Processing through encoding: Quantum circuit approaches for point-wise multiplication and convolution

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.