Codeword Stabilized Quantum Codes

Quantum error correction codes play a central role in quantum computation and quantum information. While considerable understanding has now been obtained for a broad class of quantum codes, almost all of this has focused on stabilizer codes, the quantum analogues of classical additive codes. However, such codes are strictly suboptimal in some settings—there exist nonadditive codes which encode a larger logical space than possible with a stabilizer code of the same length and capable of tolerating the same number of errors. There are only a handful of such examples, and their constructions have proceeded in an ad hoc fashion, each code working for seemingly different reasons.

We present a unifying approach to quantum error correcting code design, namely, the codeword stabilized quantum codes, that encompasses additive (stabilizer) codes, as well as all known examples of nonadditive codes with good parameters. In addition to elucidating nonadditive codes, this unified perspective promises to shed new light on additive codes as well. Our codes are described by two objects: First, the codeword stabilizer that can be taken to describe a graph state, and which transforms the quantum errors to be corrected into effectively classical errors. And second, a classical code capable of correcting the induced classical error model. With a fixed stabilizer state, finding a quantum code is reduced to finding a classical code that corrects the (perhaps rather exotic) induced error model.

We use this framework to generate new codes with superior parameters ((n,K,d)) to any previously known, the number of physical qubits being n, the dimension of the encoded space K, and the code distance d. In particular, we find ((10,18,3)) and ((10,20,3)) codes. We also show how to construct encoding circuits for all codes within our framework.

Andrew Cross, Graeme Smith, John A. Smolin, and Bei Zeng, "codeword Stabilized Quantum Codes" , arXiv:0708.1021v4