发表机构
Institute of Software, Chinese Academy of Sciences; University of Chinese Academy of Sciences（中国科学院软件研究所; 中国科学院大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明了基础设定下的精确CNOT综合问题的判定版本为NP完全、优化版本为NP难，通过从网格图哈密顿路径问题的多项式归约完成证明，还暗示了多个相关问题的难度。
AI 中文摘要
精确CNOT综合旨在找到实现可逆线性变换的最小规模CNOT电路。尽管多个相关综合模型已被证明具有计算难度，但它们的难度证明依赖于额外结构，如受限量子比特连通性、编码输入或无限制中间变量。最基础的设定——恒等输入、固定数量的带标签量子比特、无辅助量子比特以及全对全CNOT连通性——的复杂性此前尚未得到解决。在本研究中，我们证明了这种基础精确CNOT综合问题的判定版本是NP完全的，因此其优化版本是NP难的。我们的证明通过两步从网格图上的哈密顿路径问题进行多项式归约：首先，通过一元编码映射将网格图等距嵌入超立方体；然后将该超立方体哈密顿路径问题编码为基础精确CNOT综合问题。主要挑战在于，CNOT综合仅指定最终奇偶校验矩阵，无法直接强制哈密顿路径所需的中间顶点访问。为克服这一困难，我们引入额外的记录量子比特，将所需的中间顶点访问编码为最终变换，从而强制任何CNOT电路实现都必须实现预期的路径结构。除CNOT综合外，我们的结果直接暗示了几个相关问题的难度，包括GL(n,2)上的最短字问题、GL(n,2)上的Cayley图距离计算、顺序异或程序的最小化以及相位多项式电路的精确综合。
英文摘要
Exact CNOT synthesis seeks a minimum-size circuit implementing a given invertible binary linear transformation. We study its vanilla formulation: synthesis from the identity on fixed labeled qubits, without ancillas and with all-to-all connectivity. We prove that this problem is NP-hard and that its bounded decision version is NP-complete. Hardness persists even for unitriangular targets that are low-rank perturbations of the identity and under near-linear gate budgets. Our proof gives a polynomial-time reduction from the NP-complete Hamiltonian path problem on grid graphs. A unary hypercube embedding translates graph vertices into circuit parities and edge traversals into CNOT updates. The central challenge is that Hamiltonian paths require intermediate vertex visits, whereas exact synthesis constrains only the final transformation. We bridge this gap with a replicated recorder qubits construction that encodes the required parities in the final outputs, together with a tight gate budget that forces every sufficiently short implementation to trace a Hamiltonian path. Crucially, this path structure is enforced by the construction rather than imposed as a restriction on the circuit. The result directly implies NP-hardness for shortest word problem and Cayley-graph distance computation over $\mathrm{GL}(n,2)$ with elementary transvections as generators, as well as fixed-width sequential XOR program minimization. For CNOT-minimal exact phase polynomial synthesis, it yields NP-hardness with a zero phase polynomial; extending the reduction to grid-graph Hamiltonian cycle problem establishes hardness even when the final linear transformation is the identity. These complementary results show that the linear and phase components each independently suffice for NP-hardness of exact phase polynomial synthesis.
