AI 中文总结
该研究提出ZX-演算优化Clifford+T电路的精确性屏障，解释两类电路优化表现差异，在双量子比特电路上实现高比例T门不可压缩的刚性认证。
AI 中文摘要
门综合与电路优化通常被分开研究，且二者相互作用的证据相互矛盾：ZX-演算重写可移除Solovay-Kitaev电路中稳定比例的T门，却几乎无法从数论综合的电路中移除任何T门。我们证明这两种行为均源于同一个界：对于任何能精确保留所实现元素的优化器（包括所有带提取的可靠ZX重写），可实现的T门数量下界为所综合环元素的分母指数。该精确性屏障可按实例计算，且通过一个经认证的因子将精确后处理与感知近似的重新综合区分开，在递归深度为5时该因子可达101倍。这两种行为正是该屏障在距基准层不同距离处的作用结果。对于Solovay-Kitaev电路，我们证明局部ZX简化层（蜘蛛融合与恒等移除）可精确计算Z₂*Z₈的自由积范式，从而实现按实例的精确压缩；在经过校准的遍历性假设下，该压缩还具有与深度无关的极限定律，且在两个独立构建的网络上得到验证。对于数论综合的电路，基准层已被完全饱和：在单量子比特字上，自动化ZX简化可通过基于Z轴正规子群相位关联的最小T门数量闭式公式，精确达到该基准层；在双量子比特及以上，相同赋值可产生无条件刚性证书，该证书在量子香农分解加gridsynth流水线中，可证明99.4%-99.9%的综合T门数量不可压缩，且随着精度收紧，刚性会增强。这解释并预测了该流水线近期报道的近乎零优化的规模。
英文摘要
Gate synthesis and circuit optimization are usually studied separately, and evidence on their interaction is contradictory: ZX-calculus rewriting removes a stable fraction of Solovay-Kitaev circuits, yet almost nothing from number-theoretically synthesized circuits. We show both behaviours follow from a single bound. For any optimizer that preserves the implemented element exactly--including all sound ZX rewriting with extraction--the achievable T-count is bounded below by the denominator exponent of the synthesized ring element. This exactness barrier is computable per instance and separates exact post-processing from approximation-aware resynthesis by a certified factor reaching 101x at recursion depth five. The two behaviours are then the barrier operating at different distances from the floor. For Solovay-Kitaev circuits we prove that the local ZX simplification layer (spider fusion and identity removal) computes exactly the free-product normal form of Z_2 * Z_8, giving exact per-instance compression and, under a calibrated ergodicity hypothesis, a depth-independent limit law confirmed on two independently constructed nets. For number-theoretically synthesized circuits the floor is already saturated: on single-qubit words automated ZX simplification attains it exactly, via a closed-form formula for minimal T-count in terms of phase linkage through the Z-axis normalizer. At two qubits and beyond the same valuation yields unconditional rigidity certificates, which on the quantum-Shannon-decomposition plus gridsynth pipeline certify 99.4-99.9% of the synthesized T-count as incompressible, with rigidity strengthening as accuracy tightens. This explains, and predicts the size of, the near-null optimization recently reported for that pipeline.
Comments32 pages, 8 figures
