Fidelity-based distance bounds for $N$-qubit approximate quantum error correction

TL;DR

This paper introduces fidelity-based distance bounds for approximate quantum error correction in $N$-qubit systems, providing computationally efficient methods to evaluate error approximations and illustrating them with GHZ and W states.

Contribution

It develops new fidelity-based distance measures for bounding errors in approximate quantum error correction, with analytical and numerical evaluation for general $N$-qubit states.

Findings

Derived closed-form expressions for fidelity-based distances.

Validated bounds with $N$-qubit GHZ and W states.

Reduced computational complexity in error approximation evaluation.

Abstract

The Eastin-Knill theorem is a central result of quantum error correction theory and states that a quantum code cannot correct errors exactly, possess continuous symmetries, and implement a universal set of gates transversely. As a way to circumvent this result, there are several approaches in which one gives up on either exact error correction or continuous symmetries. In this context, it is common to employ a complementary measure of fidelity as a way to quantify quantum state distinguishability and benchmark approximations in error correction. Despite having useful properties, evaluating fidelity measures stands as a challenging task for quantum states with a large number of entangled qubits. With that in mind, we address two distance measures based on the sub- and superfidelities as a way to bound error approximations, which in turn require a lower computational cost. We model the lack of exact error correction to be equivalent to the action of a single dephasing channel, evaluate the proposed fidelity-based distances both analytically and numerically, and obtain a closed-form expression for a general $N$-qubit quantum state. We illustrate our bounds with two paradigmatic examples, an $N$-qubit mixed GHZ state and an $N$-qubit mixed $W$ state.