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Quantum Speedups for Group Relaxations of Integer Linear Programs
Authors:
Brandon Augustino,
Dylan Herman,
Guneykan Ozgul,
Jacob Watkins,
Atithi Acharya,
Enrico Fontana,
Junhyung Lyle Kim,
Shouvanik Chakrabarti
Abstract:
Integer Linear Programs (ILPs) are a flexible and ubiquitous model for discrete optimization problems. Solving ILPs is \textsf{NP-Hard} yet of great practical importance. Super-quadratic quantum speedups for ILPs have been difficult to obtain because classical algorithms for many-constraint ILPs are global and exhaustive, whereas quantum frameworks that offer super-quadratic speedup exploit local…
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Integer Linear Programs (ILPs) are a flexible and ubiquitous model for discrete optimization problems. Solving ILPs is \textsf{NP-Hard} yet of great practical importance. Super-quadratic quantum speedups for ILPs have been difficult to obtain because classical algorithms for many-constraint ILPs are global and exhaustive, whereas quantum frameworks that offer super-quadratic speedup exploit local structure of the objective and feasible set. We address this via quantum algorithms for Gomory's group relaxation. The group relaxation of an ILP is obtained by dropping nonnegativity on variables that are positive in the optimal solution of the linear programming (LP) relaxation, while retaining integrality of the decision variables. We present a competitive feasibility-preserving classical local-search algorithm for the group relaxation, and a corresponding quantum algorithm that, under reasonable technical conditions, achieves a super-quadratic speedup. When the group relaxation satisfies a nondegeneracy condition analogous to, but stronger than, LP non-degeneracy, our approach yields the optimal solution to the original ILP. Otherwise, the group relaxation tightens bounds on the optimal objective value of the ILP, and can improve downstream branch-and-cut by reducing the integrality gap; we numerically observe this on several practically relevant ILPs. To achieve these results, we derive efficiently constructible constraint-preserving mixers for the group relaxation with favorable spectral properties, which are of independent interest.
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Submitted 13 February, 2026;
originally announced February 2026.
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Mechanisms for Quantum Advantage in Global Optimization of Nonconvex Functions
Authors:
Dylan Herman,
Guneykan Ozgul,
Anuj Apte,
Junhyung Lyle Kim,
Anupam Prakash,
Jiayu Shen,
Shouvanik Chakrabarti
Abstract:
We present new theoretical mechanisms for quantum speedup in the global optimization of nonconvex functions, expanding the scope of quantum advantage beyond traditional tunneling-based explanations. As our main building-block, we demonstrate a rigorous correspondence between the spectral properties of Schrödinger operators and the mixing times of classical Langevin diffusion. This correspondence m…
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We present new theoretical mechanisms for quantum speedup in the global optimization of nonconvex functions, expanding the scope of quantum advantage beyond traditional tunneling-based explanations. As our main building-block, we demonstrate a rigorous correspondence between the spectral properties of Schrödinger operators and the mixing times of classical Langevin diffusion. This correspondence motivates a mechanism for separation on functions with unique global minimum: while quantum algorithms operate on the original potential, classical diffusions correspond to a Schrödinger operators with a WKB potential having nearly degenerate global minima. We formalize these ideas by proving that a real-space adiabatic quantum algorithm (RsAA) achieves provably polynomial-time optimization for broad families of nonconvex functions. First, for block-separable functions, we show that RsAA maintains polynomial runtime while known off-the-shelf algorithms require exponential time and structure-aware algorithms exhibit arbitrarily large polynomial runtimes. These results leverage novel non-asymptotic results in semiclassical analysis. Second, we use recent advances in the theory of intrinsic hypercontractivity to demonstrate polynomial runtimes for RsAA on appropriately perturbed strongly convex functions that lack global structure, while off-the-shelf algorithms remain exponentially bottlenecked. In contrast to prior works based on quantum tunneling, these separations do not depend on the geometry of barriers between local minima. Our theoretical claims about classical algorithm runtimes are supported by rigorous analysis and comprehensive numerical benchmarking. These findings establish a rigorous theoretical foundation for quantum advantage in continuous optimization and open new research directions connecting quantum algorithms, stochastic processes, and semiclassical analysis.
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Submitted 3 October, 2025;
originally announced October 2025.
