Codeword Stabilized Codes From m-Uniform Graph States

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.