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