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

2025-12-12

AI Summary

This paper addresses the low efficiency of pointwise multiplication and convolution operations in quantum signal processing with complex-valued functions. To this end, it proposes a novel “Processing through Encoding” paradigm. Methodologically, complex functions are directly encoded into auxiliary qubit states, enabling pointwise products to emerge implicitly in the final-state amplitudes; combined with Fourier-basis encoding and the inverse quantum Fourier transform (IQFT), an end-to-end quantum convolution circuit is realized. Key contributions include: (i) the first quantum pointwise multiplication scheme that requires no explicit arithmetic gates; (ii) the first integrable and verifiable quantum convolution circuit; and (iii) a theoretically complete construction, validated via numerical simulation and modular implementation using the quantumaudio toolkit—accurately generating target products and convolution outputs. This work establishes a new pathway for quantum signal processing.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

Abstract

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.

Problem

• Develop quantum circuits for pointwise multiplication and convolution of functions
• Encode multiple complex functions onto auxiliary qubits for processing
• Apply techniques to quantum audio signal processing and synthesis

Innovation

• Quantum circuits encode multiple complex functions onto qubits
• Pointwise multiplication forms naturally in quantum state coefficients
• Convolution via Fourier encoding and inverse quantum Fourier transform