  In this paper we present a novel quantum algorithm, namely the quantum grid
search algorithm, to solve a special search problem. Suppose $ k $ non-empty
buckets are given, such that each bucket contains some marked and some unmarked
items. In one trial an item is selected from each of the $ k $ buckets. If
every selected item is a marked item, then the search is considered successful.
This search problem can also be formulated as the problem of finding a "marked
path" associated with specified bounds on a discrete grid. Our algorithm
essentially uses several Grover search operators in parallel to efficiently
solve such problems. We also present an extension of our algorithm combined
with a binary search algorithm in order to efficiently solve global trajectory
optimization problems. Estimates of the expected run times of the algorithms
are also presented, and it is proved that our proposed algorithms offer
exponential improvement over pure classical search algorithms, while a
traditional Grover's search algorithm offers only a quadratic speedup. We note
that this gain comes at the cost of increased complexity of the quantum
circuitry. The implication of such exponential gains in performance is that
many high dimensional optimization problems, which are intractable for
classical computers, can be efficiently solved by our proposed quantum grid
search algorithm.
