流式 Clifford+T 编译中的记忆-魔法交换律
A Memory-Magic Exchange Law in Streaming Clifford+T Compilation
- 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、量子科学与技术交叉研究所、陕西省量子信息协同创新中心、教育部智能感知与图像理解重点实验室）
- 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）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究确定流式 Clifford+T 编译中记忆与魔法态的交换率，提出行列式方法证明 α≥17/7，并推测一般 α=3，指出常数速率律仅适用于坐标式合成。
AI中文摘要：
一个以加法片段形式到达容错处理器的相位可以被记住，直到最后一片到达，或者在到达时执行：第一种选择需要跨轮次携带的经典记忆，第二种选择则需要在相位已知之前提交魔法态。我们确定了无辅助坐标式 Clifford+T 编译中，每放弃一位记忆所对应的已提交 T 门数的交换率 α。Clifford+T 格子的 Ramanujan 界给出 α≥2 且具有显式常数，这是谱方法的平方根障碍。一种初等行列式方法，利用四元数分子在 Q(√2) 的两个实嵌入中都是球面上的格点这一事实，计数了低于该障碍的任意旋转陪集附近的字，并渐近且无条件地给出 α≥11/5，而由此产生的球面截面的高度二分法将其提升至 α≥17/7。在 Clifford 框架陪集处，体积律在亚指数因子范围内成立：除消失比例外的所有 z 旋转都需要 T 计数 (3-o(1))log₂(1/ε)，而其已提交片段接近 Clifford 框架 z 旋转的过程（包括逐旋转流水线）具有 α≥3-o(1)，在 Ross-Selinger 典型成本假设下，一个分数直通族可以达到该值。在一个由 T 计数至 22 的穷举枚举支持的一致分布猜想下，一般有 α=3，并且记忆应以整旋转为单位释放。即使相位抵消为单位元，这些界仍然成立；侧信息通过条件熵进入；概率混合使成本减半但不改变速率。使用干净辅助比特和相位梯度催化剂，跨坐标批处理的表查找将速率驱动至 O(1/log log(1/ε))，因此常数速率律特定于坐标式合成。
英文摘要：
A phase that reaches a fault-tolerant processor in additive pieces can be remembered until the last piece arrives, or executed on arrival: the first option costs classical memory carried across rounds, the second costs magic states committed before the phase is known. We determine the exchange rate $α$, committed $T$ gates per bit of memory forgone, for ancilla-free coordinatewise Clifford+$T$ compilation. The Ramanujan bound gives $α\ge2$ with explicit constants, the square-root barrier of the spectral method. An elementary determinant method, using that quaternion numerators lie on spheres in both real embeddings of $\mathbb{Q}(\sqrt2)$, counts words near any rotation coset below that barrier. With a height dichotomy for the sphere sections it produces and a fibration over rational projections, it gives every $α<5/2$ unconditionally and uniformly over frames, by an exact SMT check of its continuum case analysis; $5/2$ is where this method stops. At Clifford-framed cosets the volume law holds up to subexponential factors: all but a vanishing fraction of $z$-rotations need $T$-count $(3-o(1))\log_2(1/\varepsilon)$, and processes whose committed pieces are near Clifford-framed $z$-rotations, including per-rotation pipelines, have $α\ge3-o(1)$, which a fractional-passthrough family attains under the Ross-Selinger typical-cost hypothesis. Under an equidistribution conjecture supported by exhaustive enumeration to $T$-count 22, $α=3$ in general and memory should be shed in whole rotations. The bounds hold even when the phases cancel to the identity; side information enters through a conditional entropy; probabilistic mixing halves the costs but not the rate. With clean ancillas and a phase-gradient catalyst, table lookups batched over coordinates drive the rate to $O(1/\log\log(1/\varepsilon))$, so the constant-rate law is specific to coordinatewise synthesis.
