Abstract
The continuous-time quantum walk is a quantum version of a random walk that
evolves by Schrodinger's equation. With the Hamiltonian equal to the adjacency matrix
of a sequence of graphs, called a dynamic graph, continuous-time quantum walks
have been shown to implement quantum gates, including the T gate, Hadamard gate,
and the Controlled{NOT gate. Since these gates make up a universal set of quantum
gates, they can implement any other arbitrary quantum gate up to an arbitrary
approximation. This process, however, can be tedious and ine cient. To alleviate
this, we have developed a parameterized dynamic graph on which a continuous-time
quantum walk can implement any arbitrary single-qubit quantum gate, and we have
also extended this result to implement any two-qubit controlled-unitary quantum
gate. These dynamic graphs have at most length three. Using these, we implemented
Draper's quantum addition circuit, which is based on the quantum Fourier transform,
using a continuous-time quantum walk on a dynamic graph.