（一个变体的）Clifford 电路综合是NP完全的
(A Variant of) Clifford Circuit Synthesis is NP-Complete
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- Johannes Kepler University Linz（林茨约翰内斯·开普勒大学）
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中文总结 AI 辅助
本研究证明最优Clifford电路综合的一个变体是NP难的，通过将3边可着色性归约到CZ门电路综合，并证明额外Clifford门无法改进深度，从而明确其复杂性边界。
中文摘要 AI 辅助
最优电路综合问题是指针对预先指定的基本门集，寻找给定功能的最短深度电路表示。这是量子与经典硬件设计中的核心问题。虽然经典版本已被充分理解——无论是在启发式方法还是严格的难度断言方面——但对于最优量子电路综合的了解则少得多。我们专注于最优Clifford电路综合，已知有多种启发式方法，例如通过归约到3-SAT。我们的主要结果提供了匹配的难度结果：最优Clifford综合的一个变体是NP难的。证明分两部分进行：（i）将3正则图上的3边可着色性归约到仅涉及CZ门的电路综合问题；（ii）证明额外基本Clifford门——尤其是Hadamard、相位和CNOT——的可用性不能进一步改进最优电路深度。我们的工作加深了对Clifford电路复杂性的理解：模拟和等价检查属于P类，而判定一个Clifford酉是否能在指定深度内实现则是NP完全的。
英文摘要
Optimal circuit synthesis is the problem of finding the shortest-depth circuit representation of a given functionality with respect to a pre-specified elementary gate set. This is a central problem in both quantum and classical hardware design. While the classical version is very well understood -- both in terms of heuristics and rigorous hardness assertions -- much less is known about optimal quantum circuit synthesis. We focus on optimal Clifford circuit synthesis, for which various heuristics are known, e.g. via reduction to 3-SAT. Our main result supplies a matching hardness result: a variant of optimal Clifford synthesis is NP-hard. The proof proceeds in two parts: (i) reduce 3-edge colorability on 3-regular graphs to a circuit synthesis problem that only involves CZ gates, (ii) prove that the availability of additional elementary Clifford gates -- most notably: Hadamard, phase and CNOT -- cannot lead to further improvements of the optimal circuit depth. Our work sharpens the complexity-theoretic understanding of Clifford circuits: simulation and equivalence checking are in P, whereas deciding whether a Clifford unitary admits an implementation within a prescribed depth is NP-complete.
