Quantum Error Correction Scheme for Fully Correlated Noise

TL;DR

This paper develops and experimentally tests quantum error correction schemes for fully correlated noise channels on IBM quantum computers, emphasizing efficient operator decomposition and extending to channels with Pauli error operators.

Contribution

It presents a modified encoding/decoding scheme compatible with standard quantum gates and demonstrates its effectiveness on IBM hardware, including for channels with Pauli errors.

Findings

Efficient decomposition improves error correction accuracy.

Modified schemes outperform previous methods.

Successful correction of errors in channels with Pauli operators.

Abstract

This paper investigates quantum error correction schemes for fully-correlated noise channels on an $n$-qubit system, where error operators take the form $W^{\otimes n}$, with $W$ being an arbitrary $2\times 2$ unitary operator. In previous literature, a recursive quantum error correction scheme can be used to protect $k$ qubits using $(k+1)$-qubit ancilla. We implement this scheme on 3-qubit and 5-qubit channels using the IBM quantum computers, where we uncover an error in the previous paper related to the decomposition of the encoding/decoding operator into elementary quantum gates. Here, we present a modified encoding/decoding operator that can be efficiently decomposed into (a) standard gates available in the \texttt{qiskit} library and (b) basic gates comprised of single-qubit gates and CNOT gates. Since IBM quantum computers perform relatively better with fewer basic gates, a more efficient decomposition gives more accurate results. Our experiments highlight the importance of an efficient decomposition for the encoding/decoding operators and demonstrate the effectiveness of our proposed schemes in correcting quantum errors. Furthermore, we explore a special type of channel with error operators of the form $\sigma_x^{\otimes n}, \sigma_y^{\otimes n}$ and $\sigma_z^{\otimes n}$, where $\sigma_x, \sigma_y, \sigma_z$ are the Pauli matrices. For these channels, we implement a hybrid quantum error correction scheme that protects both quantum and classical information using IBM's quantum computers. We conduct experiments for $n = 3, 4, 5$ and show significant improvements compared to recent work.