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arXiv:2610.00502

Quantum Algorithms for OPI Variants Beyond Locality and Classical Decodability

Seyoon Ragavan, Noah Shutty

quant-phSubmitted Sep 30, 2026• Announced Oct 2, 2026• 0 citations
v1 · Sep 30, 2026v2 · Oct 2, 2026
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Abstract

Regev's reduction is a quantum algorithmic framework for finding codewords satisfying nonlinear constraints by decoding the dual code. To date, applications that have not been dequantized have relied on efficient classical decoders and coordinate-wise constraints specifying a set of allowed values for each coordinate. We push past these two restrictions separately via two separate contributions. Our first contribution uses a quantum decoder to find solutions $\mathbf{y}\in(\mathbb{F}_q\setminus\{0\})^m$ to $\mathbf{B}\mathbf{y}=0$, where $\mathbf{B}\in\mathbb{F}_q^{n\times m}$. For fixed prime $q>2$, Chen, Liu, and Zhandry (Eurocrypt 2022) solve this problem for random matrices with $m=Ω(n^2)$, a regime now covered by classical algorithms. We adapt their template to codes (spanned by the rows of $\mathbf{B}$) with a ``two-fold multiplication property'': the coordinate-wise products of pairs of codewords span a space of dimension much smaller than $m$. Our second contribution retains classical decoding but allows global constraints on symbol frequencies. We study ``histogram-local'' constraints, which specify the allowed numbers of occurrences of each symbol. For broad families of these constraints, we show that a uniformly random satisfying vector remains satisfying with constant probability after resampling a uniformly random coordinate. This stability provides sufficient Fourier mass at decodable Hamming weights, yielding efficient quantum algorithms for variants of optimal polynomial intersection (OPI).

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In recent years, a particularly interesting line of research has focused on designing quantum algorithms for code and lattice problems inspired by Regev's reduction. The core idea is to use a decoder for a given code to find short codewords in its dual. For example, Jordan et al. demonstrated how structured codes can be used in this framework to exhibit some quantum advantage. In particular, they showed how the classical decodability of Reed-Solomon codes can be leveraged to solve the Optimal Polynomial Intersection (OPI) problem quantumly. This approach was further improved by Chailloux and Tillich using stronger soft decoders, though their analysis was restricted to a specific setting of OPI. In this work, we reconcile these two approaches. We build on a recent formulation of the reduction by Chailloux and Hermouet in the lattice-based setting, which we rewrite in the language of codes. With this reduction, we show that the results of Jordan et al. can be recovered under Bernoulli noise models, simplifying the analysis. This characterization then allows us to integrate the stronger soft decoders of Chailloux and Tillich into the OPI framework, yielding improved algorithms.

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A provable quantum advantage for approximate optimization via decoded quantum interferometry

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Quantum Advantage from Soft Decoders

André Chailloux, Jean–Pierre Tillich

In the last years, Regev's reduction has been used as a quantum algorithmic tool for providing a quantum advantage for variants of the decoding problem. Following this line of work, the authors of [JSW+24] have recently come up with a quantum algorithm called Decoded Quantum Interferometry that is able to solve in polynomial time several optimization problems. They study in particular the Optimal Polynomial Interpolation (OPI) problem, which can be seen as a decoding problem on Reed-Solomon codes. In this work, we provide strong improvements for some instantiations of the OPI problem. The most notable improvements are for the $ISIS_{\infty}$ problem (originating from lattice-based cryptography) on Reed-Solomon codes but we also study different constraints for OPI. Our results provide natural and convincing decoding problems for which we believe to have a quantum advantage. Our proof techniques involve the use of a soft decoder for Reed-Solomon codes, namely the decoding algorithm from Koetter and Vardy [KV03]. In order to be able to use this decoder in the setting of Regev's reduction, we provide a novel generic reduction from a syndrome decoding problem to a coset sampling problem, providing a powerful and simple to use theorem, which generalizes previous work and is of independent interest. We also provide an extensive study of OPI using the Koetter and Vardy algorithm.

