Fidelity of an encoded [7,1,3] logical zero

Abstract

I calculate the fidelity of a $[7,1,3]$ Calderbank-Shor-Steane quantum error correction code logical zero state constructed in a nonequiprobable Pauli operator error environment for two methods of encoding. The first method is to apply fault-tolerant error correction to an arbitrary state of seven qubits utilizing Shor states for syndrome measurement. The Shor states are themselves constructed in the nonequiprobable 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 nonequiprobable 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 higher fidelity depends on which of the Pauli operators errors is dominant. Nevertheless, perfect error correction applied after the encoding suppresses errors to at least first order for both methods.