Encoding an arbitrary state in a [7,1,3] quantum error correction code

Abstract

We calculate the fidelity with which an arbitrary state can be encoded into a [7, 1, 3] Calderbank-Shor-Steane quantum error correction code in a non-equiprobable Pauli operator error environment with the goal of determining whether this encoding can be used for practical implementations of quantum computation. The determination of usability is accomplished by applying ideal error correction to the encoded state which demonstrates the correctability of errors that occurred during the encoding process. We also apply single-qubit Clifford gates to the encoded state and determine the accuracy with which these gates can be implemented. Finally, fault tolerant noisy error correction is applied to the encoded states allowing us to compare noisy (realistic) and perfect error correction implementations. We find the encoding to be usable for the states \({|0\rangle, |1\rangle}\), and \({|\pm\rangle = |0\rangle\pm|1\rangle}\). These results have implications for when non-fault tolerant procedures may be used in practical quantum computation and whether quantum error correction must be applied at every step in a quantum protocol.