Quantum algorithms for distinguishing unitary operators

Abstract

Distinguishing between unitary operators is one of the fundamental problems in the field of quantum computing. In the operator identification problem, we are given access to unknown operator U as a black-box that implements either an operator U1 or an operator U2, where U1 and U2 are arbitrary unitary operators and their operations are known to us. The goal is to determine whether U is an implementation of U1 or U2. In this thesis, two different versions of operator identification problem have been studied followed by a generalization. Firstly, we consider the case when an exact implementation of the operation of the operators U1 and U2 is given to us. We show that amplitude amplification, which is one of the important tools in quantum computing, can be used to design an efficient algorithm to solve this version of operator identification problem without error. But, in the quantum circuit theory, it may not be always possible to implement an arbitrary operator exactly and it may happen that a fabricated circuit implements a close approximation of the desired unitary operator. For the second version of the problem, we consider the case where the approximate implementation of the operation of the operators U1 and U2 is given to us; once again the goal is to design an algorithm to solve the problem. Finally, we consider a general version of the operator identification problem when the candidate set is of any size, say n. That is, U implements one of the operators present in the candidate set fU1, U2, ... , Ung and we have to identify U. We propose novel approaches to solve all these three problems in this thesis.