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arXiv:1912.04303 · Physical review. B./Physical review. B

Artificial neural network based computation for out-of-time-ordered correlators

Yukai Wu, L.-M. Duan, Dong-Ling Deng

Published Jun 22, 2020• 23 citations
arXivHTMLPDF

Abstract

Out-of-time-ordered correlators (OTOCs) are of crucial importance for studying a wide variety of fundamental phenomena in quantum physics, ranging from information scrambling to quantum chaos and many-body localization. However, apart from a few special cases, they are notoriously difficult to compute even numerically due to the exponential complexity of generic quantum many-body systems. In this paper, we introduce a machine learning approach to OTOCs based on the restricted-Boltzmann-machine architecture, which features wide applicability and could work for arbitrary-dimensional systems with massive entanglement. We show, through a concrete example involving a two-dimensional transverse field Ising model, that our method is capable of computing early-time OTOCs with respect to random pure quantum states or infinite-temperature thermal ensembles. Our results showcase the great potential for machine learning techniques in computing OTOCs, which open up numerous directions for future studies related to similar physical quantities.

DOI
10.1103/physrevb.101.214308

Related papers

Learning out-of-time-ordered correlators with classical kernel methods

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Out-of-time-ordered correlators (OTOCs) are widely used to investigate information scrambling in quantum systems. However, directly computing OTOCs with classical computers is an expensive procedure. This is due to the need to classically simulate the dynamics of quantum many-body systems, which entails computational costs that scale rapidly with system size. Similarly, exact simulation of the dynamics with a quantum computer (QC) will either only be possible for short times with noisy intermediate-scale quantum devices or require a fault-tolerant QC which is currently beyond technological capabilities. This motivates a search for alternative approaches to determine OTOCs and related quantities. In this paper, we explore four parameterized sets of Hamiltonians describing local one-dimensional quantum systems of interest in condensed matter physics. For each set, we investigate whether classical kernel methods can accurately learn the $\mathit{XZ}$-OTOC and a particular sum of OTOCs as functions of the Hamiltonian parameters. We frame the problem as a regression task, generating small batches of labeled data with classical tensor network methods for quantum many-body systems with up to 40 qubits. Using this data, we train a variety of standard kernel machines and observe that the Laplacian and radial basis function kernels perform best, achieving a coefficient of determination (${R}^{2}$) on testing sets of at least 0.7167, with averages between 0.8112 and 0.9822 for the various sets of Hamiltonians, together with small root mean squared error and mean absolute error. Hence, after training, the models can replace further uses of tensor networks for calculating an OTOC function of a system within the parameterized sets. Accordingly, the proposed method can assist with extensive evaluations of an OTOC function.

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

Shenglong Xu, Brian G. Swingle

This tutorial article introduces the physics of quantum information scrambling in quantum many-body systems. The goals are to understand how to precisely quantify the spreading of quantum information and how causality emerges in complex quantum systems. We introduce a general framework to study the dynamics of quantum information, including detection and decoding. We show that the dynamics of quantum information is closely related to operator dynamics in the Heisenberg picture, and, under certain circumstances, can be precisely quantified by the so-called out-of-time-ordered correlator (OTOC). The general behavior of the OTOC is discussed based on several toy models, including the Sachdev-Ye-Kitaev model, random circuit models, and Brownian models, in which the OTOC is analytically tractable. We introduce numerical methods, including exact diagonalization and tensor network methods, to calculate the OTOC for generic quantum many-body systems. We also survey current experimental schemes for measuring the OTOC in various quantum simulators. Published by the American Physical Society 2024

