Finding good quantum codes using the Cartan form

TL;DR

This paper introduces a fast numerical method for finding high-quality quantum error-correcting codes by optimizing worst-case fidelity, utilizing the Cartan form to efficiently explore the relevant search space.

Contribution

It presents a novel, efficient optimization approach for quantum code design that leverages the Cartan decomposition to focus on resilient nonlocal degrees of freedom.

Findings

The method effectively finds good quantum codes with reduced computational effort.

Using worst-case fidelity improves the assessment of code performance.

The Cartan form reduces the search space without significantly impacting code quality.

Abstract

We present a simple and fast numerical procedure to search for good quantum codes for storing logical qubits in the presence of independent per-qubit noise. In a key departure from past work, we use the worst-case fidelity as the figure of merit for quantifying code performance, a much better indicator of code quality than, say, entanglement fidelity. Yet, our algorithm does not suffer from inefficiencies usually associated with the use of worst-case fidelity. Specifically, using a near-optimal recovery map, we are able to reduce the triple numerical optimization needed for the search to a single optimization over the encoding map. We can further reduce the search space using the Cartan decomposition, focusing our search over the nonlocal degrees of freedom resilient against independent per-qubit noise, while not suffering much in code performance.