\ensuremath{\lambda}}_{\mathrm{eff}}$ depends on the location of bipartition boundaries as well as on the geometry of applied gates.