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arXiv:2410.23270

Generalized Short Path Algorithms: Towards Super-Quadratic Speedup over Markov Chain Search for Combinatorial Optimization

Shouvanik Chakrabarti, Dylan Herman, Guneykan Ozgul, Shuchen Zhu, Brandon Augustino, Tianyi Hao, Zichang He, Ruslan Shaydulin, Marco Pistoia

Published Oct 30, 2024• 0 citations
arXivHTMLPDF

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 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.

DOI
10.48550/arxiv.2410.23270

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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 Dalzelletal.[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 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.
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.
Designing quantum algorithms with a speedup over their classical analogs is a central challenge in quantum information science. Motivated by recent experimental observations of a superlinear quantum speedup in solving the Maximum Independent Set problem on certain unit-disk graph instances [Ebadi et al., Science 376, 6598 (2022)], we develop a theoretical framework to analyze the relative performance of the optimized quantum adiabatic algorithm and a broad class of classical Markov chain Monte Carlo algorithms. We outline conditions for the quantum adiabatic algorithm to achieve a quadratic speedup on hard problem instances featuring flat low-energy landscapes and provide example instances with either a quantum speedup or slowdown. We then introduce an additional local Hamiltonian with no sign problem to the optimized adiabatic algorithm to achieve a quadratic speedup over a wide class of classical simulated annealing, parallel tempering, and quantum Monte Carlo algorithms in solving these hard problem instances. Finally, we use this framework to analyze the experimental observations.
This chapter covers applications of quantum computing in the area of combinatorial optimization. This area is related to operations research, and it encompasses many tasks that appear in science and industry, such as scheduling, routing, and supply chain management. We cover specific problems where a quadratic quantum speedup may be available via Grover’s quantum algorithm for unstructured search. We also cover several more recent proposals for achieving superquadratic speedups, including the quantum adiabatic algorithm, the quantum approximate optimization algorithm (QAOA), and the short-path algorithm.
The short path algorithm gives a super-Grover speedup for various optimization problems under the assumption of a unique ground state and under an assumption on the density of low-energy states. Here, we remove the assumption of a unique ground state; this uses the same algorithm but a slightly different analysis and holds for arbitrary MAX-DDD-LIN-222 problems. Then, specializing to the case D=2D=2D=2, we show that for certain values of the objective function we can always achieve a super-Grover speedup (albeit a very slight one) without any assumptions on the density of states. Finally, for random instances, we give a heuristic treatment suggesting a more significant improvement.
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.