Quantum adiabatic optimization without heuristics

Quantum adiabatic optimization (QAO) is performed using a time-dependent Hamiltonian $H(s)$ with spectral gap $\gamma(s)$. Assuming the existence of an oracle $\Gamma$ such that \gamma_\min = \Theta\left(\min_s\Gamma(s)\right), we provide an algorithm that reliably performs QAO in time O\left(\gamma_\min^{-1}\right) with O\left(\log(\gamma_\min^{-1})\right) oracle queries, where \gamma_\min = \min_s \gamma(s). Our strategy is not heuristic and does not require guessing time parameters or annealing paths. Rather, our algorithm naturally produces an annealing path such that $\|dH/ds\| \approx \gamma(s)$ and chooses its own runtime to be as close as possible to optimal while promising convergence to the ground state. We then demonstrate the feasibility of this approach in practice by explicitly constructing a gap oracle $\Gamma$ for the problem of finding the minimum point $m = \mathrm{argmin}_u W(u)$ of the cost function $W:\mathcal{V}\longrightarrow [0,1]$, restricting ourselves to computational basis measurements and driving Hamiltonian $H(0)=I - |\mathcal{V}|^{-1}\sum_{u,v \in \mathcal{V}}\vert{u}\rangle\langle{v}\vert$. Requiring only that $W$ have a constant lower bound on its spectral gap and upper bound $\kappa$ on its spectral ratio, our QAO algorithm returns $m$ with probability $(1-\epsilon)(1-e^{-1/\epsilon})$ in time $\widetilde{\mathcal{O}}(\epsilon^{-1}[\sqrt{|\mathcal{V}|} + (\kappa-1)^{2/3}|\mathcal{V}|^{2/3}])$. This achieves a quantum advantage for all $\kappa$, and recovers Grover scaling up to logarithmic factors when $\kappa \approx 1$. We implement the algorithm as a subroutine in an optimization procedure that produces $m$ with exponentially small failure probability and expected runtime $\widetilde{\mathcal{O}}(\epsilon^{-1}[\sqrt{|\mathcal{V}|} + (\kappa-1)^{2/3}|\mathcal{V}|^{2/3}])$ even when $\kappa$ is not known beforehand.