Generalized Quantum Search with Parallelism

Abstract

We generalize Grover's unstructured quantum search algorithm to enable it to use an arbitrary starting superposition and an arbitrary unitary matrix simultaneously. We derive an exact formula for the probability of the generalized Grover's algorithm succeeding after n iterations. We show that the fully generalized formula reduces to the special cases considered by previous authors. We then use the generalized formula to determine the optimal strategy for using the unstructured quantum search algorithm. On average the optimal strategy is about 12% better than the naive use of Grover's algorithm. The speedup obtained is not dramatic but it illustrates that a hybrid use of quantum computing and classical computing techniques can yield a performance that is better than either alone. We extend the analysis to the case of a society of k quantum searches acting in parallel. We derive an analytic formula that connects the degree of parallelism with the optimal strategy for k-parallel quantum search. We then derive the formula for the expected speed of k-parallel quantum search.