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 -η) approximation to the optimal cost value in sub-exponential time for arbitrarily small choice of η. 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 are largely inspired by Hastings' short-path algorithm [Quantum 2 (2018) 78].