Riemannian Optimization for Holevo Capacity

TL;DR

This paper introduces a Riemannian gradient descent method to compute the Holevo capacity of quantum channels, offering a scalable and efficient approach to estimate classical capacity bounds in quantum information theory.

Contribution

It formulates the Holevo capacity computation as a Riemannian optimization problem and proposes a gradient descent algorithm that improves efficiency and scalability over existing methods.

Findings

Outperforms existing methods in numerical experiments

Provides lower bounds on quantum channel capacities

Demonstrates efficiency and scalability of the approach

Abstract

Computing the classical capacity of a noisy quantum channel is crucial for understanding the limits of communication over quantum channels. However, its evaluation remains challenging due to the difficulty of computing the Holevo capacity and the even greater difficulty of regularization. In this work, we formulate the computation of the Holevo capacity as an optimization problem on a product manifold constructed from probability distributions and their corresponding pure input states for a quantum channel. A Riemannian gradient descent algorithm is proposed to solve the problem, providing lower bounds on the classical capacity of general quantum channels and outperforming existing methods in numerical experiments in both efficiency and scale.