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arXiv:2604.12131

Dequantizing Short-Path Quantum Algorithms

François Le Gall, Suguru Tamaki

quant-phcs.CCcs.DSSubmitted Apr 13, 2026• Announced Apr 15, 2026• 0 citations
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Abstract

The short-path quantum algorithm introduced by Hastings (Quantum 2018, 2019) is a variant of adiabatic quantum algorithms that enables an easier worst-case analysis by avoiding the need to control the spectral gap along a long adiabatic path. Dalzell, Pancotti, Campbell, and Brandão (STOC 2023) recently revisited this framework and obtained a clear analysis of the complexity of the short-path algorithm for several classes of constraint satisfaction problems (MAX-$k$-CSPs), leading to quantum algorithms with complexity $2^{(1-c)n/2}$ for some constant $c>0$. This suggested a super-quadratic quantum advantage over classical algorithms. In this work, we identify an explicit classical mechanism underlying a substantial part of this line of work, and show that it leads to clean dequantizations. As a consequence, we obtain classical algorithms that run in time $2^{(1-c')n}$, for some constant $c'>c$, for the same classes of constraint satisfaction problems. This shows that current short-path quantum algorithms for these problems do not achieve a super-quadratic advantage. On the positive side, our results provide a new ``quantum-inspired'' approach to designing classical algorithms for important classes of constraint satisfaction problems.

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The short-path quantum algorithm introduced by Hastings (Quantum 2018, 2019) is a variant of adiabatic quantum algorithms that enables an easier worst-case analysis by avoiding the need to control the spectral gap along a long adiabatic path. Dalzell, Pancotti, Campbell, and Brandão (STOC 2023) recently revisited this framework and obtained a clear analysis of the complexity of the short-path algorithm for several classes of constraint satisfaction problems (MAX-kkk-CSPs), leading to quantum algorithms with complexity 2(1−c)n/22^{(1-c)n/2}2(1−c)n/2 for some constant c>0c>0c>0. This suggested a super-quadratic quantum advantage over classical algorithms.
In this work, we identify an explicit classical mechanism underlying a substantial part of this line of work, and show that it leads to clean dequantizations. As a consequence, we obtain classical algorithms that run in time 2(1−c′)n2^{(1-c')n}2(1−c′)n, for some constant c′>cc'>cc′>c, for the same classes of constraint satisfaction problems. This shows that current short-path quantum algorithms for these problems do not achieve a super-quadratic advantage. On the positive side, our results provide a new ``quantum-inspired'' approach to designing classical algorithms for important classes of constraint satisfaction problems.
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.
We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), Ising spin glasses (p-spin model), and k-local constraint satisfaction problems (k-CSP). We show that either (a) the algorithm finds the optimal solution in time O*(2(0.5−c)n) for an n-independent constant c, a 2cn advantage over Grover’s algorithm; or (b) there are sufficiently many low-cost solutions such that classical random guessing produces a (1−η) approximation to the optimal cost value in sub-exponential time for arbitrarily small choice of η. Additionally, we show that for a large fraction of random instances from the k-spin model and for any fully satisfiable or slightly frustrated k-CSP formula, statement (a) is the case. The algorithm and its analysis are largely inspired by Hastings’ short-path algorithm.
We introduce quantum tilted walks, a quantum algorithmic framework for solving exact combinatorial optimization problems. The framework applies an average of powers of a tilted Hamiltonian that biases the discriminant matrix of a base Markov chain (mixer) with the objective function. Our starting point is quantum short-path algorithms, which prepare the ground state of such a Hamiltonian and obtain super-quadratic speedups over exhaustive search for certain combinatorial optimization problems. Recently, Le Gall and Tamaki~(arXiv:2604.12131) developed a classical conditioning-and-search algorithm for weighted MAX-Ekkk-LIN2 and weighted MAX-kkk-CSP. Under the same assumptions, their algorithm is only sub-quadratically slower than quantum short-path algorithms. Consequently, existing short-path algorithms do not establish a super-quadratic speedup over this stronger classical baseline. For maximization problems, we give conditions under which tilted walks increase the amplitude on the target state with high objective value when initialized from a starting state with lower objective value. This framework captures conditioning-and-search and yields super-quadratic speedups over it for the same problems. While our framework recovers quantum short-path algorithms as a special case, it neither requires ground-state preparation nor initialization in the ground state of the base mixer. We demonstrate these advantages on a synthetic optimization problem for which tilted walks achieve a super-quadratic speedup whereas the short-path algorithms do not.
Quantum-Enhanced Constraint Satisfaction Problems (QCSP) addresses the computational bottleneck in solving constraint satisfaction problems (CSPs) by introducing quantum superposition to enhance search strategies. This paper presents a novel quantum algorithm designed to significantly reduce the computational complexity of solving CSPs, particularly those exhibiting high branching factors. We detail the algorithm's core mechanism, leveraging quantum superposition to efficiently explore the solution space and improve search performance. The proposed approach offers a fundamentally new method for tackling CSPs, promising substantial improvements in scalability and efficiency compared to classical approaches. This work is motivated by the inherent advantages of quantum computation in exploring complex solution spaces.
Quantum-inspired optimization algorithms represent a novel approach to addressing constraint satisfaction problems (CSPs) by mimicking the phenomena observed in quantum systems. This paper explores the potential of adapting quantum principles – specifically superposition and entanglement – to enhance the search process within CSPs, aiming to overcome the limitations of classical algorithms. We detail the core mechanism of this design, focusing on how the algorithms leverage quantum concepts to explore a wider range of potential solutions and improve overall performance. The paper presents a comprehensive analysis of the proposed framework and its expected impact on solving complex CSPs.
Constraint satisfaction problems (CSPs) represent a significant class of computational problems that arise in various domains, including artificial intelligence, optimization, and cryptography. Traditional algorithms for solving CSPs often struggle with scalability, particularly when dealing with complex problems and large search spaces. This research investigates the potential of utilizing quantum walks, a quantum analogue of random walks, to develop more efficient algorithms for CSPs. We explore the theoretical underpinnings of quantum walk-based approaches and demonstrate their ability to explore solution spaces more effectively than classical methods. The core claim is that by leveraging quantum mechanics, we can significantly reduce the computational cost associated with solving CSPs. We present a framework for formulating CSPs as quantum walk problems and outline key considerations for designing and implementing such algorithms. The work highlights the promise of quantum computing in tackling intractable problems within the CSP domain.