Mitigating Coherent Noise by Balancing Weight-2 $Z$-Stabilizers

Abstract

Physical platforms such as trapped ions suffer from coherent noise where errors manifest as rotations about a particular axis and can accumulate over time. We investigate passive mitigation through decoherence free subspaces, requiring the noise to preserve the code space of a stabilizer code, and to act 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, which require the weight-$2$ $Z$-stabilizers to cover all the qubits that are in the support of some $X$-component. Further, the weight-$2$ $Z$-stabilizers generate a direct product of single-parity-check codes with even block length. By adjusting the size of these components, we are able to construct a large family of QECC codes, oblivious to coherent noise, that includes the $[[4L^2,1,2L]]$ Shor codes. Moreover, given $M$ even and any $[[n,k,d]]$ stabilizer code, we can construct an $[[Mn,k,\ge d]]$ stabilizer code that is oblivious to coherent noise. If we require that transversal $Z$-rotations preserve the code space only up to some finite level $l$ in the Clifford hierarchy, then we can construct higher level gates necessary for universal quantum computation. The $Z$-stabilizers supported on each non-zero $X$-component form a classical binary code C, which is required to contain a self-dual code, and the classical Gleason's theorem constrains its weight enumerator. The conditions for a stabilizer code being preserved by transversal $2\pi /2^l$ $Z$-rotations at $4\le l\le l_{\max }<\infty$ level in the Clifford hierarchy lead to generalizations of Gleason's theorem that may be of independent interest to classical coding theorists.

arXiv commentJingzhen Hu and Qingzhong Liang contributed equally to this work. The paper was accepted to IEEE Transactions on Information Theory. The ISIT paper is available as an ancillary file

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• Mitigating Coherent Noise by Balancing Weight-2 $Z$-Stabilizers
• Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians
• High-Rate Amplitude-Damping Shor Codes with Immunity to Collective Coherent Errors
• Robust projective measurements through measuring code-inspired observables
• Quantum dual extended Hamming code immune to collective coherent errors