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arXiv:2004.11038 · Physical Review A

Tensor-network-based machine learning of non-Markovian quantum processes

Chu Guo, Kavan Kishore Modi, Dario Poletti

Published Dec 17, 2020• 51 citations
arXivHTMLPDF

Abstract

We show how a tensor-network-based machine learning algorithm can learn the structures of generic, non-Markovian, quantum stochastic processes. First, a process is represented as a matrix product operator (MPO) and trained with a database of local input states at different times and the corresponding time-nonlocal output state. We then apply the algorithm to predict the output state of a process at different times for a system that is coupled to a spin-chain environment. We then reconstruct the full process, and we quantify the non-Markovian memory by means of the bond dimension of the MPO for various properties of the system, of the environment, and of their interaction. Our study paves the way for a possible experimental investigation into the process tensor and its properties, and an effective characterization of noise in quantum devices.

DOI
10.1103/physreva.102.062414

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Every quantum system is coupled to an environment. Such system-environment interaction leads to temporal correlation between quantum operations at different times, resulting in non-Markovian noise. In principle, a full characterisation of non-Markovian noise requires tomography of a multi-time processes matrix, which is both computationally and experimentally demanding. In this paper, we propose a more efficient solution. We employ machine learning models to estimate the amount of non-Markovianity, as quantified by an information-theoretic measure, with tomographically incomplete measurement. We test our model on a quantum optical experiment, and we are able to predict the non-Markovianity measure with 90%90\%90% accuracy. Our experiment paves the way for efficient detection of non-Markovian noise appearing in large scale quantum computers.
Predicting a stochastic process' future lies at the heart of many scientific areas.A predictive model extracts information from a stochastic process' past and uses it to generate future statistics.There has been significant amount of effort expended towards finding optimal predictive models that minimize the required amount of past information.By taking advantage of quantum resources, quantum models have been shown to reduce the memory requirements beyond classical limits.Meanwhile, tensor networks are an extremely useful mathematical tool for understanding quantum many-body systems.In this thesis, we explore the connection between tensor networks and quantum predictive models.A particular class of tensor networks, matrix product states (MPS), is utilized to further improve quantum models.First, we establish a connection between matrix product states (MPS) and the optimal classical predictive models.MPS methods offer a systematic method for constructing quantum predictive models and even an improved method for computing the amount of quantum memory.Second, we show that for some families of stochastic processes our method allows us to construct quantum models with unbounded memory advantages over the optimal classical models.Third, we propose a family of divergence measures to quantify the distance between two stochastic processes.This family of divergence measures overcomes certain weaknesses of the existing measures.Moreover, we propose an efficient means of computing the divergence measure using our MPS methods.Finally, we address the problem of constructing quantum predictive models solely from the output of an observed stochastic process.We use machine learning methods to propose a systematic algorithm for learning the quantum model.This provides an alternative method for constructing quantum models with fixed amounts of amount of memory.
Non-Markovian quantum state diffusion provides a wavefunction-based framework for modeling open quantum systems. In this work, we introduce a novel machine learning approach based on an operator construction algorithm. This algorithm employs a neural network as a universal generator to reconstruct the stochastic time evolution operator from an ensemble of quantum trajectories. Unlike conventional machine learning methods that approximate time-dependent wavefunctions or expectation values, our operator-based approach offers broader applicability to stochastic processes. We benchmark the algorithm on the spin-boson model across diverse spectral densities, demonstrating its accuracy. Furthermore, we showcase the operator's utility in calculating absorption spectra and reconstructing reduced density matrices at extended timescales. These results establish a new paradigm for the application of machine learning in quantum dynamics.
Abstract Tensor Networks are non-trivial representations of high-dimensional tensors, originally designed to describe quantum many-body systems. We show that Tensor Networks are ideal vehicles to connect quantum mechanical concepts to machine learning techniques, thereby facilitating an improved interpretability of neural networks. This study presents the discrimination of top quark signal over QCD background processes using a Matrix Product State classifier. We show that entanglement entropy can be used to interpret what a network learns, which can be used to reduce the complexity of the network and feature space without loss of generality or performance. For the optimisation of the network, we compare the Density Matrix Renormalization Group (DMRG) algorithm to stochastic gradient descent (SGD) and propose a joined training algorithm to harness the explainability of DMRG with the efficiency of SGD.
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Algorithms developed to solve many-body quantum problems, like tensor networks, can turn into powerful quantum-inspired tools to tackle issues in the classical domain. This work focuses on matrix product operators, a prominent numerical technique to study many-body quantum systems, especially in one dimension. It has been previously shown that such a tool can be used for classification, learning of deterministic sequence-to-sequence processes, and generic quantum processes. We further develop a matrix product operator algorithm to learn probabilistic sequence-to-sequence processes and apply this algorithm to probabilistic cellular automata. This new approach can accurately learn probabilistic cellular automata processes in different conditions, even when the process is a probabilistic mixture of different chaotic rules. In addition, we find that the ability to learn these dynamics is a function of the bitwise difference between the rules and whether one is much more likely than the other. Published by the American Physical Society 2024