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Quantum Speedups for Markov Chain Monte Carlo Methods with Application to Optimization
Authors:
Guneykan Ozgul,
Xiantao Li,
Mehrdad Mahdavi,
Chunhao Wang
Abstract:
We propose quantum algorithms that provide provable speedups for Markov Chain Monte Carlo (MCMC) methods commonly used for sampling from probability distributions of the form $π\propto e^{-f}$, where $f$ is a potential function. Our first approach considers Gibbs sampling for finite-sum potentials in the stochastic setting, employing an oracle that provides gradients of individual functions. In th…
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We propose quantum algorithms that provide provable speedups for Markov Chain Monte Carlo (MCMC) methods commonly used for sampling from probability distributions of the form $π\propto e^{-f}$, where $f$ is a potential function. Our first approach considers Gibbs sampling for finite-sum potentials in the stochastic setting, employing an oracle that provides gradients of individual functions. In the second setting, we consider access only to a stochastic evaluation oracle, allowing simultaneous queries at two points of the potential function under the same stochastic parameter. By introducing novel techniques for stochastic gradient estimation, our algorithms improve the gradient and evaluation complexities of classical samplers, such as Hamiltonian Monte Carlo (HMC) and Langevin Monte Carlo (LMC) in terms of dimension, precision, and other problem-dependent parameters. Furthermore, we achieve quantum speedups in optimization, particularly for minimizing non-smooth and approximately convex functions that commonly appear in empirical risk minimization problems.
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Submitted 4 April, 2025;
originally announced April 2025.
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Generalized Short Path Algorithms: Towards Super-Quadratic Speedup over Markov Chain Search for Combinatorial Optimization
Authors:
Shouvanik Chakrabarti,
Dylan Herman,
Guneykan Ozgul,
Shuchen Zhu,
Brandon Augustino,
Tianyi Hao,
Zichang He,
Ruslan Shaydulin,
Marco Pistoia
Abstract:
We analyze generalizations of quantum algorithms based on the short path framework first proposed by Hastings~[\textit{Quantum} 2, 78 (2018)], which has been extended and shown by Dalzell~et~al.~[STOC~'23] to achieve super-Grover speedups for certain binary optimization problems. We demonstrate that, under some commonly satisfied technical conditions, an appropriate generalization can achieve supe…
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We analyze generalizations of quantum algorithms based on the short path framework first proposed by Hastings~[\textit{Quantum} 2, 78 (2018)], which has been extended and shown by Dalzell~et~al.~[STOC~'23] to achieve super-Grover speedups for certain binary optimization problems. We demonstrate that, under some commonly satisfied technical conditions, an appropriate generalization can achieve super-quadratic speedups not only over unstructured search but also over a classical optimization algorithm that searches for the optimum by drawing samples from the stationary distribution of a Markov chain. We employ this framework to obtain algorithms for problems including variants of Max Bisection, Max Independent Set, and finding the ground states of the Antiferromagnetic Ising Model and the Sherrington-Kirkpatrick Model, whose runtimes are asymptotically faster than those obtainable with previous short path techniques. In certain cases, our algorithms achieve super-quadratic speedups compared to the best known classical algorithms with rigorously established runtimes.
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Submitted 1 December, 2025; v1 submitted 30 October, 2024;
originally announced October 2024.
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Stochastic Quantum Sampling for Non-Logconcave Distributions and Estimating Partition Functions
Authors:
Guneykan Ozgul,
Xiantao Li,
Mehrdad Mahdavi,
Chunhao Wang
Abstract:
We present quantum algorithms for sampling from non-logconcave probability distributions in the form of $π(x) \propto \exp(-βf(x))$. Here, $f$ can be written as a finite sum $f(x):= \frac{1}{N}\sum_{k=1}^N f_k(x)$. Our approach is based on quantum simulated annealing on slowly varying Markov chains derived from unadjusted Langevin algorithms, removing the necessity for function evaluations which c…
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We present quantum algorithms for sampling from non-logconcave probability distributions in the form of $π(x) \propto \exp(-βf(x))$. Here, $f$ can be written as a finite sum $f(x):= \frac{1}{N}\sum_{k=1}^N f_k(x)$. Our approach is based on quantum simulated annealing on slowly varying Markov chains derived from unadjusted Langevin algorithms, removing the necessity for function evaluations which can be computationally expensive for large data sets in mixture modeling and multi-stable systems. We also incorporate a stochastic gradient oracle that implements the quantum walk operators inexactly by only using mini-batch gradients. As a result, our stochastic gradient based algorithm only accesses small subsets of data points in implementing the quantum walk. One challenge of quantizing the resulting Markov chains is that they do not satisfy the detailed balance condition in general. Consequently, the mixing time of the algorithm cannot be expressed in terms of the spectral gap of the transition density, making the quantum algorithms nontrivial to analyze. To overcome these challenges, we first build a hypothetical Markov chain that is reversible, and also converges to the target distribution. Then, we quantified the distance between our algorithm's output and the target distribution by using this hypothetical chain as a bridge to establish the total complexity. Our quantum algorithms exhibit polynomial speedups in terms of both dimension and precision dependencies when compared to the best-known classical algorithms.
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Submitted 17 October, 2023;
originally announced October 2023.