Quantum Algorithms and Initialization

Abstract

Abstract : We are going to show that whether or not a function is evenly distributed can be determined in quantum polynomial time without initialization. We will also show that the functional evaluation oracle and the functional phase transform are equivalent. To solve the first conjecture, we will construct a quantum algorithm to solve the reduced problem to determine whether or not a function is evenly balanced and then extend this result by employing our initialization-free quantum functional evaluation oracle. For the second conjecture, we will construct the quantum functional evaluation oracle from the quantum functional phase transform. For this purpose, we will localize the both operators on the auxiliary register and then build a quantum circuit that can transform both localized operators to each other. We will extend the domain of the functional phase transform to the Hilbert space, which is the domain of a function evaluation oracle, so that the relative phase changes in the functional phase transform can be encoded into the control register. The limitation on the size of quantum computers makes it important to reuse qubits for auxiliary registers. Our initialization-free algorithm will greatly reduce the size of quantum computers since independent computational processes can share auxiliary registers. Furthermore, our algorithm can be useful in symmetric cryptography, because to design secure block ciphers it is essential to find evenly distributed functions. Moreover, until now it is not known how to implement functional evaluation on a quantum computer. Also which one is easy or possible to implement on a quantum computer among the functional phase transform and the functional evaluation oracle has not been answered yet. Thus our result can give a flexibility to the realization of functional evaluation on quantum computers.