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arXiv:2609.39823

Improved Quantum Random Self-Reduction for Linear Problems

Vahid R. Asadi, Shuichi Hirahara, Nobutaka Shimizu

quant-phcs.CCSubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations
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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.

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quant-pharXiv:2610.01848•Oct 2, 2026•0 citations
We study quantum random self-reductions for linear problems over finite fields. Let M∈Fn×nM\in\mathbb{F}^{n\times n}M∈Fn×n be an arbitrary matrix, and let O\mathcal{O}O be an oracle that agrees with the linear map x↦Mxx\mapsto Mxx↦Mx on an ε\varepsilonε-fraction of inputs x∼Fnx\sim\mathbb{F}^nx∼Fn. Given coherent access to O\mathcal{O}O and coherent entry access to MMM, we give a uniform quantum reduction that computes MxMxMx on any prescribed input xxx with probability at least 2/32/32/3 in time O~(nT1/3)\widetilde{O}(nT^{1/3})O(nT1/3), for n≤T≤n3/2n\le T\le n^{3/2}n≤T≤n3/2 and constant field size and ε\varepsilonε, where TTT is the cost of one coherent query to O\mathcal{O}O. In particular, when T=O~(n)T=\widetilde{O}(n)T=O(n), the reduction runs in time O~(n4/3)\widetilde{O}(n^{4/3})O(n4/3), improving the O~(n3/2+T)\widetilde{O}(n^{3/2}+T)O(n3/2+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.
The linear problem specified by an n × n matrix M over a finite field is the problem of computing the product of M and a given vector x. We present optimal error-tolerant random self-reductions (also known as worst-case to average-case reductions) for all linear problems: Given a linear-size circuit that computes M x on an ε-fraction of inputs x for a positive constant ε, we construct a randomized linear-size circuit that computes M x for all inputs x with high probability. This resolves the open problem posed by Asadi, Golovnev, Gur, Shinkar, and Subramanian (SODA’24), who presented quantum n1.5-time random self-reductions for all linear problems. Somewhat surprisingly, we also demonstrate the quantum advantage of their quantum reduction over classical uniform algorithms, by proving that any classical subquadratic-time random self-reduction requires the advice complexity of Ω(log(1/ε) · logn), as long as the field size is at most 1/ε. We complement this advice complexity lower bound by presenting (1) a random self-reduction with the optimal advice complexity of O(log(1/ε) · logn) and (2) a uniform random self-reduction over a large finite field.
We study the problem of designing worst-case to average-case reductions for quantum algorithms. For all linear problems, we provide an explicit and efficient transformation of quantum algorithms that are only correct on a small (even sub-constant) fraction of their inputs into ones that are correct on all inputs. This stands in contrast to the classical setting, where such results are only known for a small number of specific problems or restricted computational models. En route, we obtain a tight Ω(n2)Ω(n^2)Ω(n2) lower bound on the average-case quantum query complexity of the Matrix-Vector Multiplication problem. Our techniques strengthen and generalise the recently introduced additive combinatorics framework for classical worst-case to average-case reductions (STOC 2022) to the quantum setting. We rely on quantum singular value transformations to construct quantum algorithms for linear verification in superposition and learning Bogolyubov subspaces from noisy quantum oracles. We use these tools to prove a quantum local correction lemma, which lies at the heart of our reductions, based on a noise-robust probabilistic generalisation of Bogolyubov's lemma from additive combinatorics.
Worst-case to average-case reductions are a cornerstone of complexity theory, providing a bridge between worst-case hardness and average-case computational difficulty. While recent works have demonstrated such reductions for fundamental problems using deep tools from ad- ditive combinatorics, these approaches often suffer from substantial complexity and suboptimal overheads. In this work, we focus on the quantum setting, and provide a new reduction for the Matrix-Vector Multiplication problem that is more efficient, and conceptually simpler than previous constructions. By adapting hardness self-amplification techniques to the quantum do- main, we obtain a quantum worst-case to average-case reduction with improved dependence on the success probability, laying the groundwork for broader applications in quantum fine-grained complexity.
