The Quantum Cyclic Rotation Gate

A circular shift operator (or cyclic rotation gate) ROTk applies a rightward (or leftward) shift to an input register of n qubits o by as many positions as encoded by an additional input k∈N. Specifically, the qubit at position x is moved to position (x+k)modn. While it is known that there exists a quantum rotation operator that can be implemented in O(log(n))-time, through the repeated parallel application of the elementary Swap operators, there is no systematic procedure that concretely constructs the quantum operator ROT for variable size n of the quantum register and a variable parameter k. We fill the gap, providing a systematic implementation of the cyclic rotation operator (denoted ROT) in a quantum circuit model of computation whose depth is O(log(n)). We show how the circular shift operator can be utilized in quantum approaches to text processing, focusing on the problem of getting all possible cyclic rotations of a string in O(log2(n)) depth.