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arXiv:2606.04843

Decoded Quantum Interferometry Beyond Hamming: Rank-Metric and Translation Association Schemes

Alexandre Krajenbrink, Colin Krawchuk, Ansis Rosmanis, Matthias Rosenkranz

quant-phSubmitted Jun 3, 2026• Announced Jun 4, 2026• 0 citations
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Abstract

Decoded Quantum Interferometry (DQI) uses coherent decoding and a quantum Fourier transform to find high-quality solutions of structured optimisation problems. Existing analyses are closely tied to Hamming space, which underlies the optimisation objective, Dicke state preparation and the decoding step of the algorithm. Here we extend the core DQI mechanism beyond Hamming space to finite geometries with translation symmetry, where points are grouped into shells by their distance from a basepoint. Mathematically, these geometries are translation association schemes. In this setting the algorithm can be analysed by tracking one amplitude per shell, and biasing the prepared state towards high-quality solutions becomes a finite tridiagonal eigenvalue problem. As a non-Hamming example, we develop an efficient DQI protocol for finding an m x n finite-field matrix with smallest rank difference to a target matrix. Initial states are uniform superpositions over fixed-rank matrices, and Gabidulin codes provide candidates for efficient low-rank decoding up to a cutoff l. For this objective, this finds solutions with an effective-rank proxy near min(m,n)-l, and the corresponding expected score can be converted into a constant-probability bound on the residual rank of a sample. For Gabidulin nearest-codeword instances, a covering-radius obstruction shows that this bound does not imply an additive guarantee for the true optimum, and we do not claim a quantum advantage for the rank-metric construction. The results instead identify the geometric and coding ingredients for DQI beyond Hamming space.

Related papers

Discovering New Problems for Decoded Quantum Interferometry

Rhik Mazumder, Pradeep Niroula

Decoded quantum interferometry (DQI) uses classical error correction to solve optimization problems, but it is hard to tell which problems could give it a quantum advantage. We develop soundness rules that screen candidate problems by their code parameters, decoding guarantees and classical baselines. For the standard objective of counting satisfied constraints, we prove that under worst-case Hamming decoding DQI's guarantee can exceed Prange completion by at most $(\sqrt{2}-1)/2$. Optimal polynomial intersection (OPI) attains this margin cap asymptotically, so no candidate in this setting can exceed OPI's margin. We also derive a performance law for real-valued objectives: the objective fixes both the solution quality and which errors the decoder must correct. An AI-agent search guided by these rules found two candidate applications. Alternant max-agreement on McEliece-type public keys tests what a secret decoder is worth to DQI against classical solvers that see only the public instance. Multiplicative polynomial intersection, a knapsack-type problem built from discrete logarithms, pairs a cosine objective with a public decoder for signed errors. For both we prove decoding guarantees and finite-length comparisons with Prange restarts, and on the larger instances we tested, DQI's expected quality exceeds what our classical solvers reach within fixed budgets.

quant-pharXiv:2610.06753•Oct 6, 2026•0 citations

Weighted performance bounds for decoded quantum interferometry with imperfect decoding

Maoxin Gao, Yan Chang, Wei Huang +3 more

Decoded quantum interferometry (DQI) offers an approach to a class of combinatorial optimization problems by reducing quantum state preparation to a decoding task. However, the existing performance bound for imperfect decoding uses the maximum failure rate over error-weight layers to bound the performance loss uniformly, which may overestimate the effect of decoding failures in layers with low probability. Building on the existing analysis, we combine the failure rate of each layer with its probability weight in the chosen state to derive tighter performance lower bounds. For binary linear constraint problems with uniformly random target values, we obtain a finite-size lower bound on the average satisfaction ratio for fixed state parameters and a fixed decoder, and establish its improvement over the existing bound. In part of the parameter region where the existing bound falls to the random-assignment baseline, the weighted bound still provides a guarantee above that baseline. We further extend the analysis to general finite fields under a model of uniformly random translations of allowed sets of equal size. Trials in individual layers using a fixed belief propagation decoder illustrate the size of the improvement and provide confidence guarantees after accounting for sampling uncertainty. These results reveal the role of retaining the correspondence between the failure distribution and the state probabilities, and suggest directions for more refined performance analyses of other DQI constructions.

