Codeword Stabilized Codes from m-Uniform Graph States

TL;DR

This paper introduces a method to construct quantum error-correcting codes using m-uniform graph states and classical codes, providing explicit examples and protocols for encoding and recovery.

Contribution

It presents a novel framework combining m-uniform graph states and classical codes to generate new quantum error-correcting codes within the CWS framework.

Findings

Constructed new families of QECCs with specific parameters.

Provided measurement-based encoding and recovery protocols.

Demonstrated the applicability of m-uniform states in QECC design.

Abstract

An m-uniform quantum state on n qubits is an entangled state in which every m-qubit subsystem is maximally mixed. Starting with an m-uniform state realized as the graph state associated with an m-regular graph, and a classical [n,k,d \ge m+1] binary linear code with certain additional properties, we show that pure [[n,k,m+1]]_2 quantum error-correcting codes (QECCs) can be constructed within the codeword stabilized (CWS) code framework. As illustrations, we construct pure [[2^{2r}-1,2^{2r}-2r-3,3]]_2 and [[(2^{4r}-1)^2, (2^{4r}-1)^2 - 32r-7, 5]]_2 QECCs. We also give measurement-based protocols for encoding into code states and for recovery of logical qubits from code states.