Efficient Computation of the Quantum Rate-Distortion Function

TL;DR

This paper introduces symmetry reduction and an inexact mirror descent algorithm to efficiently compute the quantum rate-distortion function, enabling faster and more accurate calculations for multi-qubit systems.

Contribution

It presents a novel symmetry reduction approach and an inexact mirror descent algorithm with provable convergence for quantum rate-distortion computation.

Findings

Symmetry reduction simplifies quantum rate-distortion problems.

The proposed algorithm outperforms existing methods in speed and accuracy.

First numerical computation of a multi-qubit quantum rate-distortion function.

Abstract

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.