Efficient circuit implementation of quantum walks on non-degree-regular graphs

This paper presents a set of highly efficient quantum circuits for discrete-time quantum walks on non-degree-regular graphs. In particular, we describe a general procedure for constructing highly efficient quantum circuits for quantum walks on star graphs of any degree and Cayley trees with an arbitrary number of layers, which are nonsparse in general. We also show how to modify these circuits to implement a full quantum-walk search algorithm on these graphs, without reference to a ``black-box'' oracle. This provides a practically implementable method to explore quantum-walk-based algorithms with the aim of eventual real-world applications.