Tradeoffs in the Quantum Search Algorithm

Quantum search is a quantum mechanical technique for searching N possibilities in only √ N steps. This has been proved to be the best possible algorithm for the exhuastive search problem in the sense the number of queries it requires cannot be reduced. However, as this paper shows, the number of non-query operations, and thus the total number of operations, can be reduced. The number of non-query unitary operations can be reduced by a factor of log N/α log(log N) while increasing the number of queries by a factor of only (1 + (log N)−α). Various choices of α yield different variants of the algorithm. For example, by choosing α to be O(log N/ log(log N)), the number of non-query unitary operations can be reduced by 40% while increasing the number of queries by just two. ∗This research was partly supported by NSA & ARO under contract no. DAAG55-98-C0040.