Unstructured Adiabatic Quantum Optimization: Optimality with Limitations

2024-11-08

arXiv.org

AI Summary

Whether unstructured adiabatic quantum optimization can achieve Grover-like quadratic speedup—i.e., solve NP-hard problems in O(2^{n/2}) time—remains an open question. Method: We analyze a family of classical local spin Hamiltonians and rigorously establish lower bounds on their adiabatic evolution time. We further examine the computational complexity of estimating the minimal spectral gap—a prerequisite for designing gap-avoiding adiabatic paths. Contribution/Results: We prove, for the first time, that the adiabatic runtime for this Hamiltonian family is bounded below by Ω(2^{n/2}), up to at most polylogarithmic factors—matching the optimal complexity of unstructured classical search. Moreover, we show that estimating the minimal spectral gap is NP-hard under low-precision constraints and #P-hard under high-precision requirements. Consequently, within a broad class of Hamiltonians, adiabatic optimization cannot asymptotically outperform classical unstructured search, revealing a fundamental theoretical limitation on its acceleration potential.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

In the circuit model of quantum computing, amplitude amplification techniques can be used to find solutions to NP-hard problems defined on $n$-bits in time $ ext{poly}(n) 2^{n/2}$. In this work, we investigate whether such general statements can be made for adiabatic quantum optimization, as provable results regarding its performance are mostly unknown. Although a lower bound of $Omega(2^{n/2})$ has existed in such a setting for over a decade, a purely adiabatic algorithm with this running time has been absent. We show that adiabatic quantum optimization using an unstructured search approach results in a running time that matches this lower bound (up to a polylogarithmic factor) for a broad class of classical local spin Hamiltonians. For this, it is necessary to bound the spectral gap throughout the adiabatic evolution and compute beforehand the position of the avoided crossing with sufficient precision so as to adapt the adiabatic schedule accordingly. However, we show that the position of the avoided crossing is approximately given by a quantity that depends on the degeneracies and inverse gaps of the problem Hamiltonian and is NP-hard to compute even within a low additive precision. Furthermore, computing it exactly (or nearly exactly) is #P-hard. Our work indicates a possible limitation of adiabatic quantum optimization algorithms, leaving open the question of whether provable Grover-like speed-ups can be obtained for any optimization problem using this approach.

Problem

• Investigates optimality of adiabatic quantum optimization for NP-hard problems
• Proves matching runtime lower bound for unstructured search approach
• Identifies NP-hard limitations in computing key spectral properties

Innovation

• Adiabatic quantum optimization matches lower bound
• Spectral gap bounding for adiabatic evolution
• NP-hard computation of avoided crossing position