Matthew Hastings

Abstract We demonstrate the possibility of (sub)exponential quantum speedup via a quantum algorithm that follows an adiabatic path of a gapped sparse Hamiltonian with no sign problem. This is in sharp contrast with frustration-free stoquastic Hamiltonians, where no such speedup is possible as shown by Bravyi and Terhal (2008). 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. (2003), but uses additional ideas for simultaneously achieving a spectral gap and a short adiabatic path.