Quantum error correction requires the use of error syndromes derived from measurements that may be unreliable. Recently, quantum data-syndrome (QDS) codes have been proposed as a possible approach to protect against both data and syndrome errors, in which a set of linearly dependent stabilizer measurements are performed to increase redundancy. Motivated by wanting to reduce the total number of measurements performed, we introduce QDS subsystem codes, and show that they can outperform similar QDS stabilizer codes derived from them. We also give a construction of single-error-correcting QDS stabilizer codes from impure stabilizer codes, and show that any such code must satisfy a variant of the quantum Hamming bound for QDS codes. Finally, we use this bound to prove a new bound that applies to impure, but not pure, stabilizer codes that may be of independent interest.

翻译：量子纠错需要利用从测量中获得的误差综合征，而这些测量可能不可靠。近期，量子数据-综合征（QDS）码被提出作为一种可能的方法，用于同时保护数据和综合征免受误差影响，其中通过执行一组线性相关的稳定子测量来增加冗余。出于减少总测量次数的动机，我们引入了QDS子系统码，并证明其性能优于从它们导出的类似QDS稳定子码。我们还给出了一种从非纯稳定子码构造单纠错QDS稳定子码的方法，并表明任何此类码都必须满足QDS码的量子汉明界变体。最后，我们利用该界证明了一个新的界，该界适用于非纯（而非纯）稳定子码，且可能具有独立的研究价值。