Quantum Query Complexities

The study of quantum query complexities for black-box functions (oracles) plays a central role for design methods of efficient quantum algorithms. We consider a general framework called the oracle identification problem (OIP) for black-box oracles and analyze the quantum query complexity. The OIP is, given a set S of M Boolean black-box oracles out of 2^N ones, to determine which oracle in S is the current black-box oracle. We can exploit the information that candidates of the current oracle is restricted to S. The OIP contains several concrete problems such as the Grover search problem and the Bernstein-Vazirani problem. Our interest is in the quantum query complexity, for which we present several upper bounds. They are quite general and almost optimal: (i) The upper bound of query complexity of OIP is O( {√ {N log M log N} log log M} ; ) for any S such that M = |S| > N, which is better than the obvious bound N if M < 2^{N / log3 N} and the lower bound is Ω ( {√ {N log M} log ^{ - 1} N} ; ) . Thus, the bounds are tight up to a logarithmic factor. (ii) It is O( {√ N } ; ) for any S if |S| = N, which includes the upper bound for the Grover search as a special case. One can see that this bound is tight by the well-known lower bound of the Grover search problem, which needs Ω ( {√ N } ; ) queries by any quantum algorithms. (iii) For a wide range of oracles (|S| = N) such as random oracles and balanced oracles, the query complexity is O( /line {N/K} ) , where K is a simple parameter determined by S. This is a joint work with Andris Ambainis, Akinori Kawachi, Hiroyuki Masuda, Raymond H. Putra, and Shigeru Yamashita.