Quantum Adder for Superposition States

Abstract

Quantum superposition is one of the essential features that make quantum computation surpass classical computation in space complexity and time complexity. However, it is a double-edged sword. For example, it is troublesome to add all the numbers stored in a superposition state. The usual solution is taking out and adding the numbers one by one. If there are 2 n numbers, the complexity of this scheme is O(2 n) which is the same as the complexity of the classical scheme O(2 n). Moreover, taking account to the current physical computing speed, quantum computers will have no advantage. In order to solve this problem, a new method for summing all numbers in a quantum superposition state is proposed in this paper, whose main idea is that circularly shifting the superposition state and summing the new one with the original superposition state. Our scheme can effectively reduce the time complexity to O(n).