资源可调的非线性逐元素变换量子电路实现
Resource-Tunable Quantum Circuit Implementation of Nonlinear Element-Wise Transformations
- School of Mathematical Sciences, Harbin Engineering University（哈尔滨工程大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出资源可调的量子框架，通过多项式分解实现非线性逐元素变换，在查询复杂度、门复杂度和辅助比特间权衡，适用于激活函数和图像变换。
AI中文摘要：
非线性变换在量子机器学习及其他量子数值应用中具有重要意义，此类变换可通过多项式近似归结为逐元素多项式的实现。已有工作确立了变换的可行性并提升了查询或辅助空间效率，但电路深度仍随度数线性增长。在本工作中，我们开发了一个资源可调的量子框架，将$(2^d-1)$次多项式分解为$m$个较低次因子，其中$m$控制并行度。通过将$m$从1变化到$2^d-1$，该框架在查询复杂度、额外门复杂度、辅助量子比特数量和归一化之间提供权衡，涵盖从二叉树实现到深度为$\mathcal{O}(n+d)$且具有高度并行化查询（匹配查询下界）的构造。进一步分析了多项式近似和实现误差的影响。我们针对sigmoid和tanh等非线性激活函数以及逐像素图像变换演示了该框架。
英文摘要：
Nonlinear transformations are important in quantum machine learning and other quantum numerical applications, which can be reduced to the implementation of element-wise polynomials through polynomial approximation. Existing works establish the feasibility of transformations and improve query or auxiliary-space efficiency, but circuit depth remains linear in scale with degree. In this work, we develop a resource-tunable quantum framework that decomposes a $(2^d-1)$-degree polynomial into $m$ lower-degree factors, with $m$ controlling the degree of parallelism. By varying $m$ from $1$ to $2^d-1$, the framework provides a trade-off among query complexity, additional gate complexity, ancilla count, and normalization, ranging from a binary-tree implementation to a $\mathcal{O}(n+d)$-depth construction with highly parallel oracle queries that match the query lower bound. The effects of polynomial approximation and implementation errors are further analyzed. We demonstrate the framework for nonlinear activation functions such as sigmoid and tanh, as well as pixel-wise image transformations.
