CSS Codes that are Oblivious to Coherent Noise
Physical platforms such as trapped ions suffer from coherent noise that does not follow a simple stochastic model. We view coherent errors as rotations about a particular axis, and observe that since they can accumulate coherently over time, they can be more damaging. It is natural to consider coherent noise acting transversally giving rise to an effective error, which is a Z-rotation on each qubit by some angle \theta. Rather than addressing coherent noise through active error correction, we instead investigate passive mitigation through decoherence free subspaces. In the language of stabilizer codes, we require the noise to preserve the code space, and to act trivially (as the logical identity operator) on the protected information. Thus, we develop necessary and sufficient conditions for all transversal Z-rotations to preserve the code space of a stabilizer code. These conditions require the existence of a large number of weight 2 Z-stabilizers, and together, these weight 2 Z-stabilizers generate a direct product of single-parity-check codes. By adjusting the size of these components, we are able to construct a large family of CSS codes, oblivious to coherent noise, that includes the [[4L^{2}, 1,2L]] Shor codes. Given m even and given any [[n, k, d]] CSS code, we can construct an [[mn, k, d^{\prime}\geq d]] CSS code that is oblivious to coherent noise. This result is generalized to stabilizer codes in [Hu, Liang, Rengaswamy, and Calderbank 2020]. The MacWilliams Identities play a central role in the technical analysis, and classical coding theorists may be interested in connections to classical codes with all weights divisible by some integer d.
