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arXiv:2406.06360 · Physical Review A

Non-Markov quantum belief propagation

Jack Ceroni, Ian MacCormack, Guillaume Verdon

Published Jul 12, 2024• 0 citations
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Abstract

We provide a rigorous proof of the approximate convergence of sliding-window quantum belief-propagation as outlined heuristically in the work of Bilgin and Poulin [Phys. Rev. B 81, 054106 (2010)], in the absence of the quantum Markov property. In particular, we confirm the hypothesis outlined in this work that the approximation error of each step in the belief-propagation algorithm decreases exponentially with the sliding-window size, under the assumption that the underlying state on which belief-propagation is being performed possesses a so-called thermal boundedness property: a relaxation of the Markov property required for exact convergence.

DOI
10.1103/physreva.110.012435

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We provide a rigorous proof of the approximate convergence of sliding-window quantum belief-propagation as outlined heuristically in the work of Bilgin and Poulin [Phys. Rev. B 81, 054106 (2010)], in the absence of the quantum Markov property. In particular, we confirm the hypothesis outlined in this work that the approximation error of each step in the belief-propagation algorithm decreases exponentially with the sliding-window size, under the assumption that the underlying state on which belief-propagation is being performed possesses a so-called thermal boundedness property: a relaxation of the Markov property required for exact convergence.
Belief Propagation algorithms acting on Graphical Models of classical probability distributions, such as Markov Networks, Factor Graphs and Bayesian Networks, are amongst the most powerful known methods for deriving probabilistic inferences amongst large numbers of random variables. This paper presents a generalization of these concepts and methods to the quantum case, based on the idea that quantum theory can be thought of as a noncommutative, operator-valued, generalization of classical probability theory. Some novel characterizations of quantum conditional independence are derived, and definitions of Quantum n-Bifactor Networks, Markov Networks, Factor Graphs and Bayesian Networks are proposed. The structure of Quantum Markov Networks is investigated and some partial characterization results are obtained, along the lines of the Hammersely-Clifford theorem. A Quantum Belief Propagation algorithm is presented and is shown to converge on 1-Bifactor Networks and Markov Networks when the underlying graph is a tree. The use of Quantum Belief Propagation as a heuristic algorithm in cases where it is not known to converge is discussed. Applications to decoding quantum error correcting codes and to the simulation of many-body quantum systems are described.
We present an accurate numerical algorithm, called quantum belief propagation, for simulation of one-dimensional quantum systems at nonzero temperature. The algorithm exploits the fact that quantum effects are short-range in these systems at nonzero temperature, decaying on a length scale inversely proportional to the temperature. We compare to exact results on a spin-1∕21∕21∕2 Heisenberg chain. Even a very modest calculation, requiring diagonalizing only ten-by-ten matrices, reproduces the peak susceptibility with a relative error of less than ${10}^{\ensuremath{-}5}$, while more elaborate calculations further reduce the error.
Non-Markovian effects are important in modeling the behavior of open quantum systems arising in solid-state physics, quantum optics as well as in study of biological and chemical systems. A common approach to the analysis of such systems is to approximate the non-Markovian environment by discrete bosonic modes thus mapping it to a Lindbladian or Hamiltonian simulation problem. While systematic constructions of such modes have been proposed in previous works [D. Tamascelli et al, PRL (2012), A. W. Chin et al, J. of Math. Phys (2010)], the resulting approximation lacks rigorous convergence guarantees. In this paper, we initiate a rigorous study of the convergence properties of these methods. We show that under some physically motivated assumptions on the system-environment interaction, the finite-time dynamics of the non-Markovian open quantum system computed with a sufficiently large number of modes is guaranteed to converge to the true result. Furthermore, we show that, for most physically interesting models of non-Markovian environments, the approximation error falls off polynomially with the number of modes. Our results lend rigor to numerical methods used for approximating non-Markovian quantum dynamics and allow for a quantitative assessment of classical as well as quantum algorithms in simulating non-Markovian quantum systems.
We propose a randomized algorithm to compute the log-partition function of weakly interacting fermions with polynomial runtime in both the system size and precision. Although weakly interacting fermionic systems are considered tractable for many computational methods such as the diagrammatic quantum Monte Carlo, a mathematically rigorous proof of polynomial runtime has been lacking. In this work we first extend the proof techniques developed in previous works for proving the convergence of the cumulant expansion in periodic systems to the non-periodic case. A key equation used to analyze the sum of connected Feynman diagrams, which we call the tree-determinant expansion, reveals an underlying tree structure in the summation. This enables us to design a new randomized algorithm to compute the log-partition function through importance sampling augmented by belief propagation. This approach differs from the traditional method based on Markov chain Monte Carlo, whose efficiency is hard to guarantee, and enables us to obtain a algorithm with provable polynomial runtime.
Tensor network contraction on arbitrary graphs is a fundamental computational challenge with applications ranging from quantum simulation to error correction. While belief propagation (BP) provides a powerful approximation algorithm for this task, its accuracy limitations are poorly understood and systematic improvements remain elusive. Here, we develop a rigorous theoretical framework for BP in tensor networks, leveraging insights from statistical mechanics to devise a that systematically improves the BP approximation. We prove that the cluster expansion converges exponentially fast if an object called the decays sufficiently fast with the loop size, giving a rigorous error bound on BP. We also provide a simple and efficient algorithm to compute the cluster expansion to arbitrary order. We demonstrate the efficacy of our method on the two-dimensional Ising model, where we find that our method significantly improves upon BP and existing corrective algorithms such as loop series expansion. Our work opens the door to a systematic theory of BP for tensor networks and its applications in decoding quantum error-correcting codes and simulating quantum systems.