arXiv:2411.12553•ACM Symposium on Theory of Computing (STOC)•Jun 15, 2025•4 citations
Regev's reduction is a quantum algorithmic framework for finding codewords satisfying nonlinear constraints by decoding the dual code. To date, applications that have not been dequantized have relied on efficient classical decoders and coordinate-wise constraints specifying a set of allowed values for each coordinate. We push past these two restrictions separately via two separate contributions.
Our first contribution uses a quantum decoder to find solutions y∈(Fq∖{0})m\mathbf{y}\in(\mathbb{F}_q\setminus\{0\})^my∈(Fq​∖{0})m to By=0\mathbf{B}\mathbf{y}=0By=0, where B∈Fqn×m\mathbf{B}\in\mathbb{F}_q^{n\times m}B∈Fqn×m​. For fixed prime q>2q>2q>2, Chen, Liu, and Zhandry (Eurocrypt 2022) solve this problem for random matrices with m=Ω(n2)m=Ω(n^2)m=Ω(n2), a regime now covered by classical algorithms. We adapt their template to codes (spanned by the rows of B\mathbf{B}B) with a ``two-fold multiplication property'': the coordinate-wise products of pairs of codewords span a space of dimension much smaller than mmm.
Our second contribution retains classical decoding but allows global constraints on symbol frequencies. We study ``histogram-local'' constraints, which specify the allowed numbers of occurrences of each symbol. For broad families of these constraints, we show that a uniformly random satisfying vector remains satisfying with constant probability after resampling a uniformly random coordinate. This stability provides sufficient Fourier mass at decodable Hamming weights, yielding efficient quantum algorithms for variants of optimal polynomial intersection (OPI).
In recent years, a particularly interesting line of research has focused on designing quantum algorithms for code and lattice problems inspired by Regev's reduction. The core idea is to use a decoder for a given code to find short codewords in its dual. For example, Jordan et al. demonstrated how structured codes can be used in this framework to exhibit some quantum advantage. In particular, they showed how the classical decodability of Reed-Solomon codes can be leveraged to solve the Optimal Polynomial Intersection (OPI) problem quantumly. This approach was further improved by Chailloux and Tillich using stronger soft decoders, though their analysis was restricted to a specific setting of OPI. In this work, we reconcile these two approaches. We build on a recent formulation of the reduction by Chailloux and Hermouet in the lattice-based setting, which we rewrite in the language of codes. With this reduction, we show that the results of Jordan et al. can be recovered under Bernoulli noise models, simplifying the analysis. This characterization then allows us to integrate the stronger soft decoders of Chailloux and Tillich into the OPI framework, yielding improved algorithms.
Decoded quantum interferometry (DQI) is a novel paradigm for tackling approximate optimization problems on quantum computers. This framework comes with strong performance guarantees and exploits a well-established duality between optimization and coding theory. A central question, however, is whether DQI can actually provably outperform all polynomial-time classical algorithms. In this work, we establish such an advantage in an oracle setting: we consider an optimization task called folded optimal polynomial intersection (folded OPI), where the acceptance sets are chosen randomly and accessed through membership oracles. We establish a strict gap between the approximation ratio achievable by any polynomial-time classical algorithm and the approximation ratio achieved by the DQI algorithm. Our proof builds on Jordan et al.'s DQI framework for approximate optimization and extends the classical lower-bound method underlying Yamakawa and Zhandry's exact-search oracle separation to approximation. Building on recent developments by Sun and Wootters, Horinaga and Yamakawa, and Jo, we further show that a modified version of the DQI algorithm achieves a strictly higher score guarantee on typical sampled folded OPI instances. As a concrete example, for code rate 0.30.30.3, DQI achieves expected scores of approximately 0.850.850.85. In contrast, exceeding the classical threshold of 0.650.650.65 by any fixed amount with constant probability on sampled instances requires super-polynomially many classical membership queries.