quant-pharXiv:2604.15025•Apr 17, 2026•0 citations

From Constraint to Code: DQI-Kit -- A Software Framework for Decoded Quantum Interferometry

Simon Thelen, Wolfgang Mauerer

Trying to solve hard optimisation problems with quantum techniques requires transformations of domain objectives and constraints into formats compatible with a chosen quantum algorithm. This often introduces inefficiencies and overheads that limit or even endanger potential quantum advantage for current and future approaches. To understand and mitigate these inefficiencies, software toolchains are essential for implementing transformations, analysing overheads and eventually selecting optimal transformation paths. Decoded Quantum Interferometry (DQI) is a novel approach that achieves apparent quantum advantage for certain algebraic optimisation problems. It natively operates on Max-LINSAT, which is unusual for combinatorial optimisation, and creates the need for software solutions that alleviate the burden of manually transforming problems of interest into this format. We present DQI-Kit, a software framework that provides a unified, extensible interface for automatically encoding constrained optimisation problems into Max-LINSAT. Users can describe the various types of objectives and constraints that are common in industrial optimisation problems. Our framework converts these into Max-LINSAT instances via a series of problem transformations and computes an estimate of the expected performance of DQI on these instances. We provide an initial analysis of the implemented transformations, discussing inefficiencies and ways to mitigate them. DQI-Kit is the basis for our ultimate goal of establishing a standardised framework that will enable further investigations to identify practical use cases for which quantum advantage with DQI can be achieved.

quant-pharXiv:2605.16955•May 19, 2026•0 citations

Cycle Codes and Decoded Quantum Interferometry

Anuj Apte, Shouvanik Chakrabarti, Andi Gu +6 more

Decoded Quantum Interferometry (DQI) reduces optimization problems with two-variable constraints to decoding cycle codes. For one such problem, namely MaxCut, prior work showed that DQI achieves a nontrivial satisfaction fraction guarantee only on linear-girth graphs, for which MaxCut is classically easy. However, these no-go results rely on minimum distance assumptions that do not represent true decodability thresholds for common noise channels, thus underestimating actual DQI performance. To estimate the true performance, we derive DQI satisfaction guarantees in the presence of imperfect decoding for fixed instances and generalize prior results for random instances. We then study homological cycle codes over arbitrary finite fields, and show that minimum-weight decoding is NP-hard for every field of size $q>2$, in contrast with known results for binary cycle codes. On the Linial--Simkin ensemble of regular graphs with logarithmic girth, we prove maximum-likelihood recovery bounds and a strong converse for a family of non-uniform additive noise channels. Moreover, LP decoders provide polynomial-time recovery of a positive fraction of randomly located errors with arbitrary values. These results allow us to prove upper and lower bounds on the approximate optima achievable by DQI when restricted to classical decoders. These bounds rule out quantum advantage in the regimes we analyze but also identify a family of regular Max-$k$-Cut instances on which DQI efficiently achieves a nontrivial satisfaction fraction guarantee.

quant-pharXiv:2609.40036•Oct 1, 2026•0 citations

A provable quantum advantage for approximate optimization via decoded quantum interferometry

Maximilian J. Kramer, Elies Gil-Fuster, Benjamin D. M. Jones +2 more

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 $0.3$, DQI achieves expected scores of approximately $0.85$. In contrast, exceeding the classical threshold of $0.65$ by any fixed amount with constant probability on sampled instances requires super-polynomially many classical membership queries.

