A recent paper by Jordan et al. introduced Decoded Quantum Interferometry (DQI), a novel quantum algorithm that uses the quantum Fourier transform to reduce linear optimization problems -- max-XORSAT and max-LINSAT -- to decoding problems. In this paper, we extend DQI to optimization problems involving quadratic constraints, which we call max-QUADSAT. Leveraging a connection to quadratic Gauss sums, we give an efficient algorithm to prepare the DQI state for max-QUADSAT (DISCLAIMER: A mistake was found in one of the algorithm's steps, which invalidates this result unless a fix is found). To demonstrate that our algorithm achieves a quantum advantage, we introduce the Quadratic Optimal Polynomial Intersection (quadratic-OPI) problem, a restricted variant of OPI for which, to our knowledge, the standard DQI framework offers no algorithmic speedup. We show that quadratic-OPI is an instance of max-QUADSAT and use our algorithm to optimize it. Lastly, we present a new generalized proof of the "semicircle law" for the fraction of satisfied constraints, generalizing it to any DQI state of problems where the distribution of the number of satisfied constraints for a random assignment is sufficiently close to a binomial distribution. This condition holds exactly for the DQI state of max-LINSAT, and approximately holds in the max-QUADSAT case, with the approximation becoming exponentially better as the problem size increases. This establishes performance guarantees for our algorithm.
A recent paper by Jordan et al. introduced Decoded Quantum Interferometry (DQI), a novel quantum algorithm that uses the quantum Fourier transform to reduce linear optimization problems -- max-XORSAT and max-LINSAT -- to decoding problems. In this paper, we extend DQI to optimization problems involving quadratic constraints, which we call max-QUADSAT. Leveraging a connection to quadratic Gauss sums, we give an efficient algorithm to prepare the DQI state for max-QUADSAT (DISCLAIMER: A mistake was found in one of the algorithm's steps, which invalidates this result unless a fix is found). To demonstrate that our algorithm achieves a quantum advantage, we introduce the Quadratic Optimal Polynomial Intersection (quadratic-OPI) problem, a restricted variant of OPI for which, to our knowledge, the standard DQI framework offers no algorithmic speedup. We show that quadratic-OPI is an instance of max-QUADSAT and use our algorithm to optimize it. Lastly, we present a new generalized proof of the "semicircle law" for the fraction of satisfied constraints, generalizing it to any DQI state of problems where the distribution of the number of satisfied constraints for a random assignment is sufficiently close to a binomial distribution. This condition holds exactly for the DQI state of max-LINSAT, and approximately holds in the max-QUADSAT case, with the approximation becoming exponentially better as the problem size increases. This establishes performance guarantees for our algorithm.
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.
Attaining a quantum speedup in solving practically useful optimization problems has been one of the holy grails in the field of quantum computing. While prior approaches have demonstrated speedups for certain structured problem classes, establishing a clear and scalable advantage on broadly useful practical optimization problems remains challenging. Recently, a new approach to solving the max-LINSAT class of optimization problems has emerged, called Decoded Quantum Interferometry (DQI). In DQI, a combination of techniques rooted in (classical) coding theory and interferometry are used to obtain the solution of max-LINSAT. In the special problem instance of the optimal polynomial intersection (OPI) problem, strong evidence exists to show that an superpolynomial speedup exists over the best classical methods in obtaining an approximate solution. In this review, we give a self-contained description of DQI and the necessary background to understand the algorithm. Specifically, we give the essentials of Galois fields, optimization problems such as max-LINSAT and OPI, and coding theory, followed by a step-by-step walkthrough of the quantum algorithm and its operating principle.
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=1 whenever the rate satisfies R>0.75. This matches the previous average-case bound, whereas DQI cannot achieve s=1 unless R=1. Our existential bound guarantees the existence of a solution when R>0.7158, improving over the previous threshold R>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.
Decoded Quantum Interferometry (DQI) is a recently introduced quantum algorithm that reduces discrete optimization to decoding with potential advantages over the best-known polynomial-time classical algorithms for certain Max-LINSAT problems. In its original formulation, however, DQI treats all constraints uniformly and cannot exploit the weight structure present in most optimization problems of interest. In this work, we develop multivariate Decoded Quantum Interferometry (multivariate DQI) for weighted optimization problems, focusing on the weighted Max-LINSAT problem over a prime field. Grouping constraints into N blocks by distinct weights, we introduce multivariate DQI states built from N-variable polynomials of bounded total degree, and derive a closed-form asymptotic expression for both their optimal expectation value and their concentration behavior. We give an explicit preparation circuit using a single decoder call, and extend the analysis to imperfect decoding. We also show that, for certain weighted OPI problems, multivariate DQI outperforms a natural weighted analogue of Prange's algorithm, which serves as the weighted counterpart of the classical benchmark used in the unweighted setting. Finally, we extend the ideas to Hamiltonian DQI, obtaining approximate Gibbs states for commuting Pauli Hamiltonians with block structure.
In 2025, Jordan et al. introduced Decoded Quantum Interferometry (DQI), a quantum algorithm for combinatorial optimization based on decoding. They considered Reed-Solomon decoding and its associated optimization problem, called Optimal Polynomial Intersection (OPI), which may be viewed as a polynomial regression problem over a finite field. DQI exhibits provable speedups on certain problem instances and establishes a connection between decoding problems and optimization tasks. Leveraging the well-understood dual structure and decoding theory of algebraic geometry codes from other curve families, we introduce COFI: Curve-based Optimal Function Intersection. By considering two-point Hermitian codes, Suzuki codes, and extended norm-trace codes, we broaden the range of algebraic geometry codes used in DQI and identify families that offer further improvements over one-point Hermitian codes in the Hermitian Optimal Polynomial Intersection framework considered by Jordan and Gu. Depending on the family and parameter regime, these curves can reduce quantum resource requirements or increase the number of constraints that can be considered.
The Optimal Polynomial Intersection (OPI) problem is the following: Given sets S1,…,Sm⊆F and evaluation points a1,…,am∈F, find a polynomial Q∈F[x] of degree less than n so that Q(ai)∈Si for as many i∈{1,2,…,m} as possible. Decoded Quantum Interferometry (DQI) is a quantum algorithm that efficiently returns good solutions to the problem, even on worst-case instances (Jordan et. al., 2025). The quality of the solutions returned follows a semicircle law, which outperforms known efficient classical algorithms. But does DQI obtain the best possible solutions? That is, are there solutions better than the semicircle law for worst-case OPI instances? Surprisingly, before this work, the best existential results coincide with (and follow from) the best algorithmic results.
In this work, we show that there are better solutions for worst-case OPI instances over prime fields. In particular, DQI and the semicircle law are not optimal. For example, when the lists Si have size ρp for ρ∼1/2, our results imply the existence of a solution that asymptotically beats the semicircle law whenever n/m≥0.6225, and we show that an asymptotically perfect solution exists whenever n/m≥0.7496. Our results generalize to Max-LINSAT problems derived from any Maximum Distance Separable (MDS) code, and to any ρ∈(0,1). The key insight to our improvement is a connection to local leakage resilience of secret sharing schemes. Along the way, we recover several re-proofs of the existence of solutions achieving the semicircle law.