Summary: Lower Bounds for Local Search by Quantum Arguments
Scott Aaronson #
ABSTRACT
The problem of finding a local minimum of a black­box func­
tion is central for understanding local search as well as quan­
tum adiabatic algorithms. For functions on the Boolean
hypercube {0, 1} n , we show a lower bound
of# # 2 n/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# # 2 n/2 /n 2
# on the problem's classical ran­
domized 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
F.1.2 [Computation by Abstract Devices]: Modes of

Source: Aaronson, Scott - Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (MIT)

Collections: Physics; Computer Technologies and Information Sciences