Quantum Circuit Design for Multi-Object Search

We study the quantum circuit design using 1-bit and 2-bit unitary gates for the iterations of the multi-object quantum search algorithm. The oracle block is designed in order to efficiently implement any sign-flipping operations. A chief ingredient in the design is the permutation operator which maps a set of search targets to another set on which the sign-flipping operation can be easily done. Such a proposed algorithmic approach implicates a minimal symmetric group generation problem: how to generate elements of a symmetric group using the smallest number of concatenations with a set of given generators. For the general case, this is an open problem. We indicate how the complexity issues depend on the solution of this problem through simple examples. Department of Mathematics, Ohio University-Eastern, St. Clairsville, OH 43950; [email protected]. Supported in part by an Ohio University Research Challenge Award. Department of Mathematics, Texas A&M University, College Station, TX 77743-3368; [email protected]. Supported in part by a DARPA QuIST grant and a TAMU TITF initiative. Department of Mathematical Science, University of Nevada-Las Vegas, Las Vegas, NV 89154; [email protected].