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

Understand

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.