Structured error recovery for code-word-stabilized quantum codes

Abstract

Code-word-stabilized (CWS) codes are, in general, nonadditive quantum codes that can correct errors by an exhaustive search of different error patterns, similar to the way that we decode classical nonlinear codes. For an $n$-qubit quantum code correcting errors on up to $t$ qubits, this brute-force approach consecutively tests different errors of weight $t$ or less and employs a separate $n$-qubit measurement in each test. In this article, we suggest an error grouping technique that allows one to simultaneously test large groups of errors in a single measurement. This structured error recovery technique exponentially reduces the number of measurements by about ${3}^{t}$ times. While it still leaves exponentially many measurements for a generic CWS code, the technique is equivalent to syndrome-based recovery for the special case of additive CWS codes.