用量子电路实现函数乘法与卷积，通过编码直接生成结果。
Processing through encoding: Quantum circuit approaches for point-wise multiplication and convolution
- 将多个复函数编码到辅助量子比特，直接生成点乘结果
- 利用傅里叶系数点乘加逆QFT，实现卷积运算
- 已集成至quantumaudio包，适用于量子音频处理
本文提出基于量子电路的复函数点乘与卷积方法，称为“通过编码进行处理”。通过将多个复函数编码至辅助量子比特，其点乘 $f(x)g(x)$ 自然表现为量子态部分系数。依据卷积定理，进一步展示如何构造卷积 $f*g$：先编码傅里叶系数 $\ ext{\mathcal{F}}[f]$ 与 $\ ext{\mathcal{F}}[g]$，进行点乘，再执行逆量子傅里叶变换。本文讨论了这些技术的模拟实现，将其集成至扩展版 \ exttt{quantumaudio} 包用于音频信号处理，并提供了初步实验验证。该工作为量子信号处理开辟新路径，有望应用于量子增强音频操作与合成等领域。
原文摘要 · Abstract (English)
This paper introduces quantum circuit methodologies for pointwise multiplication and convolution of complex functions, conceptualized as "processing through encoding". Leveraging known techniques, we describe an approach where multiple complex functions are encoded onto auxiliary qubits. Applying the proposed scheme for two functions $f$ and $g$, their pointwise product $f(x)g(x)$ is shown to naturally form as the coefficients of part of the resulting quantum state. Adhering to the convolution theorem, we then demonstrate how the convolution $f*g$ can be constructed. Similarly to related work, this involves the encoding of the Fourier coefficients $\mathcal{F}[f]$ and $\mathcal{F}[g]$, which facilitates their pointwise multiplication, followed by the inverse Quantum Fourier Transform. We discuss the simulation of these techniques, their integration into an extended \verb|quantumaudio| package for audio signal processing, and present initial experimental validations. This work offers a promising avenue for quantum signal processing, with potential applications in areas such as quantum-enhanced audio manipulation and synthesis.
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