Asymmetric Linear-Combination-of-Unitaries Realization of Quantum Convolution via Modular Adders

TL;DR

This paper introduces a novel quantum circuit framework for implementing discrete circular convolution using an asymmetric linear combination of unitaries, enabling efficient quantum convolution operations with phase preservation.

Contribution

It presents an asymmetric-LCU formulation of quantum circular convolution that preserves phase information and simplifies kernel state preparation, along with a recursive operator algebra compatible with quantum workflows.

Findings

Hermitian operator for real kernels facilitates spectral transformations

Recursive construction and optimized compilation of the convolution circuit

Analysis of resource scaling and implementation trade-offs

Abstract

Discrete circular convolution over $\mathbb{Z}/N\mathbb{Z}$ is a linear operator and can be implemented on quantum hardware within the linear-combination-of-unitaries (LCU) framework. In this work, we make this connection explicit through an asymmetric-LCU formulation: circular convolution is the postselected block of a circuit whose controlled-shift unitary is modular addition on computational-basis states. The asymmetry is essential: fixing the postselection state to the uniform state $|u\rangle$ while supplying the kernel state $|\mathbf{b}\rangle$ as the input ancilla naturally preserves the complex coefficients $b_i$ within the block, whereas a symmetric overlap would yield $|b_i|^2$ weights and erase their phases. Accordingly, when $|\mathbf{a}\rangle$ and $|\mathbf{b}\rangle$ are supplied by upstream quantum routines, the convolution subroutine requires only the fixed uncompute $\mathrm{PREP}_u^\dagger$, completely avoiding the need for a kernel-dependent inverse preparation $\mathrm{PREP}_b^\dagger$. We then introduce a reversal matrix $J_n=X^{\otimes n}$ and define reflected shifts $\widetilde{L}_{i,n}=L_{i,n}J_n$. This symmetrization yields a recursive operator algebra for convolution that is natively compatible with LCU/block-encoding workflows. The resulting symmetrized operator differs from circular convolution only by one known input-side $J_n$ layer. Crucially, for real-valued kernels, the resulting operator $H_n(\mathbf{b})=\sum_i b_i\widetilde{L}_{i,n}$ is Hermitian, providing a direct Hermitian interface for quantum singular value transformation (QSVT) and related spectral transformations. Based on this framework, we present a transparent recursive construction, paired with an exactly equivalent optimized bitwise compilation of the same $\mathrm{SELECT}$ block. Finally, we evaluate implementation trade-offs and resource scaling under explicit cost-model conventions.