quant-pharXiv:2610.02145•Oct 2, 2026•0 citations

Optimization by decoded quantum interferometry

Stephen P. Jordan, Noah Shutty, Mary K. Wootters +6 more

Achieving superpolynomial speed-ups for optimization has long been a central goal for quantum algorithms1. Here we introduce decoded quantum interferometry (DQI), a quantum algorithm that uses the quantum Fourier transform to reduce optimization problems to decoding problems. When approximating optimal polynomial fits over finite fields, DQI achieves a superpolynomial speed-up over known classical algorithms. The speed-up arises because the algebraic structure of the problem is reflected in the decoding problem, which can be solved efficiently. We then investigate whether this approach can achieve a speed-up for optimization problems that lack an algebraic structure but have sparse clauses. These problems reduce to decoding low-density parity-check codes, for which powerful decoders are known2,3. To test this, we construct a max-XORSAT instance for which DQI finds an approximate optimum substantially faster than general-purpose classical heuristics, such as simulated annealing. Although a tailored classical solver can outperform DQI on this instance, our results establish that combining quantum Fourier transforms with powerful decoding primitives provides a promising new path towards quantum speed-ups for hard optimization problems. Decoded quantum interferometry is a quantum algorithm that uses the quantum Fourier transform to reduce optimization problems to decoding problems.

