The Fidelity of an Encoded [7,1,3] Logical Zero

TL;DR

This paper evaluates the fidelity of a [7,1,3] quantum error correction code's logical zero state under non-equiprobable Pauli errors, comparing fault-tolerant syndrome measurement with direct encoding, and finds error suppression at first order in both methods.

Contribution

It provides a comparative analysis of two encoding methods for the [7,1,3] code in a realistic error environment, highlighting conditions for optimal fidelity.

Findings

Fidelity depends on the dominant Pauli error type.

Verifications of Shor states can reduce fidelity.

Perfect error correction suppresses errors to first order.

Abstract

I calculate the fidelity of a [7,1,3] CSS quantum error correction code logical zero state constructed in a non-equiprobable Pauli operator error environment for two methods of encoding. The first method is to apply fault tolerant error correction to an arbitrary state of 7 qubits utilizing Shor states for syndrome measurement. The Shor states are themselves constructed in the non-equiprobable Pauli operator error environment and their fidelity depends on the number of verifications done to ensure multiple errors will not propagate into the encoded quantum information. Surprisingly, performing these verifications may lower the fidelity of the constructed Shor states. The second encoding method is to simply implement the $[7,1,3]$ encoding gate sequence also in the non-equiprobable Pauli operator error environment. Perfect error correction is applied after both methods to determine the correctability of the implemented errors. I find that which method attains a higher fidelity depends on the which of Pauli operator errors is dominant. Nevertheless, perfect error correction supresses errors to at least first order for both methods.