The Optimal Polynomial Intersection (OPI) problem asks us to find a low-degree polynomial over a finite field whose values lie in prescribed subsets on as many given inputs as possible. Decoded quantum interferometry (DQI) gives a quantum algorithm for OPI in parameter regimes beyond those achieved by the best known classical heuristics. Follow-up works improve the parameter regimes, but their analyses are limited to average-case settings. Recently, Sun and Wootters showed that, even in the worst case, OPI has a solution in a larger parameter regime than the one covered by DQI. However, they left open whether one can design a quantum algorithm that solves OPI in the worst-case beyond the DQI regime. We give such a quantum algorithm. As a byproduct, we also improve the existential bound of Sun and Wootters in certain parameter regimes. In particular, when each subset contains roughly half of the field elements, our algorithm finds a solution with satisfaction rate s=1s=1s=1 whenever the rate satisfies R>0.75R>0.75R>0.75. This matches the previous average-case bound, whereas DQI cannot achieve s=1s=1s=1 unless R=1R=1R=1. Our existential bound guarantees the existence of a solution when R>0.7158R> 0.7158R>0.7158, improving over the previous threshold R>0.7495R>0.7495R>0.7495. More generally, our existential results extend to the Max-LINSAT problem with respect to arbitrary maximum distance separable (MDS) codes. The corresponding algorithmic results apply only to MDS codes whose dual admits an efficient list decoder. Our results are obtained through a novel application of a Brascamp--Lieb-type inequality in the MDS setting, which may have further applications.
In recent years, a particularly interesting line of research has focused on designing quantum algorithms for code and lattice problems inspired by Regev's reduction. The core idea is to use a decoder for a given code to find short codewords in its dual. For example, Jordan et al. [JSW + 24] demonstrated how structured codes can be used in this framework to exhibit some quantum advantage. In particular, they showed how the classical decodability of Reed-Solomon codes can be leveraged to solve the Optimal Polynomial Intersection (OPI) problem quantumly. This approach was further improved by Chailloux and Tillich [CT25] using stronger soft decoders, though their analysis was restricted to a specific setting of OPI. In this work, we reconcile these two approaches. We build on a recent formulation of the reduction by Chailloux and Hermouet [CH25] in the lattice-based setting, which we rewrite in the language of codes. With this reduction, we show that the results of Jordan et al. can be recovered under Bernoulli noise models, simplifying the analysis. This characterization then allows us to integrate the stronger soft decoders of Chailloux and Tillich into the OPI framework, yielding improved algorithms.
Recently, Jordan et al. (Nature, 2025) introduced a novel quantum-algorithmic technique called Decoded Quantum Interferometry (DQI) for solving specific combinatorial optimization problems associated with classical codes. They presented a constraint-satisfaction problem called Optimal Polynomial Intersection (OPI) and showed that, for this problem, a DQI algorithm running in polynomial time can satisfy a larger fraction of constraints than any known polynomial-time classical algorithm. In this work, we propose several improvements to the DQI algorithm, including sidestepping the quadratic-time Dicke state preparation. Given random access to the input, we show how these improvements result in a nearly linear-time DQI algorithm for the OPI problem. Concurrently and independently with this work, Khattar et al. (arXiv:2510:10967) also construct a nearly linear-time DQI algorithm for OPI using slightly different techniques.
In the last years, Regev's reduction has been used as a quantum algorithmic tool for providing a quantum advantage for variants of the decoding problem. Following this line of work, the authors of [JSW+24] have recently come up with a quantum algorithm called Decoded Quantum Interferometry that is able to solve in polynomial time several optimization problems. They study in particular the Optimal Polynomial Interpolation (OPI) problem, which can be seen as a decoding problem on Reed-Solomon codes. In this work, we provide strong improvements for some instantiations of the OPI problem. The most notable improvements are for the ISIS∞ISIS_{\infty}ISIS∞​ problem (originating from lattice-based cryptography) on Reed-Solomon codes but we also study different constraints for OPI. Our results provide natural and convincing decoding problems for which we believe to have a quantum advantage. Our proof techniques involve the use of a soft decoder for Reed-Solomon codes, namely the decoding algorithm from Koetter and Vardy [KV03]. In order to be able to use this decoder in the setting of Regev's reduction, we provide a novel generic reduction from a syndrome decoding problem to a coset sampling problem, providing a powerful and simple to use theorem, which generalizes previous work and is of independent interest. We also provide an extensive study of OPI using the Koetter and Vardy algorithm.