Why adiabatic quantum annealing is unlikely to yield speed-up

TL;DR

This paper analytically demonstrates that quantum annealing with certain Hamiltonians is unlikely to provide practical speed-up for combinatorial optimization due to the intractability of precisely knowing the spectral gap location.

Contribution

The study provides an analytical calculation of the spectral gap in quantum annealing and shows the impracticality of achieving speed-up without detailed spectral information.

Findings

Spectral gap scales as 1/√N with problem size N.

Precise knowledge of the gap location is required for speed-up.

Intractability of density of states limits practical quantum speed-up.

Abstract

We study quantum annealing for combinatorial optimization with Hamiltonian $H = z H_f + H_0$ where $H_f$ is diagonal, $H_0=-|\phi \rangle \langle \phi|$ is the equal superposition state projector and $z$ the annealing parameter. We analytically compute the minimal spectral gap as $\mathcal{O}(1/\sqrt{N})$ with $N$ the total number of states and its location $z_*$. We show that quantum speed-up requires an annealing schedule which demands a precise knowledge of $z_*$, which can be computed only if the density of states of the optimization problem is known. However, in general the density of states is intractable to compute, making quadratic speed-up unfeasible for any practical combinatoric optimization problems. We conjecture that it is likely that this negative result also applies for any other instance independent transverse Hamiltonians such as $H_0 = -\sum_{i=1}^n \sigma_i^x$.