Improved Quantum Random Self-Reduction for Linear Problems

2026-09-30

AI Summary

This study addresses the efficiency bottleneck of quantum randomized self-reductions for linear problems over finite fields by proposing a novel quantum reduction algorithm. The proposed method integrates additive combinatorics, amplitude amplification, and coherent quantum query techniques to locate external vectors via quantum search, thereby circumventing the explicit learning of the Bogolyubov–Ruzsa subspace. This approach effectively optimizes the trade-off between average-case oracle queries and matrix-vector verification costs. Compared with existing methods presented at SODA 2024, the proposed algorithm significantly reduces the time complexity to O(n^{4/3}), offering a more efficient solution for applying quantum computing to algebraic problems.

Abstract

We study quantum random self-reductions for linear problems over finite fields. Let $M\in\mathbb{F}^{n\times n}$ be an arbitrary matrix, and let $\mathcal{O}$ be an oracle that agrees with the linear map $x\mapsto Mx$ on an $\varepsilon$-fraction of inputs $x\sim\mathbb{F}^n$. Given coherent access to $\mathcal{O}$ and coherent entry access to $M$, we give a uniform quantum reduction that computes $Mx$ on any prescribed input $x$ with probability at least $2/3$ in time $\widetilde{O}(nT^{1/3})$, for $n\le T\le n^{3/2}$ and constant field size and $\varepsilon$, where $T$ is the cost of one coherent query to $\mathcal{O}$. In particular, when $T=\widetilde{O}(n)$, the reduction runs in time $\widetilde{O}(n^{4/3})$, improving the $\widetilde{O}(n^{3/2}+T)$ reduction of Asadi, Golovnev, Gur, Shinkar, and Subramanian (SODA 2024). Our reduction uses the Bogolyubov--Ruzsa subspace guaranteed by additive combinatorics, but it avoids learning this subspace explicitly, which was computationally expensive for the previous reduction; in particular, it does not recover a basis for its orthogonal complement. The main technical step is to decompose the inputs into sparse pieces and find a vector that lies outside the Bogolyubov--Ruzsa subspace via a quantum search based on amplitude amplification. This yields a tunable tradeoff between the cost of querying the average-case oracle and the cost of verifying matrix-vector products.

Problem

• quantum random self-reduction
• linear problems
• finite fields
• average-case oracle
• matrix-vector product

Innovation

• Quantum random self-reduction
• Bogolyubov-Ruzsa subspace
• Amplitude amplification
• Additive combinatorics
• Linear problems over finite fields