Unstructured Adiabatic Quantum Optimization: Optimality with Limitations

TL;DR

This paper investigates the limitations of unstructured adiabatic quantum optimization, showing it can match known lower bounds for certain problems but faces computational hardness in predicting optimal evolution points, indicating fundamental constraints.

Contribution

It demonstrates that adiabatic quantum optimization can achieve known lower bounds but is limited by NP-hardness in predicting critical evolution parameters, revealing fundamental limitations.

Findings

Adiabatic quantum optimization matches the $ ext{poly}(n) 2^{n/2}$ lower bound for certain problems.

Computing the spectral gap and avoided crossing positions is NP-hard and ext{ extsterling}P-hard.

Limitations suggest challenges in achieving Grover-like speed-ups with adiabatic algorithms.

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 $\text{poly}(n) 2^{n/2}$. 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.