In 2005, H{\o}yer and \v{S}palek showed that constant-depth quantum circuits augmented with multi-qubit Fanout gates are quite powerful, able to compute a wide variety of Boolean functions as well as the quantum Fourier transform. They also asked what other multi-qubit gates could rival Fanout in terms of computational power, and suggested that the quantum Threshold gate might be one such candidate. Threshold is the gate that indicates if the Hamming weight of a classical basis state input is greater than some target value. We prove that Threshold is indeed powerful--there are polynomial-size constant-depth quantum circuits with Threshold gates that compute Fanout to high fidelity. Our proof is a generalization of a proof by Rosenthal that exponential-size constant-depth circuits with generalized Toffoli gates can compute Fanout. Our construction reveals that other quantum gates able to "weakly approximate" Parity can also be used as substitutes for Fanout.

翻译：2005年，Høyer与Špalek证明了增强多量子比特扇出门的常数深度量子电路具有强大计算能力，能够计算多种布尔函数以及量子傅里叶变换。他们同时提出疑问：还有哪些多量子比特门能在计算能力上与扇出门相媲美，并指出量子阈值门可能是候选之一。阈值门的功能是判断经典基态输入的汉明权重是否超过特定目标值。我们证明阈值门确实具有强大计算能力——存在包含阈值门的多项式规模常数深度量子电路，能以高保真度实现扇出计算。我们的证明推广了Rosenthal关于含广义托佛利门的指数规模常数深度电路可实现扇出计算的论证。该构造方法揭示：其他能够"弱近似"奇偶校验的量子门同样可作为扇出门的替代方案。