Implementing quantum gates using length-3 dynamic quantum walks

It is well known that any quantum gate can be decomposed into the universal gate set {T, H, CNOT}, and recent results have shown that each of these gates can be implemented using a dynamic quantum walk, which is a continuous-time quantum walk on a sequence of graphs. This procedure for converting a quantum gate into a dynamic quantum walk, however, can result in long sequences of graphs. To alleviate this, in this paper, we develop a length-3 dynamic quantum walk that implements any single-qubit gate. Furthermore, we extend this result to give length-3 dynamic quantum walks that implement any single-qubit gate controlled by any number of qubits. Using these, we implement Draper's quantum addition circuit, which is based on the quantum Fourier transform, using a dynamic quantum walk.