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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

TL;DR

The paper develops a generalized quantum short-path framework that extends Hastings' and Dalzell et al.'s ideas to arbitrary Markov chains, enabling super-quadratic speedups over Markov-chain search for combinatorial optimization. By introducing a short-path Hamiltonian H_b via the discriminant of the base Markov chain and a concave transformation g_eta, the authors realize efficient short jumps and amplitude-augmented long jumps under stability and spectral-density conditions. They demonstrate concrete speedups for problems such as MaxCut with fixed Hamming weight, MIS on bounded-degree graphs, antiferromagnetic Ising, and Sherrington-Kirkpatrick models, with both theoretical guarantees and numerical validation. The work balances rigorous functional-inequality-based analysis with practical algorithmic prescriptions, showing how non-uniform priors and constrained feasible sets can yield meaningful quantum advantages, and discusses the limits related to Groverization. Overall, the framework broadens the landscape of quantum speedups in combinatorial optimization by leveraging structured Markov chains and problem-specific constraints.

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.

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

TL;DR

The paper develops a generalized quantum short-path framework that extends Hastings' and Dalzell et al.'s ideas to arbitrary Markov chains, enabling super-quadratic speedups over Markov-chain search for combinatorial optimization. By introducing a short-path Hamiltonian H_b via the discriminant of the base Markov chain and a concave transformation g_eta, the authors realize efficient short jumps and amplitude-augmented long jumps under stability and spectral-density conditions. They demonstrate concrete speedups for problems such as MaxCut with fixed Hamming weight, MIS on bounded-degree graphs, antiferromagnetic Ising, and Sherrington-Kirkpatrick models, with both theoretical guarantees and numerical validation. The work balances rigorous functional-inequality-based analysis with practical algorithmic prescriptions, showing how non-uniform priors and constrained feasible sets can yield meaningful quantum advantages, and discusses the limits related to Groverization. Overall, the framework broadens the landscape of quantum speedups in combinatorial optimization by leveraging structured Markov chains and problem-specific constraints.

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.

Paper Structure

This paper contains 37 sections, 70 theorems, 290 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Contributions
  4. Contribution 1 -- Quantum Improvements over Non-trivial Classical Algorithms :
  5. Contribution 2 -- Technical Improvements with Generalized Short-Path Framework :
  6. Related Works
  7. Organization
  8. Preliminaries
  9. Probability Toolbox
  10. Short Path Algorithms
  11. Technical Overview
  12. Framework
  13. Applications
  14. Quantum Improvements that Lead to Speedups over Classical State-of-the-art
  15. Optimization with Fixed Hamming Weight
  16. ...and 22 more sections

Key Result

Theorem 1.1

The the following optimization problems exhibit a quantum runtime of $\mathcal{O}^\ast\left(T(n)^{0.5 - c(n)}\right)$, where $T(n)$ is the runtime of a classical algorithm based on Markov chain search, and $c(n) > 0$ quantifies the degree of improvement over quadratic speedup:

Figures (3)

  • Figure 1: Empirical selection of $b$. A Quartiles of $b$ values that minimize the effective runtime of the algorithm for \ref{['e:MaxCut']}. As $n$ increases, the runtime-optimal $b$ converges to a range approximately between 0.8 and 1.2. The red dot line shows the converged value of $b \approx 0.78$ of phase transition where the overlap with the initial state crosses $0.99$. B Quartiles of $b$ values that minimize the spectral gap for \ref{['e:MaxCut']}. For most instances tested, the spectral gap is minimized when $b$ is larger than the phase transition value, rendering the phase transition $b$ a safe choice. C The overlap values with the initial state and the ground state (optimal solution) for one $n=30$\ref{['e:MaxCut']} instance with varying $b$. The dotted verticle line denotes the phase transition $b$.
  • Figure 2: The inverse of the ground state overlap versus the feasible space size $\binom{n}{k}$ for \ref{['e:MaxCut']} with $n$ varying from 10 to 30 and $b=$ 0.78. The worst-case instances are fitted using an exponential function with base $\binom{n}{k}$ with an error bar denoting one standard deviation of the fitted exponent. The 95% confidence interval on the fitted exponent is $[0.391, 0.408]$.
  • Figure 3: Empirical fitting of the inverse overlap of the ground state $\lvert \langle \psi_b|z\rangle\rvert^{-1}$ and the runtime \ref{['eqn: runtime']} of \ref{['e:MaxCut']}, \ref{['e:MaxBisection']}, and MIS with difference choices of $b$. For the left column \ref{['e:MaxCut']}, we use data with $n$ ranging from 10 to 30. Top: $b=0.8$ the fitted exponent is 0.392 with 95% confidence interval $[0.383, 0.402]$, indicating there could exist $b$ that is better than the phase transition one in Figure \ref{['fig: mkc-runtime']}; Bottom: $b=1$ the fitted exponent is 0.348 with 95% confidence interval $[0.271, 0.426]$. For the middle column \ref{['e:MaxBisection']}, we use data with $n$ ranging from 16 to 22. Top: $b=0.7$ the fitted exponent is 0.444 with 95% confidence interval $[0.436, 0.452]$; Bottom: $b=1$ the fitted exponent is 0.873 with 95% confidence interval $[0.842, 0.905]$. For the right column (MIS), we use data with $n$ ranging from 10 to 21. Top: $b=0.6$ the fitted exponent is 0.400 with 95% confidence interval $[0.386, 0.415]$; Bottom: $b=0.8$ the fitted exponent is 0.392 with 95% confidence interval $[0.122, 0.663]$.

Theorems & Definitions (136)

  • Theorem 1.1: Applications of Generalized Short Path Framework (informal Theorem \ref{['thm:formal_speedup']})
  • Definition 2.1: Discriminant Matrix
  • Definition 2.2: Markov Functionals
  • Definition 2.3: Poincaré Inequality
  • Definition 2.4: Log-Sobolev Inequality
  • Definition 2.5: Modified Log-Sobolev Inequality
  • Definition 2.6: $P$-pseudo Lipschitz Norm
  • Definition 3.1: $\Delta_{P}$ stability
  • Definition 3.2: $\gamma$ Spectral Density
  • Theorem 3.3: Informal Theorem \ref{['thrm:MTG_RT']}
  • ...and 126 more