用傅里叶分析揭示变分量子电路的函数表达能力，可预测哪些电路适合特定数据。
Fourier Analysis of Variational Quantum Circuits for Supervised Learning
- 将变分量子电路的输出建模为傅里叶级数，分析其频谱特性。
- 发现变分参数会抑制某些频率成分，导致部分傅里叶系数恒为零。
- 提出算法精确计算电路频谱，可提前判断最佳候选电路。
变分量子电路（VQC）的功能空间可通过截断傅里叶和描述。我们发现，该截断傅里叶和的频谱不仅取决于编码门，还受变分电路结构约束，部分系数被强制为零，等效于移除对应频率。据我们所知，首次给出了傅里叶系数关于变分参数的三角多项式函数依赖关系。基于此，我们提出一种算法，可精确计算任意给定电路的频谱及对应傅里叶系数。最后，通过比较数据集的傅里叶变换与可用频谱，可预测从一组候选电路中哪个最能拟合数据。
原文摘要 · Abstract (English)
VQC can be understood through the lens of Fourier analysis. It is already well-known that the function space represented by any circuit architecture can be described through a truncated Fourier sum. We show that the spectrum available to that truncated Fourier sum is not entirely determined by the encoding gates of the circuit, since the variational part of the circuit can constrain certain coefficients to zero, effectively removing that frequency from the spectrum. To the best of our knowledge, we give the first description of the functional dependence of the Fourier coefficients on the variational parameters as trigonometric polynomials. This allows us to provide an algorithm which computes the exact spectrum of any given circuit and the corresponding Fourier coefficients. Finally, we demonstrate that by comparing the Fourier transform of the dataset to the available spectra, it is possible to predict which VQC out of a given list of choices will be able to best fit the data.
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