Quantum error correction for non-maximally entangled states

TL;DR

This paper introduces an 8-qubit error correction code tailored for non-maximally entangled states, reducing resource requirements compared to traditional codes, and demonstrates its applicability to larger non-maximally entangled systems.

Contribution

It presents a novel error correction scheme for non-maximally entangled states, showing fewer qubits are needed for single-error correction than existing codes.

Findings

Error states for non-maximally entangled qubits map to only 8 distinct errors.

An 8-qubit code can correct a single error in non-maximally entangled states.

Fewer than 5n qubits are sufficient to correct a single error in n-qubit non-maximally entangled states.

Abstract

Quantum states have high affinity for errors and hence error correction is of utmost importance to realise a quantum computer. Laflamme showed that 5 qubits are necessary to correct a single error on a qubit. In a Pauli error model, four different types of errors can occur on a qubit. Maximally entangled states are orthogonal to each other and hence can be uniquely distinguished by a measurement in the Bell basis. Thus a measurement in Bell basis and a unitary transformation is sufficient to correct error in Bell states. However, such a measurement is not possible for non-maximally entangled states. In this work we show that the 16 possible errors for a non-maximally entangled two qubit system map to only 8 distinct error states. Hence, it is possible to correct the error without perfect knowledge of the type of error. Furthermore, we show that the possible errors can be grouped in such a way that all 4 errors can occur on one qubit, whereas only bit flip error can occur on the second qubit. As a consequence, instead of 10, only 8 qubits are sufficient to correct a single error. We propose an 8-qubit error correcting code to correct a single error in a non-maximally entangled state. We further argue that for an $n$-qubit non-maximally entangled state of the form $a|0>^{n} + b|1>^{n}$, it is always possible to correct a single error with fewer than $5n$ qubits, in fact only $3n+2$ qubits suffice.