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

Timothe Presles, Cyrille Enderli, Gilles Burel, El Houssain Baghious

TL;DR

The paper addresses the hard problem of estimating the partition function $Z_\Omega$ of a Markov Random Field used for radar anomaly localization, where exact computation is intractable for large graphs. It proposes a circuit-oriented quantum algorithm in the one-clean-qubit (DQC1) model that encodes the Hamiltonian as a linear combination of unitaries, uses block-encoding and a Chebyshev-approximation of the exponential to form a walk operator, and then estimates the partition function via trace estimation. The authors provide a concrete formulation for binary quadratic forms, apply it to pairwise Markov networks, and report initial simulation results showing feasible accuracy at small scales, along with discussions on parameter choices ($K$, $Q$) and hardware constraints. The work offers a potential quantum route to accelerate partition-function estimation in radar MRFs, with future directions including continuous-variable extensions and gate-optimization for NISQ-era hardware.

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.

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

TL;DR

The paper addresses the hard problem of estimating the partition function of a Markov Random Field used for radar anomaly localization, where exact computation is intractable for large graphs. It proposes a circuit-oriented quantum algorithm in the one-clean-qubit (DQC1) model that encodes the Hamiltonian as a linear combination of unitaries, uses block-encoding and a Chebyshev-approximation of the exponential to form a walk operator, and then estimates the partition function via trace estimation. The authors provide a concrete formulation for binary quadratic forms, apply it to pairwise Markov networks, and report initial simulation results showing feasible accuracy at small scales, along with discussions on parameter choices (, ) and hardware constraints. The work offers a potential quantum route to accelerate partition-function estimation in radar MRFs, with future directions including continuous-variable extensions and gate-optimization for NISQ-era hardware.

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.
Paper Structure (17 sections, 29 equations, 4 figures, 2 tables)

This paper contains 17 sections, 29 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Previous work on mixed graphical model learning
  4. Chebyshev approximation of the partition function
  5. Quantum algorithm for partition function estimation
  6. The one clean qubit model
  7. Block encoding of a linear combination of unitaries
  8. Quantum estimation of Chebyshev Polynomials
  9. Trace estimation of the walk operator
  10. Application to a pairwise undirected graphical model
  11. Formulation
  12. Application to a Markov network
  13. Results
  14. Experimental setup
  15. Results for the partition function estimation algorithm
  16. ...and 2 more sections

Figures (4)

  • Figure 1: Circuit for estimating Re(Tr($U$))$/2^q$ in the one clean qubit model.
  • Figure 2: Circuit to implement $W_H$. $R_0$ denotes the zero-reflection operator $(2\ketbra{0}{0}_{m'} - I_{m'})$ as defined in Grover's algorithm Grover.
  • Figure 3: Circuit to implement $U_k$. $m'$ controlled-$\mathcal{X}$ gates are applied to restore the pure state on the ancillary qubits used in $P'$.
  • Figure 4: Example of graph representation of a MRF for $n=5$