Encoding an Arbitrary State in a [7,1,3] Quantum Error Correction Code

TL;DR

This paper evaluates the fidelity of encoding arbitrary quantum states into a [7,1,3] CSS code under realistic error conditions, analyzing error correction and gate application accuracy for practical quantum computing.

Contribution

It provides a detailed analysis of encoding fidelity, error correction, and gate accuracy in a non-equiprobable Pauli error environment, informing practical quantum computation strategies.

Findings

Encoding fidelity remains high for certain error probabilities.

Fault-tolerant error correction improves state fidelity under noise.

Non-fault-tolerant procedures may be viable in specific regimes.

Abstract

We calculate the fidelity with which an arbitrary state can be encoded into a [7,1,3] CSS 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. This determination is accomplished by applying ideal error correction to the encoded state which demonstrates the correctability of errors that occurred during the encoding process. We then apply single-qubit Clifford gates to the encoded state and determine the accuracy with which these gates can be applied. Finally, fault tolerant noisy error correction is applied to the encoded states in the non-equiprobable Pauli operator error environment allowing us to compare noisy (realistic) and perfect error correction implementations. We note that this maintains the fidelity of the encoded state for certain error-probability values. These results have implications for when non-fault tolerant procedures may be used in practical quantum computation and whether quantum error correction should be applied at every step in a quantum protocol.