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

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.

Figures (5)

• Figure 1: General $n$-qubit recursive LCU circuit (schematic). The middle block is a $\mathrm{SELECT}_{\widetilde{L}}$ in the $U_n$-nesting form; the first and last controlled blocks are shown explicitly, and $\cdots$ denotes the intermediate ones. The ancilla is uncomputed by applying $\mathrm{PREP}_u^\dagger$.
• Figure 2: General $n$-qubit asymmetric-LCU convolution circuit with a QFT adder realization of $\mathrm{SELECT}_L$. The middle block is implemented as $\mathrm{QFT}_N\rightarrow \Phi_{A\to D}\rightarrow \mathrm{QFT}_N^\dagger$ on the data register $D$, with control from ancilla/index register $A$.
• Figure 3: Asymmetric-LCU convolution circuit with a ripple-carry adder realization of $\mathrm{SELECT}_L$. The middle block implements $\mathrm{ADD}^{\mathrm{rc}}_N:(i,k)\mapsto(i,k+i \bmod N)$. Ancillary carry wires required by a specific ripple-carry variant are omitted for schematic clarity.
• Figure 4: Recursive construction of the reflected generator $U_n$. The operation on the upper register is conditioned on the LSB ($q_0$): if $q_0=\ket{0}$, $U_{n-1}$ is applied; if $q_0=\ket{1}$, $J_{n-1}$ is applied.
• Figure 5: Compiled $n$-qubit bitwise realization with explicit internal structure. Top panel: $\mathrm{SELECT}_{\widetilde{L}}=\left(\prod_{m=0}^{n-1}W_{A_m}\!\left[\mathrm{INC}_{[m:n-1]}^{\mathrm{(cmp)}}\right]\right)(I_A\otimes J_n)$ inside the asymmetric-LCU pipeline. Bottom panel: decomposition of one block $W_{A_m}\!\left[\mathrm{INC}_{[m:n-1]}^{\mathrm{(cmp)}}\right]$ into a chain of multi-controlled $X$ operations $W_{A_m,D_m,\ldots,D_{j-1}}[X_{D_j}]$ ending with $W_{A_m}[X_{D_m}]$.

Theorems & Definitions (28)

• Theorem 1: Asymmetric LCU realization of circular convolution
• proof
• Remark 1: Asymmetry and the state-input specialization
• Remark 2: Normalization of $\bm{a}$ and $\bm{b}$
• Lemma 1: Modular adder realization of $\mathrm{SELECT}_L$
• proof
• Proposition 1: Structural recursion of $U_n$
• Remark 3: Polynomial synthesis of $\widetilde{L}_{i,n}$ within the recursive framework
• Remark 4: Complexity of the direct recursive realization
• Remark 5: Equivalent optimized bitwise recursive compilation