Efficient Computation of the Quantum Rate-Distortion Function

Kerry He1, James Saunderson1, and Hamza Fawzi2

1Department of Electrical and Computer System Engineering, Monash University, Clayton VIC 3800, Australia
2Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom

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The quantum rate-distortion function plays a fundamental role in quantum information theory, however there is currently no practical algorithm which can efficiently compute this function to high accuracy for moderate channel dimensions. In this paper, we show how symmetry reduction can significantly simplify common instances of the entanglement-assisted quantum rate-distortion problems. This allows us to better understand the properties of the quantum channels which obtain the optimal rate-distortion trade-off, while also allowing for more efficient computation of the quantum rate-distortion function regardless of the numerical algorithm being used. Additionally, we propose an inexact variant of the mirror descent algorithm to compute the quantum rate-distortion function with provable sublinear convergence rates. We show how this mirror descent algorithm is related to Blahut-Arimoto and expectation-maximization methods previously used to solve similar problems in information theory. Using these techniques, we present the first numerical experiments to compute a multi-qubit quantum rate-distortion function, and show that our proposed algorithm solves faster and to higher accuracy when compared to existing methods.

The quantum rate-distortion function describes the maximum amount a quantum information source can be compressed by a quantum channel, without exceeding a maximum allowable distortion. In general, this function needs to be computed numerically by solving a convex optimization problem. However, this can be challenging for two reasons. First, the problem dimension of the optimization problem quickly become large as the size of the quantum channel increases. For common methods for measuring the distortion of the quantum information source, we show how symmetries in the optimization problem can be exploited to significantly reduce the dimension of the optimization problem, allowing us to solve the problem much faster. Second, standard gradient descent algorithms do not work well when computing the quantum rate-distortion function, due to the quantum entropy-like functions arising in the optimization problem. Instead, we show how an entropic variation of gradient descent, known as entropic mirror descent, can be employed to derive an efficient algorithm to compute the quantum rate-distortion function.

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Cited by

[1] Mehdi Karimi and Levent Tuncel, "Efficient Implementation of Interior-Point Methods for Quantum Relative Entropy", arXiv:2312.07438, (2023).

[2] Masahito Hayashi and Geng Liu, "Generalized quantum Arimoto-Blahut algorithm and its application to quantum information bottleneck", arXiv:2311.11188, (2023).

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