Efficient Approximation of Quantum Channel Capacities

TL;DR

This paper introduces an iterative convex programming approach to efficiently approximate the capacities of classical-quantum channels, including those with continuous inputs, with proven bounds and practical examples.

Contribution

It develops a novel iterative algorithm based on convex duality for approximating quantum channel capacities with explicit complexity bounds.

Findings

The algorithm provides $ ext{O}(rac{(N ext{ or } M) M^3 ext{log}(N)^{1/2}}{ ext{epsilon}})$ complexity for capacity approximation.

The method extends to channels with continuous inputs and quantum inputs/outputs, reducing capacity estimation to multidimensional integration.

For certain channel families, the complexity is subexponential or polynomial, enabling efficient capacity approximation.

Abstract

We propose an iterative method for approximating the capacity of classical-quantum channels with a discrete input alphabet and a finite dimensional output, possibly under additional constraints on the input distribution. Based on duality of convex programming, we derive explicit upper and lower bounds for the capacity. To provide an $\varepsilon$-close estimate to the capacity, the presented algorithm requires $O(\tfrac{(N \vee M) M^3 \log(N)^{1/2}}{\varepsilon})$, where $N$ denotes the input alphabet size and $M$ the output dimension. We then generalize the method for the task of approximating the capacity of classical-quantum channels with a bounded continuous input alphabet and a finite dimensional output. For channels with a finite dimensional quantum mechanical input and output, the idea of a universal encoder allows us to approximate the Holevo capacity using the same method. In particular, we show that the problem of approximating the Holevo capacity can be reduced to a multidimensional integration problem. For families of quantum channels fulfilling a certain assumption we show that the complexity to derive an $\varepsilon$-close solution to the Holevo capacity is subexponential or even polynomial in the problem size. We provide several examples to illustrate the performance of the approximation scheme in practice.