Improved Quantum Random Self-Reduction for Linear Problems

arXiv:2609.39823v1 Announce Type: new 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.

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.

Quantum Physics (quant-ph); Computational Complexity (cs.CC)

arXiv:2609.39823 [quant-ph]

(or arXiv:2609.39823v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.39823 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Improved Quantum Random Self-Reduction for Linear Problems

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