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 {0, 1}n , we show a lower bound of 2n/4 /n 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 2n/2 /n2 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 3. Categories and Subject Descriptors

• Type: Conference
• Year: 2004
• Where: STOC
• Authors: Scott Aaronson
• Keywords: Algorithms, Lower Bound, Nontrivial Lower Bounds, Quantum Adiabatic Algorithms, STOC 2004