量子编译中表示依赖的可恢复性

中文摘要 AI 辅助

容错编译会分散高层结构：调度将累积的相位分散到各轮中，门合成将一个角度替换为Clifford+$T$字，随机化编译将旋转分散到符号随机化的片段中。它们保留了计算，但改变了下游编译器恢复聚合相位数据的代价。我们通过两种通道来形式化这种表示依赖的可恢复性：在后缀到达之前提交的输出，以及跨越切口的序列化重启状态。对于由$m$个生成元组成的交换层在精度$\epsilon$（$K_\epsilon=\Theta(1/\epsilon)$）下的$r$轮累积，任何在每条有效流上正确且失败概率至多为$\delta$的$p$遍编译器满足$\overline{A}_p+(2p-1)S\geq(1-\delta)m\log_2 K_\epsilon-h_2(\delta)$，其中$\overline{A}_p$是提交的输出，$S$是跨越状态的上限。该界限涵盖了编译时随机化编译；一个内存受限的块编译器在常数因子内达到该界限，而直接语义聚合以对数状态达到紧凑输出。这两种通道并不总是可互换的。一个在切口处执行其提交前缀的编译器需要支付状态通道从不支付的代价：即掩码的熵$\log_2\binom{m}{k}$，该掩码命名了它提前提交的$k$个坐标。延迟或缓冲的编译器则无需支付任何此类代价。我们证明了两个方向，并通过在预注册协议下使用仪器化编译器，测量了每个生成元的代价，该代价匹配$\log_2\binom{m}{k}/k$，对于公开掩码和缓冲控制，该代价消失。在公开的固定总误差Clifford+$T$和表面码模型下，先物化流水线比先语义流水线在逻辑$T$态和时空体积上花费多个数量级。因此，只要接口允许，相位语义应保留到聚合之前。

英文摘要

Fault-tolerant compilation can disperse high-level structure: schedules split an accumulated phase across rounds, gate synthesis replaces an angle by a Clifford+$T$ word, and randomized compiling spreads rotations over sign-randomized fragments. They preserve the computation but change how cheaply a downstream compiler can recover the aggregate phase data. We formalize this representation-dependent recoverability by charging two channels: output committed before the suffix arrives, and a serialized restart state crossing the cut. For an $r$-round accumulation of a commuting layer with $m$ generators at accuracy $ε$ ($K_ε=Θ(1/ε)$), any $p$-pass compiler correct on every valid stream with failure probability at most $δ$ obeys $\overline{A}_p+(2p-1)S\geq(1-δ)m\log_2 K_ε-h_2(δ)$, where $\overline{A}_p$ is committed output and $S$ is the crossing-state cap. The bound covers compile-time randomized compiling; a memory-capped block compiler attains it within a constant factor, and direct semantic aggregation reaches compact output with logarithmic state. The two channels are not always interchangeable. A compiler whose committed prefix is executed at the cut pays a toll the state channel never pays: the entropy $\log_2\binom{m}{k}$ of the mask naming which $k$ coordinates it commits early. Compilers that defer or buffer pay none of it. We prove both directions and, with instrumented compilers under a preregistered protocol, measure a per-generator toll matching $\log_2\binom{m}{k}/k$ that vanishes for a public mask and a buffered control. Under a disclosed fixed-total-error Clifford+$T$ and surface-code model, a materialize-first pipeline costs orders of magnitude more logical $T$ states and spacetime volume than a semantic-first one. Phase semantics should therefore be preserved until aggregation whenever the interface permits.

发表机构

• School of Physics, Xidian University（西安电子科技大学物理学院）
• Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education of China, Collaborative Innovation Center of Quantum Information of Shaanxi Province, Institute of Interdisciplinary Quantum Science and Technology (IIQST), School of Artificial Intelligence, Xidian University（西安电子科技大学人工智能学院）
• Shenzhen International Quantum Academy（深圳国际量子研究院）
• Shenzhen Branch, Hefei National Laboratory（合肥国家实验室深圳分部）

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