In the quantum linear systems problem (QLSP), we are given query access to a ddd-sparse N×NN\times NN×N matrix AAA with condition number κκκ, and the ability to prepare a quantum state proportional to a vector b⃗\vec bb. The goal is to output an εεε-approximation to the quantum state proportional to the solution x⃗\vec xx of Ax⃗=b⃗A\vec{x}=\vec{b}Ax=b. Following a long line of work, the best previously known quantum algorithms for the QLSP had query complexities O(κdlog⁡(1/ε))O(κd\log(1/ε))O(κdlog(1/ε)) and κd(κd/ε)o(1)κ\sqrt d(κd/ε)^{o(1)}κd​(κd/ε)o(1), while the best known lower bounds were Ω(κlog⁡(1/ε))Ω(κ\log(1/ε))Ω(κlog(1/ε)) and Ω(κd)Ω(κ\sqrt d)Ω(κd​). We improve these bounds and show that the complexity of the QLSP is Θ(κdlog⁡(1/ε))Θ(κ\sqrt d\log(1/ε))Θ(κd​log(1/ε)). We also resolve an open problem of Berry and Childs by showing that any N×NN\times NN×N unitary can be implemented with bounded error using O(N)O(\sqrt N)O(N​) queries to its matrix entries.
Matrix scaling is a simple to state, yet widely applicable linear-algebraic problem: the goal is to scale the rows and columns of a given non-negative matrix such that the rescaled matrix has prescribed row and column sums. Motivated by recent results on first-order quantum algorithms for matrix scaling, we investigate the possibilities for quantum speedups for classical second-order algorithms, which comprise the state-of-the-art in the classical setting. We first show that there can be essentially no quantum speedup in terms of the input size in the high-precision regime: any quantum algorithm that solves the matrix scaling problem for n×nn \times nn×n matrices with at most mmm non-zero entries and with ℓ2\ell_2ℓ2​-error ε=Θ~(1/m)\varepsilon=\widetildeΘ(1/m)ε=Θ(1/m) must make Ω~(m)\widetildeΩ(m)Ω(m) queries to the matrix, even when the success probability is exponentially small in nnn. Additionally, we show that for ε∈[1/n,1/2]\varepsilon\in[1/n,1/2]ε∈[1/n,1/2], any quantum algorithm capable of producing ε100\frac{\varepsilon}{100}100ε​-ℓ1\ell_1ℓ1​-approximations of the row-sum vector of a (dense) normalized matrix uses Ω(n/ε)Ω(n/\varepsilon)Ω(n/ε) queries, and that there exists a constant ε0>0\varepsilon_0>0ε0​>0 for which this problem takes Ω(n1.5)Ω(n^{1.5})Ω(n1.5) queries. To complement these results we give improved quantum algorithms in the low-precision regime: with quantum graph sparsification and amplitude estimation, a box-constrained Newton method can be sped up in the large-ε\varepsilonε regime, and outperforms previous quantum algorithms. For entrywise-positive matrices, we find an ε\varepsilonε-ℓ1\ell_1ℓ1​-scaling in time O~(n1.5/ε2)\widetilde O(n^{1.5}/\varepsilon^2)O(n1.5/ε2), whereas the best previously known bounds were O~(n2polylog(1/ε))\widetilde O(n^2\mathrm{polylog}(1/\varepsilon))O(n2polylog(1/ε)) (classical) and O~(n1.5/ε3)\widetilde O(n^{1.5}/\varepsilon^3)O(n1.5/ε3) (quantum).
Random Clifford operators have numerous applications in quantum computing, including randomized benchmarking, classical shadows, and quantum authentication. However, sampling and implementing uniformly random nnn-qubit Clifford incur near-quadratic complexity due to the size of Clifford group. We introduce a cryptographic way to overcome these barriers: trapdoored Clifford operator distributions whose samples are computationally indistinguishable from uniformly random Cliffords, yet implementing them can be much faster given the trapdoor. We construct a distribution of trapdoored Clifford operators whose elements can be sampled and implemented in near-linear time under a variant of the learning parity with noise assumption. Our constructions allow fast tableau action on Pauli labels for classical simulation, and also can be optimized to admit polylogarithmic-depth implementation. Along the way, we construct trapdoored matrices over finite fields that support efficient multiplication by both a matrix and its inverse, resolving an open question left by Vaikuntanathan and Zamir [SODA'26]. We use these constructions to obtain faster protocols based on random Cliffords. We also explore their applications to the worst-case to average-case reductions for matrix and Clifford problems including the iterated matrix multiplication and Clifford circuit synthesis. In particular, we show the hardness of batching Clifford circuits: synthesizing circuits that apply the same Clifford to multiple registers is at least as hard as worst-case matrix multiplication, even when synthesis succeeds on a small constant fraction of random Cliffords. This extends to approximate implementations by general quantum circuits.