Mitigating Coherent Noise by Balancing Weight-2 Z-Stabilizers

Abstract

Physical platforms such as trapped ions suffer from coherent noise that does not follow a simple stochastic model. Stochastic errors in quantum systems occur randomly but coherent errors are more damaging since they can accumulate in a particular direction. We consider coherent noise acting transversally, giving rise to an effective error which is a <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula>-rotation on each qubit by some angle <inline-formula> <tex-math notation="LaTeX">$\theta $ </tex-math></inline-formula>. Rather than address coherent noise through active error correction, we 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 <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula>-rotations to preserve the code space of a stabilizer code. These conditions require the weight-<inline-formula> <tex-math notation="LaTeX">$2~Z$ </tex-math></inline-formula>-stabilizers to cover all the qubits that are in the support of the <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula>-component of some stabilizer. Furthermore, the weight-<inline-formula> <tex-math notation="LaTeX">$2~Z$ </tex-math></inline-formula>-stabilizers generate a direct product of single-parity-check codes with even block length. By adjusting the sizes of these components, we are able to construct a large family of QECC codes oblivious to coherent noise, one that includes the <inline-formula> <tex-math notation="LaTeX">$[[4L^{2}, 1, 2L]]$ </tex-math></inline-formula> Shor codes. The Shor codes are examples of constant excitation codes, where logical qubits are encoded as a code state that is a sum of physical states indexed by binary vectors with the same weight. Constant excitation codes are oblivious to coherent noise since a transversal <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula>-rotation acts as a global phase. We prove that a CSS code is oblivious to coherent noise if and only if it is a constant excitation code, and that if the code is error-detecting, then the (constant) weights in different cosets of the <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula>-stabilizers are identical.