Quantum Gate Fidelity in Terms of Choi Matrices

TL;DR

This paper introduces new methods for analyzing quantum gate fidelity using Choi matrices, providing explicit characterizations of channels with equal fidelity and practical formulas for minimum fidelity computation.

Contribution

It offers a comprehensive characterization of quantum channels with equal gate fidelity and introduces a semidefinite program for calculating minimum fidelity in qubit channels.

Findings

Equal gate fidelity implies specific relations between channels in low dimensions

A formula for minimum gate fidelity based on Choi matrix norms

Semidefinite programming approach for qubit channels

Abstract

We provide new results for computing and comparing the quantum gate fidelity of quantum channels via their Choi matrices. We extend recent work that showed there exist non-dual pairs of quantum channels with equal gate fidelity by providing an explicit characterization of all such channels. We use our characterization to show that when the dimension is 2 (or 3, under slightly stronger hypotheses), the gate fidelity of two channels is equal if and only if their difference equals the difference of some unital map and its dual -- a fact that has been shown to be false when the dimension is 4 or larger. We also present a formula for the minimum gate fidelity of a channel in terms of a well-studied norm on a compression of its Choi matrix. As a consequence, several new ways of bounding and approximating the minimum gate fidelity follow, including a simple semidefinite program to compute it for qubit channels.