arXiv:2202.07060•PRX Quantum•Jan 22, 2024•141 citations
Out-of-time-ordered correlators (OTOCs) are of crucial importance for studying a wide variety of fundamental phenomena in quantum physics, ranging from information scrambling to quantum chaos and many-body localization. However, apart from a few special cases, they are notoriously difficult to compute even numerically due to the exponential complexity of generic quantum many-body systems. In this paper, we introduce a machine learning approach to OTOCs based on the restricted-Boltzmann-machine architecture, which features wide applicability and could work for arbitrary-dimensional systems with massive entanglement. We show, through a concrete example involving a two-dimensional transverse field Ising model, that our method is capable of computing early-time OTOCs with respect to random pure quantum states or infinite-temperature thermal ensembles. Our results showcase the great potential for machine learning techniques in computing OTOCs, which open up numerous directions for future studies related to similar physical quantities.
Out-of-time-ordered correlators (OTOCs) are widely used to investigate information scrambling in quantum systems. However, directly computing OTOCs with classical computers is an expensive procedure. This is due to the need to classically simulate the dynamics of quantum many-body systems, which entails computational costs that scale rapidly with system size. Similarly, exact simulation of the dynamics with a quantum computer (QC) will either only be possible for short times with noisy intermediate-scale quantum devices or require a fault-tolerant QC which is currently beyond technological capabilities. This motivates a search for alternative approaches to determine OTOCs and related quantities. In this paper, we explore four parameterized sets of Hamiltonians describing local one-dimensional quantum systems of interest in condensed matter physics. For each set, we investigate whether classical kernel methods can accurately learn the XZ\mathit{XZ}XZ-OTOC and a particular sum of OTOCs as functions of the Hamiltonian parameters. We frame the problem as a regression task, generating small batches of labeled data with classical tensor network methods for quantum many-body systems with up to 40 qubits. Using this data, we train a variety of standard kernel machines and observe that the Laplacian and radial basis function kernels perform best, achieving a coefficient of determination (R2{R}^{2}R2) on testing sets of at least 0.7167, with averages between 0.8112 and 0.9822 for the various sets of Hamiltonians, together with small root mean squared error and mean absolute error. Hence, after training, the models can replace further uses of tensor networks for calculating an OTOC function of a system within the parameterized sets. Accordingly, the proposed method can assist with extensive evaluations of an OTOC function.
Out-of-time-order correlators (OTOCs) are the main tool for probing quantum chaos and scrambling, and have become crucial probes in many areas of quantum computing. However, the measurement of OTOCs is difficult to implement on analog quantum computers due to the requirement of backward time evolution. In this paper, we develop and implement a randomized measurement protocol to compute OTOCs on Aquila by QuEra Computing. Unlike traditional methods that require backward time evolution, our approach utilizes a sequence of global randomized quenches that approximates the unitary 2-design properties necessary for extracting infinite-temperature OTOCs from statistical correlations. We demonstrate the protocol's success by explicitly observing the lightcone of information propagation in 1D Rydberg chains, and compare hardware results to both state-vector simulations and matrix product state (MPS) tensor network calculations. This work establishes the first demonstration of fully analog randomized OTOC measurements in neutral-atom simulators, providing a scalable pathway to probe quantum chaos in complex many-body systems.
Out-of-time-ordered correlators (OTOCs) are a key observable in a wide range of interconnected fields including many-body physics, quantum information science, and quantum gravity. Measuring OTOCs using near-term quantum simulators will extend our ability to explore fundamental aspects of these fields and the subtle connections between them. Here, we demonstrate an experimental method to measure OTOCs at finite temperatures and use the method to study their temperature dependence. These measurements are performed on a digital quantum computer running a simulation of the transverse field Ising model. Our flexible method, based on the creation of a thermofield double state, can be extended to other models and enables us to probe the OTOC's temperature-dependent decay rate. Measuring this decay rate opens up the possibility of testing the fundamental temperature-dependent bounds on quantum information scrambling.
Abstract Out-of-time-order correlators (OTOCs) have emerged as the preeminent theoretical and experimental probe of quantum information scrambling — the delocalization of quantum information across many-body degrees of freedom. Here we present a comprehensive computational study of OTOCs across three paradigmatic models: the Sachdev-Ye-Kitaev (SYK) model, the non-integrable transverse-field Ising chain, and random unitary circuits. Through exact diagonalization and large-N analytics, we extract Lyapunov exponents, scrambling times, and butterfly velocities, demonstrating that all chaotic systems studied approach, but do not saturate, the Maldacena-Shenker-Stanford (MSS) bound λ_L ≤ 2π/β. Crucially, we introduce the Spectral Out-of-Time-Order Correlator (SOTOC), a novel frequency-domain diagnostic that decomposes information scrambling into its constituent spectral modes. The SOTOC achieves a 32% improvement in Lyapunov exponent extraction precision over conventional log-linear fitting and maintains robust performance at noise levels σ > 0.30, where traditional methods fail. Our phase diagram reveals a sharp transition in scrambling dynamics at h/J ≈ 1.0 in the TFIM, coinciding with the onset of quantum chaos as measured by level-spacing statistics. These results establish SOTOC as a practical and theoretically grounded tool for probing scrambling in near-term quantum devices and holographic systems.
Out-of-time-ordered correlators (OTOCs) have emerged as a diagnostic of information scrambling and quantum chaos in many-body systems. We investigate the imprints of chaos in the dynamics of OTOCs in the Heisenberg spin-1/21/21/2 chain with random fields. The system is parameterized to exhibit a crossover from integrable to chaotic dynamics. We demonstrate numerically that the approach to saturation of the OTOC can distinguish between integrable and chaotic regimes, with a power-law (1/t)(1/t)(1/t) relaxation for integrable systems and a higher-degree power-law decay (1/tα;α≥1)(1/t^α; α\ge 1)(1/tα;α≥1) followed by an exponential relaxation for the chaotic regime. We further show that long-range spectral statistics, such as the number variance, are more effective in characterizing quantum chaos in the regime near saturation of OTOC. We also demonstrate that the relaxation and initial scrambling regimes exhibit distinct and universal features, with the former being sensitive and the latter being robust against different realizations of random-fields. The long-time saturation of OTOC also fluctuates with different realizations, and its exact expression is derived through the Eigenstate Thermalization Hypothesis.
This tutorial article introduces the physics of quantum information scrambling in quantum many-body systems. The goals are to understand how to precisely quantify the spreading of quantum information and how causality emerges in complex quantum systems. We introduce a general framework to study the dynamics of quantum information, including detection and decoding. We show that the dynamics of quantum information is closely related to operator dynamics in the Heisenberg picture, and, under certain circumstances, can be precisely quantified by the so-called out-of-time-ordered correlator (OTOC). The general behavior of the OTOC is discussed based on several toy models, including the Sachdev-Ye-Kitaev model, random circuit models, and Brownian models, in which the OTOC is analytically tractable. We introduce numerical methods, including exact diagonalization and tensor network methods, to calculate the OTOC for generic quantum many-body systems. We also survey current experimental schemes for measuring the OTOC in various quantum simulators. Published by the American Physical Society 2024