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arXiv:2603.24441

Benchmarking Techniques for Decoded Quantum Interferometry

Leon Bollmann, Maximilian Hess

quant-phSubmitted Mar 25, 2026• Announced Mar 26, 2026• 0 citations
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Abstract

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.

AI Summary

Decoded Quantum Interferometry (DQI) converts structured optimization into syndrome decoding, but practical performance depends on imperfect decoding and coherent implementation costs. This paper studies how to estimate the probability of obtaining an optimal solution, rather than only expected objective values, and applies the resulting benchmarks to the Binary Paint Shop Problem (BPSP). The proposed exact benchmark evaluates a chosen decoder on every error pattern of Hamming weight at most the DQI polynomial degree $l$, then computes probabilities for classically identified optimal solutions. An approximate version samples 2,000 errors per Hamming weight to estimate decoding failure rates and assumes correctly decoded errors are uniformly distributed within each Hamming shell. Small-instance comparisons, using 10 random instances at each of six sizes between 5 and 15 car pairs, find this approximation somewhat conservative. For BPSP, two max-XORSAT encodings yield codes whose distance remains approximately constant with problem size. For max-2-XORSAT, decoding is expressed as a graph $T$-join problem. A greedy matching decoder is implemented using CNOT and Toffoli gates with $O(n^2)$ auxiliary qubits. Benchmarks on 10 random instances per size from 10 to 80 car pairs suggest an optimal degree $l\approx 2n/5$. Accounting for decoder cost favors greedy decoding over a hypothetical blossom-based decoder, but both DQI estimates scale less favorably than Gurobi. Limitations include exponential-cost exact benchmarking, an unproved uniform-shell approximation, no quantum blossom implementation, and incomparable absolute resource units for quantum gates and Gurobi work. The tested sizes do not establish larger-instance behavior.

No sign of AI-generated textIncrementalStrong assumptionsPossible errorSmall numerics only

Central claim may not hold: The purported exact DQI measurement probabilities omit cross terms between different Hamming weights, contradicting the immediately preceding coherent-state formula even on a one-bit perfectly decoded example. The uniform-subset approximation squares an expected signed sum rather than averaging its square, omitting a generally nonzero variance contribution and invalidating its interpretation as the expected measurement probability. The gate-cost estimate uses success probabilities conditioned on postselection without charging for failed postselection attempts, whose success probability is R. The supplied greedy-decoder circuit retains syndrome-dependent path-register garbage, so without an additional coherent uncomputation it does not prepare the interference-preserving state assumed by the benchmarks. The stated conversion from objective value to satisfied-constraint count is incorrect: the definitions imply s(x)=(f(x)+m)/2.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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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.
Decoded Quantum Interferometry (DQI) converts structured optimization into syndrome decoding, but practical performance depends on imperfect decoding and coherent implementation costs. This paper studies how to estimate the probability of obtaining an optimal solution, rather than only expected objective values, and applies the resulting benchmarks to the Binary Paint Shop Problem (BPSP).
The proposed exact benchmark evaluates a chosen decoder on every error pattern of Hamming weight at most the DQI polynomial degree lll, then computes probabilities for classically identified optimal solutions. An approximate version samples 2,000 errors per Hamming weight to estimate decoding failure rates and assumes correctly decoded errors are uniformly distributed within each Hamming shell. Small-instance comparisons, using 10 random instances at each of six sizes between 5 and 15 car pairs, find this approximation somewhat conservative.
For BPSP, two max-XORSAT encodings yield codes whose distance remains approximately constant with problem size. For max-2-XORSAT, decoding is expressed as a graph TTT-join problem. A greedy matching decoder is implemented using CNOT and Toffoli gates with O(n2)O(n^2)O(n2) auxiliary qubits. Benchmarks on 10 random instances per size from 10 to 80 car pairs suggest an optimal degree l≈2n/5l\approx 2n/5l≈2n/5. Accounting for decoder cost favors greedy decoding over a hypothetical blossom-based decoder, but both DQI estimates scale less favorably than Gurobi.
Limitations include exponential-cost exact benchmarking, an unproved uniform-shell approximation, no quantum blossom implementation, and incomparable absolute resource units for quantum gates and Gurobi work. The tested sizes do not establish larger-instance behavior.
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.
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.
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.
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.
Optimization problems are among the leading candidates for industrially relevant quantum advantage. Decoded quantum interferometry (DQI) has been proposed to tackle approximate optimization, establishing a connection to classical decoding problems. While previous work has primarily focused on the theoretical complexity of DQI, comparatively little is known about its empirical performance relative to classical algorithms. In this work, we shed further light on the complexity of DQI and investigate numerically whether classical sampling methods can emulate the optimization capabilities of DQI. We first present a simplified analytical characterization of DQI that connects its expected performance to binomial statistics, and we identify concrete obstacles in further studying the complexity of DQI. Exploiting the fact that DQI output probabilities are efficiently computable, we apply Markov chain Monte Carlo (MCMC) techniques, particularly block-Gibbs sampling, to sample from the induced distribution. We study the runtime scaling of these methods for two optimization problems called max-XORSAT, where we reach beyond 100010001000 effective qubits; and OPI, where we reach beyond 150150150 effective qubits. Our results show that MCMC algorithms can reliably attain the approximation ratios expected from DQI across a broad range of problem sizes. In OPI, in the regime where a super-polynomial advantage is claimed for DQI, we observe an empirical runtime for MCMC that scales approximately as 1.1n1.1^{n}1.1n, indicating exponential growth with a comparatively small base. Our findings do not refute existing quantum advantage claims but provide new empirical evidence that classical sampling algorithms can closely match DQI's optimization performance, offering a more nuanced perspective on the practical advantage of DQI.
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.