发表机构
Korea University; Sookmyung Women’s University; Purdue University; Indiana University; Seoul National University Hospital（韩国大学; 淑明女子大学; 普渡大学; 印第安纳大学; 首尔大学医院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种傅里叶-几何电路设计准则，通过分解算子空间为频率平面，指导局部旋转与纠缠层放置，使目标傅里叶系数可达，并经消融实验和物理建模验证其有效性。
AI 中文摘要
参数化量子电路（PQC）的输出可以表示为有限傅里叶级数，其可访问的频率由数据编码门固定。虽然编码器决定了哪些频率可以出现，但相应的傅里叶系数取决于可训练门和纠缠门的排列方式。尽管现有研究提供了表征门结构如何影响傅里叶系数的度量，但这些研究并未将分析转化为明确的设计准则，以规定如何排列门使目标系数可达。在本文中，我们提供了这样的准则。利用编码生成器的伴随作用，我们将算子空间分解为以频率索引的二维不变平面，并表明每个傅里叶系数恰好是有效态和可观测量在该频率平面上的双线性投影之和。由于对于泡利编码，高频平面由混合多量子比特泡利字符串张成，因此只有当局部旋转和纠缠层被排列为使有效态和可观测量都在其某个平面上获得支撑时，目标系数才能对输出有贡献。对于泡利读出和交换的双量子比特纠缠门，这产生了一个电路设计规则，该规则针对给定的交互图，指定了局部旋转和纠缠层的放置，使目标傅里叶系数可达。使用单编码层，我们通过放置消融实验、来自PDEBench的回归任务和物理信息麦克斯韦场建模验证了所提出的设计，并进一步证明了其对中等模拟门噪声和耗散的鲁棒性。
英文摘要
The output of a parameterized quantum circuit (PQC) can be expressed as a finite Fourier series whose accessible frequencies are fixed by the data-encoding gates. While the encoder determines which frequencies can appear, the corresponding Fourier coefficients depend on how the trainable and entangling gates are arranged. Although existing studies provide metrics for characterizing how gate structure affects Fourier coefficients, they do not translate these analyses into an explicit design criterion specifying how gates should be arranged to make a target coefficient reachable. In this paper, we provide such a criterion. Using the adjoint action of the encoding generator, we decompose operator space into two-dimensional invariant planes indexed by frequency and show that each Fourier coefficient is exactly a sum of bilinear projections of the effective state and observable onto the planes at that frequency. Because, for Pauli encodings, high-frequency planes are spanned by mixed multi-qubit Pauli strings, a target coefficient can contribute to the output only when local rotations and entangling layers are arranged so that both the effective state and observable acquire support on one of its planes. For Pauli readouts and commuting two-qubit entanglers, this yields a circuit design rule that specifies, for a given interaction graph, a placement of local rotations and entangling layers that makes a target Fourier coefficient reachable. Using a single encoding layer, we validate the proposed design through a placement ablation, regression tasks from PDEBench and physics-informed Maxwell field modeling, and further demonstrate its robustness to moderate simulated gate noise and damping.
