A dynamic quantum walk is a continuous-time quantum walk on a sequence of graphs, and they can implement the universal gate set T, H, CNOT. The procedure for decomposing a quantum gate into this universal gate set, and then implementing each gate as a dynamic quantum walk, however, can result in long sequences of graphs. To alleviate this, we give six scenarios under which a dynamic graph can be simplified, and they exploit commuting graphs, identical graphs, perfect state transfer, complementary graphs, isolated vertices, and uniform mixing on the hypercube. Furthermore, we develop a length-3 dynamic quantum walk that implements any singlequbit gate, and we extend this result to give length-3 dynamic quantum walks that implement any single-qubit gate controlled by any number of qubits.

This is joint work with Rebekah Herrman and Ibukunoluwa Adisa.