Quantum Error Correction Scheme for Fully Correlated Noise
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.
Chi-Kwong Li Yuqiao Li Diane Christine Pelejo Sage Stanish