Where is the true boundary of the quantum advantage region of decoded quantum interferometry (DQI)? The best existing answer is provided by Theorem 10.1 of Jordan et al., yet we show that this answer systematically underestimates the extent of quantum advantage. On the standard partial-win LDPC benchmark instance, there exist 26 consecutive parameter points (l in [642, 667]) at which Jordan's analysis declares no quantum advantage (<s>/m < 0.5), while quantum advantage is in fact present with an approximation ratio reaching 0.66. The root cause is that Jordan's bound penalizes the entire system with the worst-case Hamming-layer decoding failure rate epsilon = max_k epsilon_k, discarding the spectral structure of the DQI tridiagonal matrix. Exploiting the concentration of the Perron eigenvector, we replace the uniform penalty with the weighted average epsilon_bar = sum_k epsilon_k w_k^2 and establish a unified lower bound (Master Theorem) valid over arbitrary finite fields F_q, proving that it strictly improves upon the original bound from three independent sources.
Where is the true boundary of the quantum advantage region of decoded quantum interferometry (DQI)? The best existing answer is provided by Theorem 10.1 of Jordan et al., yet we show that this answer systematically underestimates the extent of quantum advantage. On the standard partial-win LDPC benchmark instance, there exist 26 consecutive parameter points (l in [642, 667]) at which Jordan's analysis declares no quantum advantage (<s>/m < 0.5), while quantum advantage is in fact present with an approximation ratio reaching 0.66. The root cause is that Jordan's bound penalizes the entire system with the worst-case Hamming-layer decoding failure rate epsilon = max_k epsilon_k, discarding the spectral structure of the DQI tridiagonal matrix. Exploiting the concentration of the Perron eigenvector, we replace the uniform penalty with the weighted average epsilon_bar = sum_k epsilon_k w_k^2 and establish a unified lower bound (Master Theorem) valid over arbitrary finite fields F_q, proving that it strictly improves upon the original bound from three independent sources.
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.
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.
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) 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.
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. 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.
For general max-k-XORSAT with k≥3, no polynomial-time algorithm can do substantially better than random guessing on worst-case instances unless P=NP: approximating beyond the random-assignment value of 1/2 is NP-hard. The picture changes when each variable appears in at most D constraints. In that bounded-degree setting, polynomial-time algorithms can provably beat the random baseline by an additive amount of order 1/D. For Boolean instances, this scaling is known to be optimal: the matching hardness result is due to Trevisan, while the corresponding algorithmic guarantee was established by Barak et al. Whether the same holds over general finite fields, and what it implies for quantum algorithms, has not been established. We make this connection explicit and extend the hardness to max-Ek-LINSAT(q,r) with bounded degree D and over arbitrary finite fields Fq, proving that it is NP-hard to exceed r/q+Oq,r(1/D). These results provide the complexity-theoretic benchmark for the bounded-degree instances targeted by decoded quantum interferometry (DQI), QAOA, and classical heuristics. Any quantum advantage on bounded-degree instances is therefore confined to the constant prefactor. We further show that in the context of DQI and on (k,D)-regular instances, this prefactor is sensitive to the nature of the decoder: DQI with classical decoders faces an information-theoretic 1/DlogD barrier that prevents it from matching the hardness scaling, while DQI with quantum decoders is compatible with the 1/D scaling -- identifying quantum decoding as the key ingredient for matching the complexity-theoretic scaling with DQI.