Improved algorithms for quantum identification of Boolean oracles

The oracle identification problem (OIP) was introduced by Ambainis et al. [3]. 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 [3] by providing a mostly optimal upper bound of query complexity for this problem: (i) For any oracle set S such that $|S| \le 2^{N^d}$ (d < 1), we design an algorithm whose query complexity is $O(\sqrt{N\log{M}/\log{N}})$, matching the lower bound proved in [3]. (ii) Our algorithm also works for the range between $2^{N^d}$ and 2N/logN (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 the noisy oracles as also shown in the literatures [2, 11, 18] for special cases of OIP

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