Efficient Computation of the Quantum Rate-Distortion Function

TL;DR

This work tackles the practical computation of the quantum rate-distortion function by reframing Blahut–Arimoto-style algorithms as mirror-descent methods under relative smoothness and strong convexity. It introduces symmetry-based reductions to dramatically lower problem dimensionality and develops an inexact, backtracking-enabled MD algorithm that preserves convergence guarantees, achieving sublinear rates and, locally, linear convergence for rate-distortion problems. The authors unify classical and quantum capacity approaches within a single optimization framework and provide extensive numerical experiments up to 8-qubit channels, demonstrating faster convergence and higher accuracy than prior methods. The combination of symmetry, dual Subproblem solving, and principled stopping criteria yields a scalable, practical tool for analyzing rate-distortion trade-offs in quantum information processing, with analytic results in symmetric cases and guidance for future constrained settings.

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.

Figures (5)

• Figure 1: Convergence of various of step size strategies to solve for classical and quantum channel capacities for randomly generated channels with input and output dimensions $n$, and an input alphabet with $n$ elements. Note that the constant step size strategy, which is not all shown on the plot, convergences to an accuracy of $10^{-5}$ in $984$, $1117$ and $37$ iterations for the classical, cq and ea channels respectively. Additionally, the constant step size strategy convergences to an accuracy of $10^{-10}$ in $8318$, $25093$ and $69$ iterations for the classical, cq and ea channels respectively.
• Figure 2: Convergence of mirror descent for computing the classical and quantum rate-distortion functions for Hamming distortion and entanglement fidelity distortion and for randomly generated inputs of dimension $32$.
• Figure 3: Local relative strong convexity parameters $\mu$ for the classical and quantum rate-distortion problem for the (a) uniform input distribution with Hamming distortion and (b) maximally mixed input state with entanglement fidelity distortion at different problem dimensions $n$ and dual variables $\kappa$.
• Figure 4: Comparison between experimental convergence (solid) and theoretical local convergence (dashed) rates at different values of $\kappa$ for solving the (a) classical rate-distortion function for a $4$-dimensional uniform input distribution with Hamming distortion, and (b) quantum rate-distortion function for a 2-qubit maximally mixed state with entanglement fidelity distortion.
• Figure 5: Comparison between inexact (solid) and exact (dashed) methods for solving the mirror descent subproblem. Shown are results for computing the rate-distortion function for a randomly generated 8-qubit quantum channel. Not shown is that for $\kappa=7.5$, the exact method converged to an accuracy of $10^{-10}$ in $1307$ seconds.

Theorems & Definitions (74)

• Definition 2.1: Legendre function teboulle2018simplified
• Definition 2.2: Bregman divergence bregman1967relaxation
• Example 1
• Example 2
• Example 3
• Definition 2.3: Relative smoothness
• Definition 2.4: Relative strong convexity
• Proposition 2.1
• Proposition 2.2: teboulle2018simplified
• Proposition 2.3