Enlarging the gap between quantum and classical query complexity of multifunctions

Quantum computing aims to use quantum mechanical effects for the efficient performance of computational tasks. A popular research direction is enlarging the gap between classical and quantum algorithm complexity of the same computational problem. We present new results in quantum query algorithm design for multivalued functions that allow to achieve a large quantum versus classical complexity separation. To compute a basic finite multifunction in a quantum model only one query is enough while classically three queries are required. Then, we present two generalizations and a modification of the original algorithm, and obtain the following complexity gaps: Q_UD(M^´) ≤ N versus C_UD(M^´) ≥ 3N, and Q_RD(M^´´) = 1 versus N/2 + 1 ≤ C_RD(M^´´) ≤ N.