发表机构
University of Houston; Miami University（休斯顿大学; 迈阿密大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将后选择逻辑测量的魔法约束为对应资源，针对特定代码块与自适应协议，得出接受魔法的约束及指数级抑制结果，明确阈值与代码距离相关。
AI 中文摘要
我们将后选择逻辑测量的魔法（magic）约束为产生该测量的资源。研究场景为：一个包含1个逻辑量子比特的代码块，以及一个执行测量、前馈和接受操作的自适应协议。见证（witness）逐结果地对照魔法资源理论的自由集读取接受效应，而非基于平均信道，因为信道可以是自由的，但其某一结果可能测量魔法轴。我们的第一个约束是无条件的：接受的魔法最多为自由集单元的总距离乘以一个常数。第二个是主要结果：当资源单元位于有界扩散的精确恢复骨架（exact-recovery skeleton）内时，恢复过程逐转录本（transcript）将每个插入历史低于阈值的情况置于单个自由分支上，因此只有达到阈值的连通簇会产生贡献，且精确分量展开（exact-component expansion）控制其权重。对于与代码距离d成线性关系的阈值，当插入度有界、每个单元的 dilation 振幅为O(1/d)的多项式数量单元，接受的魔法乘以接受概率最多为exp[-Ω(d log d)]。在对接受转录本求和前已去除后选择，因此消失概率的分支无法被放大为魔法效应。该阈值由电路认证，我们在一个精确的稳定子测量（stabilizer measurement）轮次后运行它，接着进行分裂读出，该读出对接受纤维测量逻辑X，对拒绝纤维测量逻辑Z。这为弱Z旋转的每单层图案（每个数据量子比特一个）认证了等于代码距离的阈值，因此假设由一个族而非单个设计满足。一个成员仅当其支撑包含逻辑Z串时才携带魔法，某一成员在接受效应层面再次达到指数级。抑制由阈值而非代码块的拓扑结构决定。
英文摘要
We bound the magic of a post-selected logical measurement by the resource that produced it. The setting is one code block with one logical qubit and an adaptive protocol that measures, feeds forward and accepts. The witness reads the accepted effect against the free set of the resource theory of magic, outcome by outcome and not on the averaged channel, since a channel can be free while one of its outcomes measures the magic axis. Our first bound is unconditional. The accepted magic is at most a constant times the summed distance of the cells from the free set. The second is the main result. When the resource cells sit inside a bounded-spread exact-recovery skeleton, the recovery puts every insertion history below a threshold onto a single free branch, transcript by transcript, so only connected clusters reaching the threshold contribute and an exact-component expansion controls their weight. With a threshold linear in the code distance, polynomially many cells of bounded insertion degree and per-cell dilation amplitude $O(1/d)$, the accepted magic times the acceptance probability is at most $\exp[-Ω(d\log d)]$. Post-selection is disposed of before accepted transcripts are summed, so a branch of vanishing probability cannot be amplified into a magic effect. The threshold is certified from a circuit, and we run it on one exact round of stabilizer measurement followed by a split readout, which measures logical $X$ on the accepted fibre and logical $Z$ on the rejected ones. That certifies a threshold equal to the code distance for every single-layer pattern of weak $Z$-rotations, one per data qubit, so the hypotheses are met by a family and not one design. A member carries magic only if its support contains a logical $Z$ string. One member attains the exponent, again at the level of the accepted effect. Suppression is set by the threshold and not by the topology of the block.
