(Sub)Exponential advantage of adiabatic quantum computation with no sign problem

TL;DR

This paper demonstrates a (sub)exponential quantum speedup using an adiabatic quantum algorithm on a specific graph-based Hamiltonian without a sign problem, strengthening previous superpolynomial separation results.

Contribution

It introduces a new graph-based Hamiltonian with a spectral gap enabling efficient adiabatic quantum computation, extending Hastings' superpolynomial separation proof.

Findings

Quantum algorithm finds 'EXIT' vertex efficiently

Classical algorithms require exponential queries

Construction uses Hastings' ideas for spectral gap

Abstract

We demonstrate the possibility of (sub)exponential quantum speedup via a quantum algorithm that follows an adiabatic path of a gapped Hamiltonian with no sign problem. This strengthens the superpolynomial separation recently proved by Hastings. The Hamiltonian that exhibits this speed-up comes from the adjacency matrix of an undirected graph, and we can view the adiabatic evolution as an efficient $\mathcal{O}(\mathrm{poly}(n))$-time quantum algorithm for finding a specific "EXIT" vertex in the graph given the "ENTRANCE" vertex. On the other hand we show that if the graph is given via an adjacency-list oracle, there is no classical algorithm that finds the "EXIT" with probability greater than $\exp(-n^\delta)$ using at most $\exp(n^\delta)$ queries for $\delta= \frac15 - o(1)$. Our construction of the graph is somewhat similar to the "welded-trees" construction of Childs et al., but uses additional ideas of Hastings for achieving a spectral gap and a short adiabatic path.