Lower Bounds for Local Search by Quantum Arguments

The problem of finding a local minimum of a black-box function is central for understanding local search as well as quantum adiabatic algorithms. For functions on the Boolean hypercube $\left\{0,1\right\}^n$, we show a lower bound of $\Omega\left(2^{n/4}/n\right)$ on the number of queries needed by a quantum computer to solve this problem. More surprisingly, our approach, based on Ambainis's quantum adversary method, also yields a lower bound of $\Omega\left(2^{n/2}/n^2\right)$ on the problem's classical randomized query complexity. This improves and simplifies a 1983 result of Aldous. Finally, in both the randomized and quantum cases, we give the first nontrivial lower bounds for finding local minima on grids of constant dimension $d\geq3$.