发表机构
QudeLeap Research, Shanghai 200030, China; Thrust of Artificial Intelligence, Information Hub, The Hong Kong University of Science and Technology (Guangzhou), Guangzhou 511453, China; Centre for Quantum Technologies, National University of Singapore, Singapore（量子跃迁研究; 香港科技大学（广州）信息枢纽人工智能领域; 新加坡国立大学量子技术中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种实现任意多变量三角多项式的量子电路，其查询复杂度达到最优，并扩展到交换酉算子，为多变量量子算法与量子学习模型提供可扩展的实用方案。
AI 中文摘要
量子操作中编码数据的多项式变换是量子算法的基础构件，其查询复杂度衡量了这些操作被使用的频率。对于多变量情形，现有方法要么实现受限的多项式族，要么以查询成本随变量数量指数增长的代价分别实现单项式。我们提出了一种量子电路，可实现任意多变量三角多项式直至显式归一化。该电路在一个变量的频率分量上相干地处理另一个变量，使得所有分量能够共享相同的信号查询。每个变量被查询的次数恰好等于其在多项式中的次数，并且我们证明了该查询复杂度是最优的。我们进一步将该构造扩展到交换酉算子。所得电路在其共享本征基上同时应用多变量多项式，使用对每个酉算子的前向和逆向查询的最小数量。我们还研究了具有固定制备和最后一个输入局部读出的可训练旋转。在查询深度至少为2时，独立的均匀初始化使每个活跃读出梯度的方差下界由其平方分支权重和深度的平方根倒数设定，且该下界对数据状态和维度一致成立。对于监督平方损失，保留样本相关性给出了基于目标和输入谱的初始化梯度方差界。这也界定了在指定步长下第一次精确梯度步后的期望损失减少。这些结果为开发多变量量子算法和量子学习模型提供了可扩展且实用的解决方案。
英文摘要
Polynomial transformations of data encoded in quantum operations are a building block of quantum algorithms, and their query complexity measures how often those operations are used. For multiple variables, existing methods either realize restricted polynomial families or implement monomials separately at a query cost exponential in the number of variables. We present a quantum circuit that implements every multivariate trigonometric polynomial up to an explicit normalization. The circuit processes one variable coherently over the frequency components of the remaining variables, allowing all components to share the same signal queries. Each variable is queried exactly as many times as its degree in the polynomial, and we prove that this query complexity is optimal. We further extend the construction to commuting unitaries. The resulting circuit applies the multivariate polynomial simultaneously across their shared eigenbasis, using the minimum numbers of forward and inverse queries to each unitary. We also study trainable rotations with fixed preparation and a local readout of the last input. At query depth at least two, independent uniform initialization gives every active readout gradient a variance lower bound set by its squared branch weight and the inverse square root of the depth, uniformly over data states and dimensions. For supervised squared loss, retaining sample correlations gives an initialization gradient-variance bound in terms of targets and input spectra. This also bounds the expected loss decrease after the first exact gradient step with a specified step size. These results provide a scalable and practical solution for developing multivariate quantum algorithms and quantum learning models.
Comments30 pages including appendix
