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

• URL: http://arxiv.org/abs/2212.04368v2
• Date: Sat, 25 Mar 2023 19:35:27 GMT
• Title: Fidelity-based distance bounds for $N$-qubit approximate quantum error correction
• Authors: Guilherme Fiusa, Diogo O. Soares-Pinto, Diego Paiva Pires
• Abstract summary: Eastin-Knill theorem states that a quantum code cannot correct errors exactly, possess continuous symmetries, and implement a universal set of gates transversely. It is common to employ a complementary measure of fidelity as a way to quantify quantum state distinguishability and benchmark approximations in error correction. 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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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