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arXiv:2501.01154 · International Radar Conference (RADAR)

Quantum Computing for Partition Function Estimation of a Markov Random Field in a Radar Anomaly Detection Problem

Timothé Presles, Cyrille Enderli, Gilles Burel, El Houssaïn Baghious

Published Oct 21, 2024• 1 citation
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Abstract

In probability theory, the partition function is a factor used to reduce any probability function to a density function with total probability of one. Among other statistical models used to represent joint distribution, Markov random fields (MRF) can be used to efficiently represent statistical dependencies between variables. As the number of terms in the partition function scales exponentially with the number of variables, the potential of each configuration cannot be computed exactly in a reasonable time for large instances. In this paper, we aim to take advantage of the exponential scalability of quantum computing to speed up the estimation of the partition function of a MRF representing the dependencies between operating variables of an airborne radar. For that purpose, we implement a quantum algorithm for partition function estimation in the one clean qubit model. After proposing suitable formulations, we discuss the performances and scalability of our approach in comparison to the theoretical performances of the algorithm.

DOI
10.1109/radar58436.2024.10994112

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In probability theory, the partition function is a factor used to reduce any probability function to a density function with total probability of one. Among other statistical models used to represent joint distribution, Markov random fields (MRF) can be used to efficiently represent statistical dependencies between variables. As the number of terms in the partition function scales exponentially with the number of variables, the potential of each configuration cannot be computed exactly in a reasonable time for large instances. In this paper, we aim to take advantage of the exponential scalability of quantum computing to speed up the estimation of the partition function of a MRF representing the dependencies between operating variables of an airborne radar. For that purpose, we implement a quantum algorithm for partition function estimation in the one clean qubit model. After proposing suitable formulations, we discuss the performances and scalability of our approach in comparison to the theoretical performances of the algorithm.
Monte Carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically. One important example is the use in statistical physics of rapidly mixing Markov chains to approximately compute partition functions. In this work, we describe a quantum algorithm which can accelerate Monte Carlo methods in a very general setting. The algorithm estimates the expected output value of an arbitrary randomized or quantum subroutine with bounded variance, achieving a near-quadratic speedup over the best possible classical algorithm. Combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions, which use multiple-stage Markov chain Monte Carlo techniques. The quantum algorithm can also be used to estimate the total variation distance between probability distributions efficiently.
We present a method to approximate partition functions of quantum systems using mixed-state quantum computation. For positive-semidefinite Hamiltonians, our method has an expected running-time that is almost linear in ${[M/({\ensuremath{\epsilon}}_{\mathrm{rel}}\mathcal{Z})]}^{2}$, where MMM is the dimension of the quantum system, Z\mathcal{Z}Z is the partition function, and ${\ensuremath{\epsilon}}_{\mathrm{rel}}$ is the relative precision. It is based on approximations of the exponential operator as linear combinations of certain operators related to block-encoding of Hamiltonians or Hamiltonian evolutions. The trace of each operator is estimated using a standard algorithm in the one-clean-qubit model. For large values of Z\mathcal{Z}Z, our method may run faster than exact classical methods, whose complexities are polynomial in MMM. We also prove that a version of the partition function estimation problem within additive error is complete for the so-called DQC1 complexity class, suggesting that our method provides a superpolynomial speedup for certain parameter values. To attain a desired relative precision, we develop a classical procedure based on a sequence of approximations within predetermined additive errors that may be of independent interest.
Abstract Probabilistic graphical models play a crucial role in machine learning and have wide applications across various fields. A pivotal subset within this realm is undirected graphical models, also known as Markov random fields. In this work, we explore the application of quantum computing to the field of machine learning, specifically focusing on the task of structure learning in Markov random fields. We propose a quantum algorithm for structure learning of an r-wise Markov Random Field with a bounded degree underlying graph, based on a nearly optimal classical greedy algorithm. Our quantum algorithm provides a polynomial speed-up over the classical counterpart, with a runtime of O ˜ 2 L M n r + 1 log ( 1 / w ) compared to the classical O ˜ M n r , where n is the number of variables. It is nearly a quadratic speedup in terms of n. Through this research, we aim to showcase the potential advantages of quantum computation in tackling complex machine-learning problems. Furthermore, the sample complexity O ˜ 2 2 L τ 2 δ 2 L ( log ( 1 / w ) + log n ) remains consistent between classical and quantum approaches, ensuring the efficiency and robustness of our method. By demonstrating the efficiency and scalability of our quantum algorithm for structure learning, we contribute to the growing body of work that highlights the role of quantum computing in advancing the field of machine learning and probabilistic graphical models.
We present two algorithms, one quantum and one classical, for estimating partition functions of quantum spin Hamiltonians. The first is a DQC1 (deterministic quantum computation with one clean qubit) algorithm. The second, for real temperatures, achieves performance comparable to a state-of-the-art DQC1 algorithm [A. N. Chowdhury, R. D. Somma, and Y. Suba\ifmmode \mbox{\c{s}}\else \c{s}\fi{}i, Phys. Rev. A 103, 032422 (2021)]. Both our algorithms take as input the Hamiltonian decomposed as a linear combination Pauli operators. We show this decomposition to be DQC1-hard for a given Hamiltonian, providing insight into the hardness of estimating partition functions.
Markov random fields (MRFs) appear in many problems in machine learning and statistics. From a computational learning theory point of view, a natural problem of learning MRFs arises: given samples from an MRF from a restricted class, learn the structure of the MRF, that is the neighbors of each node of the underlying graph. In this work, we start at a known near-optimal classical algorithm for this learning problem and develop a modified classical algorithm. This classical algorithm retains the run time and guarantee of the previous algorithm and enables the use of quantum subroutines. Adapting a previous quantum algorithm, the Quantum Sparsitron, we provide a polynomial quantum speedup in terms of the number of variables for learning the structure of an MRF, if the MRF has bounded degree.
Abstract The partition function is an essential quantity in statistical mechanics, and its accurate computation is a key component of any statistical analysis of quantum systems and phenomena. However, for interacting many-body quantum systems, its calculation generally involves summing over an exponential number of terms and can thus quickly grow to be intractable. Accurately and efficiently estimating the partition function of its corresponding system Hamiltonian then becomes the key in solving quantum many-body problems. In this paper we develop a hybrid quantum–classical algorithm to estimate the partition function, utilising a novel quantum Clifford sampling technique. Note that previous works on the estimation of partition functions require O ( 1 / ϵ Δ ) -depth quantum circuits (Srinivasan et al 2021 IEEE Int. Conf. on Quantum Computing and Engineering (QCE) pp 112–22; Montanaro 2015 Proc. R. Soc. A 471 20150301), where Δ is the minimum spectral gap of stochastic matrices and ϵ is the multiplicative error. Our algorithm requires only a shallow O ( 1 ) -depth quantum circuit, repeated O ( n / ϵ 2 ) times, to provide a comparable ϵ approximation. Shallow-depth quantum circuits are considered vitally important for currently available noisy intermediate-scale quantum devices.