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 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. 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.
翻译:诸如被困的离子体等物理平台会受到一致的噪音, 错误表现为对特定轴的旋转, 并且可以随时间累积。 我们通过不协调的无基空间来调查被动的缓解, 需要噪音来保存稳定器代码的空间, 并且作为保护信息上的逻辑身份操作者。 因此, 我们为所有跨端的 $Z 旋转创造了条件, 以保存稳定器代码的代码空间, 其中包括 $[4L2, 1, 2L] 的重量- 平流器, 以覆盖在某种X美元构成中支持的所有正方块。 此外, 重量- $ Z 平流的平流空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间, 以美元为美元, 美元为美元, 以连锁的平价代码生成一个直接的产品。 通过调整这些组件的重量, ASECC C- dC 代码可以构建一个大型的代码, 以 $xxxxx 的平流 。