AI 中文总结
该研究解决了2006年的开放问题，证明肖尔算法所需的量子傅里叶变换（QFT）在常深度下近似时必然需要扇出（Fanout）操作，将肖尔算法在NISQ电路中的实现可行性与Fanout关联。
AI 中文摘要
肖尔算法是量子优越性的典型目标，其核心操作依赖量子傅里叶变换（QFT）。我们解决了Fang、Fenner、Green、Homer和Zhang在2006年提出的开放问题，证明对于任意n量子比特模数，在常深度下近似QFT必然需要n量子比特扇出（Fanout）操作。形式上，令$\boldsymbol{\text{QFT}}_q$为作用于$n = \boldsymbol{\text{ceil}}(\boldsymbol{\text{log}} q)$个量子比特、计算模数为$q$的QFT的门。已知任意n量子比特$\text{QFT}_q$可使用$\boldsymbol{\text{FANOUT}}_n$在常深度下实现，即$\text{QFT}_q \boldsymbol{\text{∈ QAC}}^0_f$。我们通过使用$\text{QFT}_q$门构造出“不可忽略的‘猫态’”来证明其逆命题，因此$\text{QFT}_q \boldsymbol{\text{∈ QAC}}^0 \boldsymbol{\text{当且仅当 FANOUT}}_n \boldsymbol{\text{∈ QAC}}^0$。在$q = 2^n$的情况（如肖尔算法中的情况）下，我们使用单个$\text{QFT}_{2^n}$门和$O(1)$个两量子比特局部门来近似$\text{FANOUT}_n$，从而将用噪声中等规模量子（NISQ）电路实现肖尔算法的可行性与扇出（Fanout）的可行性关联起来。
英文摘要
Shor's algorithm is a canonical quantum supremacy target whose core operation relies on the Quantum Fourier Transform (QFT). In this note, we resolve an open question of Fang, Fenner, Green, Homer and Zhang from 2006 by showing that approximating QFT in constant depth, for any $n$-qubit modulus, necessarily requires the $n$-qubit Fanout operation. Formally, let $\mathsf{QFT}_q$ be the gate acting on $n = \lceil \log q \rceil$ qubits that computes the QFT under modulus $q$. It is known that any $n$-qubit $\mathsf{QFT}_q$ can be implemented in constant depth using $\mathsf{FANOUT}_n$, i.e. $\mathsf{QFT}_q \in \mathsf{QAC}^0_f$. We prove the converse by using a $\mathsf{QFT}_q$ gate to construct a state of "non-negligible felinity". Consequently, $\mathsf{QFT}_q \in \mathsf{QAC}^0 \iff \mathsf{FANOUT}_n \in \mathsf{QAC^0}$. In the case of $q = 2^n$, such as in Shor's, we approximate $\mathsf{FANOUT}_n$ using a single $\mathsf{QFT}_{2^n}$ gate and $O(1)$ two-qubit local gates, thus tying the feasibility of realizing Shor's algorithm with NISQ circuits to that of Fanout.
