Unitary Coined Discrete-Time Quantum Walks on Directed Multigraphs

Unitary Coined Discrete-Time Quantum Walks (UC-DTQW) constitute a universal model of quantum computation, meaning that any computation done by a general purpose quantum computer can either be done using the UC-DTQW framework. In the last decade,s great progress has been done in this field by developing quantum walk-based algorithms that can outperform classical ones. However, current quantum computers work based on the quantum circuit model of computation, and the general mapping from one model to the other is still an open problem. In this work we provide a matrix analysis of the unitary evolution operator of UC-DTQW, which is composed at the time of two unitary operators: the shift and coin operators. We conceive the shift operator of the system as the unitary matrix form of the adjacency matrix associated to the graph on which the UC-DTQW takes place, and provide a set of equations to transform the latter into the former and vice-versa. However, this mapping modifies the structure of the original graph into a directed multigraph, by splitting single edges or arcs of the original graph into multiple arcs. Thus, the fact that any unitary operator has a quantum circuit representation means that any adjacency matrix that complies with the transformation equations will be automatically associated to a quantum circuit, and any quantum circuit acting on a bipartite system will be always associated to a multigraph. Finally, we extend the definition of the coin operator to a superposition of coins in such a way that each coin acts on different vertices of the multigraph on which the quantum walk takes place, and provide a description of how this can be implemented in circuit form.
Comment: 20 pages