发表机构
Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究角度编码电路中输入分布对输出方差的影响，通过代数输入纯度分析，证明matchgate电路保持1/n方差，非对角族在计算基输入下为零输出，并提出预处理证书以在电路执行前认证平均方差。
AI 中文摘要
贫瘠高原通过使损失梯度随量子比特数量呈指数级变小，阻碍了参数化量子电路的训练。对于角度编码的乘积态，我们展示了输入分布如何通过代数输入纯度（即输入与电路动力学李代数的重叠）影响输出变化。在指定的读出和Haar或精确群2-设计采样下，n个量子比特上的matchgate电路对每个纯乘积输入保持1/n阶的输出方差。相反，非对角族在任意深度和每个参数选择下，对计算基输入具有零输出。独立的均匀角度产生与matchgate相同阶的平均方差。对角泡利字符串的计数识别了这些零输出端点。对于任何具有至少2n个坐标的非零输入固定数据集，我们构造了一个经典预处理证书。仅当数据集的平均纯度通过可计算的阈值时，才接受共享的随机旋转。这保证了在电路执行前，非对角族的平均输出方差为1/n阶，且期望试验次数为常数。二进制和三元加权编码也从单个均匀标量输入中恢复了独立角度的平均纯度。数值实验考察了训练行为和对更大李代数的扩展。该保证涉及输入平均的输出方差，而成功的学习还取决于任务和编码保留的信息。
英文摘要
Barren plateaus hinder training of parameterized quantum circuits by making loss gradients exponentially small in the number of qubits. For angle-encoded product states, we show how the input distribution affects output variation through algebraic input purity, i.e. the input's overlap with the circuit's dynamical Lie algebra. With the specified readouts and Haar or exact group 2-design sampling, matchgate circuits on $n$ qubits retain output variance of order $1/n$ for every pure product input. The off-diagonal family instead has zero output on computational-basis inputs at any depth and for every parameter choice. Independent uniform angles yield mean variance of the same order as matchgates. A count of diagonal Pauli strings identifies these zero-output endpoints. For any fixed dataset of nonzero inputs with at least $2n$ coordinates, we construct a classical preprocessing certificate. A shared random rotation is accepted only when the dataset's mean purity passes a computable threshold. This certifies mean output variance of order $1/n$ for the off-diagonal family before circuit execution, with a constant expected number of trials. Binary and ternary weighted encodings also recover the independent-angle mean purity from one uniform scalar input. Numerical experiments examine training behavior and extensions to larger algebras. The guarantee concerns output variance averaged over inputs, while successful learning also depends on the task and information preserved by the encoding.
