AI 中文总结
研究多控制门的量子电路合成，提出有效方案减少基本门数量，改进了现有复杂度界限，给出等距变换和量子通道的电路构造，首次提出特定条件下单控制门合成方案并建立理论下限。
AI 中文摘要
多控制门的电路合成对量子计算至关重要。本文提出了有效的合成方案以减少多控制单量子比特门的基本门数量。对于在n个量子比特上合成一般的(n - 1)控制酉矩阵，将控制增量（CINC）和广义控制-X（GCX）门数量减少到O(n²)。对于(n - 1)控制特殊酉矩阵，复杂度进一步降至O(n)。利用所提电路给出了从n到m量子比特的等距变换和量子通道的基于量子比特的电路构造。还首次给出了d为素数时使用SUM门和单量子比特门的单控制门电路合成方案，建立了合成一般n量子比特酉矩阵所需SUM和CINC门数量的理论下限。
英文摘要
Circuit synthesis of multi-controlled gates is crucial for qudit ($d$-level) quantum computing. This paper presents efficient synthesis schemes that reduce the elementary gate count for multi-controlled single-qudit gates. For synthesizing general $(n-1)$-controlled unitaries on $n$ qudits, we reduce the controlled-increment (CINC) and generalized controlled-$X$ (GCX) gate counts to $O(n^2)$, improving upon existing $O(n^{2+\log_2 d})$ CINC and $O(n^3)$ GCX bounds. For $(n-1)$-controlled special unitaries, this complexity is further reduced to $O(n)$. Furthermore, we present a method for approximately synthesizing multi-controlled unitary gates with a linear complexity in the number of control qudits. By utilizing the proposed circuit, we present qudit-based circuit constructions for isometries and quantum channels from $n$ to $m$ qudits. Moreover, for the first time, we present a circuit synthesis scheme for single-controlled gates using SUM gates and single-qudit gates when $d$ is prime. This enables all CINC-based circuits for various quantum operations to be converted into SUM-gate circuits while preserving the same asymptotic complexity. Finally, we establish a theoretical lower bound on the number of SUM and CINC gates required to synthesize general $n$-qudit unitaries.
