# Improved algorithms for quantum identification of Boolean oracles

## Abstract

The oracle identification problem (OIP) was introduced by Ambainis et al. [A. Ambainis, K. Iwama, A. Kawachi, H. Masuda, R.H. Putra, S. Yamashita, Quantum identification of boolean oracles, in: Proc. of STACS'04, in: LNCS, vol. 2996, 2004, pp. 105-116]. It is given as a set S of M oracles and a blackbox oracle f. Our task is to figure out which oracle in S is equal to the blackbox f by making queries to f. OIP includes several problems such as the Grover Search as special cases. In this paper, we improve the algorithms in [A. Ambainis, K. Iwama, A. Kawachi, H. Masuda, R.H. Putra, S. Yamashita, Quantum identification of boolean oracles, in: Proc. of STACS'04, in: LNCS, vol. 2996, 2004, pp. 105-116] by providing a mostly optimal upper bound of query complexity for this problem: (i) For any oracle set S such that | S | ≤ 2Nd (d < 1), we design an algorithm whose query complexity is O (sqrt(N log M / log N)), matching the lower bound proved in [A. Ambainis, K. Iwama, A. Kawachi, H. Masuda, R.H. Putra, S. Yamashita, Quantum identification of boolean oracles, in: Proc. of STACS'04, in: LNCS, vol. 2996, 2004, pp. 105-116]. (ii) Our algorithm also works for the range between 2Nd and 2N / log N (where the bound becomes O (N)), but the gap between the upper and lower bounds worsens gradually. (iii) Our algorithm is robust, namely, it exhibits the same performance (up to a constant factor) against noisy oracles as also shown in the literature [M. Adcock, R. Cleve, A quantum Goldreich-Levin theorem with cryptographic applications, in: Proc. of STACS'02, in: LNCS, vol. 2285, 2002, pp. 323-334; H. Buhrman, I. Newman, H. Röhrig, R. deWolf, Robust quantum algorithms and polynomials, in: Proc. of STACS'05, in: LNCS, vol. 3404, 2005, pp. 593-604; P. Høyer, M. Mosca, R. de Wolf, Quantum search on bounded-error inputs, in: Proc. of ICALP'03, in: LNCS, vol. 2719, 2003, pp. 291-299] for special cases of OIP. © 2006 Elsevier Ltd. All rights reserved.