Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end

We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), Ising spin glasses ($p$-spin model), and $k$-local constraint satisfaction problems ($k$-CSP). We show that either (a) the algorithm finds the optimal solution in time $O^*(2^{(0.5-c)n})$ for an $n$-independent constant $c$, a $2^{cn}$ advantage over Grover's algorithm; or (b) there are sufficiently many low-cost solutions such that classical random guessing produces a $(1-\eta)$ approximation to the optimal cost value in sub-exponential time for arbitrarily small choice of $\eta$. Additionally, we show that for a large fraction of random instances from the $k$-spin model and for any sufficiently close-to-regular, fully satisfiable (or slightly frustrated) $k$-CSP formula, statement (a) is the case. The algorithm and its analysis is largely inspired by Hastings' short-path algorithm [$\textit{Quantum}$ $\textbf{2}$ (2018) 78].