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arXiv:2509.08328 · Quantum Science and Technology

Towards solving industrial integer linear programs with decoded quantum interferometry

Francesc Sabater, Ouns El Harzli, Geert-Jan Besjes, Marvin Erdmann, Johannes Klepsch, Jonas Hiltrop, Jean-François Bobier, Yudong Cao, Carlos A. Riofrío

Published Apr 28, 2026• 0 citations
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Abstract

Abstract Optimization via decoded quantum interferometry (DQI) has recently gained a great deal of attention as a promising avenue for solving optimization problems using quantum computers. In this paper, we apply DQI to an industrial optimization problem in the automotive industry: the vehicle option-package pricing problem. Our main contributions are (1) formulating the industrial problem as an integer linear program (ILP), (2) converting the ILP into instances of max-XORSAT, and (3) developing a detailed quantum circuit implementation for belief propagation, a heuristic algorithm for decoding low density parity-check codes. Thus, we provide a full implementation of the DQI algorithm using Belief Propagation, which can be applied to any industrially relevant ILP by first transforming it into a max-XORSAT instance. We also evaluate the effectiveness of our implementation by benchmarking it against both Gurobi and a random sampling baseline.

DOI
10.1088/2058-9565/ae65d8

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Abstract Optimization via decoded quantum interferometry (DQI) has recently gained a great deal of attention as a promising avenue for solving optimization problems using quantum computers. In this paper, we apply DQI to an industrial optimization problem in the automotive industry: the vehicle option-package pricing problem. Our main contributions are (1) formulating the industrial problem as an integer linear program (ILP), (2) converting the ILP into instances of max-XORSAT, and (3) developing a detailed quantum circuit implementation for belief propagation, a heuristic algorithm for decoding low density parity-check codes. Thus, we provide a full implementation of the DQI algorithm using Belief Propagation, which can be applied to any industrially relevant ILP by first transforming it into a max-XORSAT instance. We also evaluate the effectiveness of our implementation by benchmarking it against both Gurobi and a random sampling baseline.
We develop a new benchmarking scheme for the Decoded Quantum Interferometry (DQI) algorithm quantifying the number of quantum gates required to obtain an optimal solution to a problem amenable to DQI. We apply the benchmarking scheme to the Binary Paint Shop Problem (BPSP) in order to benchmark the performance of DQI against a state of the art classical solver. To do so, we provide an explicit construction of a quantum circuit implementation of a greedy decoder for low-density parity check codes arising from max-2-XORSAT problems.
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.
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.
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.
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.
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 NNN blocks by distinct weights, we introduce multivariate DQI states built from NNN-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.