arXiv:2408.08292•Nature•Oct 22, 2025•18 citations
Decoded Quantum Interferometry (DQI) uses coherent decoding and a quantum Fourier transform to find high-quality solutions of structured optimisation problems. Existing analyses are closely tied to Hamming space, which underlies the optimisation objective, Dicke state preparation and the decoding step of the algorithm. Here we extend the core DQI mechanism beyond Hamming space to finite geometries with translation symmetry, where points are grouped into shells by their distance from a basepoint. Mathematically, these geometries are translation association schemes. In this setting the algorithm can be analysed by tracking one amplitude per shell, and biasing the prepared state towards high-quality solutions becomes a finite tridiagonal eigenvalue problem. As a non-Hamming example, we develop an efficient DQI protocol for finding an m x n finite-field matrix with smallest rank difference to a target matrix. Initial states are uniform superpositions over fixed-rank matrices, and Gabidulin codes provide candidates for efficient low-rank decoding up to a cutoff l. For this objective, this finds solutions with an effective-rank proxy near min(m,n)-l, and the corresponding expected score can be converted into a constant-probability bound on the residual rank of a sample. For Gabidulin nearest-codeword instances, a covering-radius obstruction shows that this bound does not imply an additive guarantee for the true optimum, and we do not claim a quantum advantage for the rank-metric construction. The results instead identify the geometric and coding ingredients for DQI beyond Hamming space.
Decoded quantum interferometry (DQI) uses classical error correction to solve optimization problems, but it is hard to tell which problems could give it a quantum advantage. We develop soundness rules that screen candidate problems by their code parameters, decoding guarantees and classical baselines. For the standard objective of counting satisfied constraints, we prove that under worst-case Hamming decoding DQI's guarantee can exceed Prange completion by at most (2−1)/2(\sqrt{2}-1)/2(2​−1)/2. Optimal polynomial intersection (OPI) attains this margin cap asymptotically, so no candidate in this setting can exceed OPI's margin. We also derive a performance law for real-valued objectives: the objective fixes both the solution quality and which errors the decoder must correct. An AI-agent search guided by these rules found two candidate applications. Alternant max-agreement on McEliece-type public keys tests what a secret decoder is worth to DQI against classical solvers that see only the public instance. Multiplicative polynomial intersection, a knapsack-type problem built from discrete logarithms, pairs a cosine objective with a public decoder for signed errors. For both we prove decoding guarantees and finite-length comparisons with Prange restarts, and on the larger instances we tested, DQI's expected quality exceeds what our classical solvers reach within fixed budgets.
Decoded quantum interferometry (DQI) offers an approach to a class of combinatorial optimization problems by reducing quantum state preparation to a decoding task. However, the existing performance bound for imperfect decoding uses the maximum failure rate over error-weight layers to bound the performance loss uniformly, which may overestimate the effect of decoding failures in layers with low probability. Building on the existing analysis, we combine the failure rate of each layer with its probability weight in the chosen state to derive tighter performance lower bounds. For binary linear constraint problems with uniformly random target values, we obtain a finite-size lower bound on the average satisfaction ratio for fixed state parameters and a fixed decoder, and establish its improvement over the existing bound. In part of the parameter region where the existing bound falls to the random-assignment baseline, the weighted bound still provides a guarantee above that baseline. We further extend the analysis to general finite fields under a model of uniformly random translations of allowed sets of equal size. Trials in individual layers using a fixed belief propagation decoder illustrate the size of the improvement and provide confidence guarantees after accounting for sampling uncertainty. These results reveal the role of retaining the correspondence between the failure distribution and the state probabilities, and suggest directions for more refined performance analyses of other DQI constructions.
Trying to solve hard optimisation problems with quantum techniques requires transformations of domain objectives and constraints into formats compatible with a chosen quantum algorithm. This often introduces inefficiencies and overheads that limit or even endanger potential quantum advantage for current and future approaches. To understand and mitigate these inefficiencies, software toolchains are essential for implementing transformations, analysing overheads and eventually selecting optimal transformation paths. Decoded Quantum Interferometry (DQI) is a novel approach that achieves apparent quantum advantage for certain algebraic optimisation problems. It natively operates on Max-LINSAT, which is unusual for combinatorial optimisation, and creates the need for software solutions that alleviate the burden of manually transforming problems of interest into this format. We present DQI-Kit, a software framework that provides a unified, extensible interface for automatically encoding constrained optimisation problems into Max-LINSAT. Users can describe the various types of objectives and constraints that are common in industrial optimisation problems. Our framework converts these into Max-LINSAT instances via a series of problem transformations and computes an estimate of the expected performance of DQI on these instances. We provide an initial analysis of the implemented transformations, discussing inefficiencies and ways to mitigate them. DQI-Kit is the basis for our ultimate goal of establishing a standardised framework that will enable further investigations to identify practical use cases for which quantum advantage with DQI can be achieved.
Decoded Quantum Interferometry (DQI) reduces optimization problems with two-variable constraints to decoding cycle codes. For one such problem, namely MaxCut, prior work showed that DQI achieves a nontrivial satisfaction fraction guarantee only on linear-girth graphs, for which MaxCut is classically easy. However, these no-go results rely on minimum distance assumptions that do not represent true decodability thresholds for common noise channels, thus underestimating actual DQI performance. To estimate the true performance, we derive DQI satisfaction guarantees in the presence of imperfect decoding for fixed instances and generalize prior results for random instances. We then study homological cycle codes over arbitrary finite fields, and show that minimum-weight decoding is NP-hard for every field of size q>2q>2q>2, in contrast with known results for binary cycle codes. On the Linial--Simkin ensemble of regular graphs with logarithmic girth, we prove maximum-likelihood recovery bounds and a strong converse for a family of non-uniform additive noise channels. Moreover, LP decoders provide polynomial-time recovery of a positive fraction of randomly located errors with arbitrary values. These results allow us to prove upper and lower bounds on the approximate optima achievable by DQI when restricted to classical decoders. These bounds rule out quantum advantage in the regimes we analyze but also identify a family of regular Max-kkk-Cut instances on which DQI efficiently achieves a nontrivial satisfaction fraction guarantee.
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.
Achieving superpolynomial speed-ups for optimization has long been a central goal for quantum algorithms1. Here we introduce decoded quantum interferometry (DQI), a quantum algorithm that uses the quantum Fourier transform to reduce optimization problems to decoding problems. When approximating optimal polynomial fits over finite fields, DQI achieves a superpolynomial speed-up over known classical algorithms. The speed-up arises because the algebraic structure of the problem is reflected in the decoding problem, which can be solved efficiently. We then investigate whether this approach can achieve a speed-up for optimization problems that lack an algebraic structure but have sparse clauses. These problems reduce to decoding low-density parity-check codes, for which powerful decoders are known2,3. To test this, we construct a max-XORSAT instance for which DQI finds an approximate optimum substantially faster than general-purpose classical heuristics, such as simulated annealing. Although a tailored classical solver can outperform DQI on this instance, our results establish that combining quantum Fourier transforms with powerful decoding primitives provides a promising new path towards quantum speed-ups for hard optimization problems. Decoded quantum interferometry is a quantum algorithm that uses the quantum Fourier transform to reduce optimization problems to decoding problems.