多量子比特魔法态培育的编码 Clifford 测量

中文摘要 AI 辅助

魔法态培育通过测量逻辑 Clifford 对称性并丢弃不一致的结果来抑制错误。对于纠缠资源，多个分支被接受，因此保留的分支携带一个多位经典记录，其损坏会产生逻辑框架错误。我们证明该记录层恰好是一个二进制线性码。对于任何第三级 Clifford 层级酉算子 $U \in \mathcal{C}_3$，分支 $UZ(a)|+\rangle^{\otimes k}$ 由可交换的 Hermitian Clifford 可观测量 $C(v) = UX(v)U^{\dagger}$ 解析，因此测量调度是生成矩阵，对有效记录的后选择给出 $P_{\mathrm{wv}} = O(q^{d})$，其中记录距离为 $d$。Plotkin 界随后限制了每个二进制记录，无论是否为线性。物理上可实现的 Clifford 族已经饱和该界：在距离四时，六个逻辑测量足以用于 $|CS\rangle$，七个用于 $|CCZ\rangle$，而独立重复则需要八个和十二个。限制为容错可测量的奇偶校验因此在记录长度上不产生代价。该节省在编译后仍然存在：在原生 CZZ 辅助的 Steane 实现中，$[6,2,4]$ CS 调度是唯一的最小成本距离四解，将培育核心的活动位置减少 26.6%，并且没有最终理想码空间投影的精确态矢量模拟确认了更高的接受率和大约一半的残余 Steane 边界权重。因此，编码同时减少了逻辑冗余和编译开销。

英文摘要

Magic-state cultivation suppresses errors by measuring logical Clifford checks and discarding inconsistent outcomes. For an entangled resource, several measurement branches are usable, so their outcomes form a multibit classical record whose corruption can produce a logical-frame error. We show that this measurement record can be protected as a binary linear code. For any third-level Clifford-hierarchy unitary $U\in\mathcal C_3$, every parity of the branch bits can be measured by a commuting Hermitian Clifford check $C(v)=UX(v)U^\dagger$. Choosing which parities to measure therefore defines a binary code $y(a)=aG$. If the valid records have minimum distance $d$, at least $d$ readout-bit flips are required to confuse one valid branch with another, giving $P_{\rm wv}=O(q^d)$. A Plotkin bound limits the length of any binary branch record, and the Clifford schedules attain this limit: at distance four, six logical measurements suffice for $|CS\rangle$ and seven for $|CCZ\rangle$, compared with eight and twelve under independent repetition. Thus restricting the logical schedule to Clifford parities requires no additional measurements. In a native-CZZ-assisted Steane implementation, the $[6,2,4]$ CS schedule is also the unique minimum-cost distance-four solution within the compiled Clifford-check family, reducing the cultivation core by $26.6\%$ in active locations. State-vector simulations without a final ideal code-space projection further show higher acceptance and approximately half the residual error weight on the two Steane blocks. Coding the logical measurement record therefore reduces both measurement redundancy and compiled fault-tolerant overhead.

发表机构

• School of Electrical Engineering, Korea University（韩国大学电气工程学院）

机构由 AI 辅助整理，请以论文原文为准。