Amplitude Amplification for Operator Identification and Randomized Classes

Amplitude amplification (AA) is tool of choice for quantum algorithm designers to increase the success probability of query algorithms that reads its input in the form of oracle gates. Geometrically speaking, the technique can be understood as rotation in a specific two-dimensional space. We study and use a generalized form of this rotation operator to design algorithms in a geometric manner. Specifically, we apply AA to algorithms that take their input in the form of input states and in which rotations with different angles and directions are used in a unified manner. We show that AA can be used to sequentially discriminate between two unitary operators, both without error and with bounded-error, in an asymptotically optimal manner. We also show how to reduce error probability in one and two-sided bounded error algorithms more efficiently than the usual parallel repetitions technique; in particular, errors can be completely eliminated from the exact error algorithms.