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arXiv:2101.08591 · Physical Review Research

Machine learning time-local generators of open quantum dynamics

Paolo P. Mazza, Dominik Zietlow, Federico Carollo, Sabine Andergassen, Georg Martius, Igor Lesanovsky

Published Apr 30, 2021• 29 citations
arXivHTMLPDF

Abstract

In the study of closed many-body quantum systems, one is often interested in the evolution of a subset of degrees of freedom.On many occasions it is possible to approach the problem by performing an appropriate decomposition into a bath and a system.In the simplest case the evolution of the reduced state of the system is governed by a quantum master equation with a time-independent, i.e., Markovian, generator.Such evolution is typically emerging under the assumption of a weak coupling between the system and an infinitely large bath.Here we are interested in understanding to which extent a neural network function approximator can predict open quantum dynamics-described by time-local generators-from an underlying unitary dynamics.We investigate this question using a class of spin models, which is inspired by recent experimental setups.We find that indeed time-local generators can be learned.In certain situations they are even time independent and allow to extrapolate the dynamics to unseen times.This might be useful for situations in which experiments or numerical simulations do not allow to capture long-time dynamics and for exploring thermalization occurring in closed quantum systems.

DOI
10.1103/physrevresearch.3.023084

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In the study of closed many-body quantum systems, one is often interested in the evolution of a subset of degrees of freedom.On many occasions it is possible to approach the problem by performing an appropriate decomposition into a bath and a system.In the simplest case the evolution of the reduced state of the system is governed by a quantum master equation with a time-independent, i.e., Markovian, generator.Such evolution is typically emerging under the assumption of a weak coupling between the system and an infinitely large bath.Here we are interested in understanding to which extent a neural network function approximator can predict open quantum dynamics-described by time-local generators-from an underlying unitary dynamics.We investigate this question using a class of spin models, which is inspired by recent experimental setups.We find that indeed time-local generators can be learned.In certain situations they are even time independent and allow to extrapolate the dynamics to unseen times.This might be useful for situations in which experiments or numerical simulations do not allow to capture long-time dynamics and for exploring thermalization occurring in closed quantum systems.
Abstract Full information about a many-body quantum system is usually out-of-reach due to the exponential growth—with the size of the system—of the number of parameters needed to encode its state. Nonetheless, in order to understand the complex phenomenology that can be observed in these systems, it is often sufficient to consider dynamical or stationary properties of local observables or, at most, of few-body correlation functions. These quantities are typically studied by singling out a specific subsystem of interest and regarding the remainder of the many-body system as an effective bath. In the simplest scenario, the subsystem dynamics, which is in fact an open quantum dynamics, can be approximated through Markovian quantum master equations. Here, we formulate the problem of finding the generator of the subsystem dynamics as a variational problem, which we solve using the standard toolbox of machine learning for optimization. This dynamical or ‘Lindblad’ generator provides the relevant dynamical parameters for the subsystem of interest. Importantly, the algorithm we develop is constructed such that the learned generator implements a physically consistent open quantum time-evolution. We exploit this to learn the generator of the dynamics of a subsystem of a many-body system subject to a unitary quantum dynamics. We explore the capability of our method to recover the time-evolution of a two-body subsystem and exploit the physical consistency of the generator to make predictions on the stationary state of the subsystem dynamics.
In experimentally realistic situations, quantum systems are never perfectly isolated and the coupling to their environment needs to be taken into account. Often, the effect of the environment can be well approximated by a Markovian master equation. However, solving this master equation for quantum many-body systems becomes exceedingly hard due to the high dimension of the Hilbert space. Here we present an approach to the effective simulation of the dynamics of open quantum many-body systems based on machine-learning techniques. We represent the mixed many-body quantum states with neural networks in the form of restricted Boltzmann machines and derive a variational Monte Carlo algorithm for their time evolution and stationary states. We document the accuracy of the approach with numerical examples for a dissipative spin lattice system.
Interacting quantum Hamiltonians are fundamental to quantum computing. Data-based tomography of time-independent quantum Hamiltonians has been achieved, but an open challenge is to ascertain the structures of time-dependent quantum Hamiltonians using time series measurements taken locally from a small subset of the spins. Physically, the dynamical evolution of a spin system under time-dependent driving or perturbation is described by the Heisenberg equation of motion. Motivated by this basic fact, we articulate a physics-enhanced machine-learning framework whose core is Heisenberg neural networks. In particular, we develop a deep learning algorithm according to some physics-motivated loss function based on the Heisenberg equation, which ``forces'' the neural network to follow the quantum evolution of the spin variables. We demonstrate that, from local measurements, not only can the local Hamiltonian be recovered, but the Hamiltonian reflecting the interacting structure of the whole system can also be faithfully reconstructed. We test our Heisenberg neural machine on spin systems of a variety of structures. In the extreme case in which measurements are taken from only one spin, the achieved tomography fidelity values can reach about 9090%90. The developed machine-learning framework is applicable to any time-dependent systems whose quantum dynamical evolution is governed by the Heisenberg equation of motion.
Reconstructing the effective equation of motion for the time evolution of a subset of degrees of freedom of a larger system remains a problem of interest in quantum physics. Many methods have been developed, but they either rely on an ad hoc ansatz, demand data that is not experimentally accessible, or lack physical interpretability. The authors employ machine-learning methods to infer the effective dynamical generator from a noisy, finite set of local measurements. Their method yields interpretable results that may be used to infer noise models on quantum simulators, or to study thermalization dynamics in quantum many-body systems.
Neural networks have emerged as a powerful way to approach many practical problems in quantum physics. In this work, we illustrate the power of deep learning to predict the dynamics of a quantum many-body system, where the training is based purely on monitoring expectation values of observables under random driving . The trained recurrent network is able to produce accurate predictions for driving trajectories entirely different than those observed during training. As a proof of principle, here we train the network on numerical data generated from spin models, showing that it can learn the dynamics of observables of interest without needing information about the full quantum state. This allows our approach to be applied eventually to actual experimental data generated from a quantum many-body system that might be open, noisy, or disordered, without any need for a detailed understanding of the system. This scheme provides considerable speedup for rapid explorations and pulse optimization. Remarkably, we show the network is able to extrapolate the dynamics to times longer than those it has been trained on, as well as to the infinite-system-size limit.
The dynamics of open quantum systems play a crucial role in quantum information science. However, obtaining numerically exact solutions for the Lindblad master equation is often computationally expensive. Recently, machine learning techniques have gained considerable attention for simulating open quantum system dynamics. In this paper, we propose a deep learning model based on time series prediction (TSP) to forecast the dynamical evolution of open quantum systems. We employ the positive operator-valued measure (POVM) approach to convert the density matrix of the system into a probability distribution and construct a TSP model based on Transformer neural networks. This model effectively captures the historical evolution patterns of the system and accurately predicts its future behavior. Our results show that the model achieves high-fidelity predictions of the system's evolution trajectory in both short- and long-term scenarios, and exhibits robust generalization under varying initial states and coupling strengths. Moreover, we successfully predicted the steady-state behavior of the system, further proving the practicality and scalability of the method.