Efficient quantum circuit implementation of quantum walks

Quantum walks, being the quantum analog of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the ``glued trees'' algorithm, which provides an exponential speedup over classical methods, relative to a particular quantum oracle. Here, we discuss the possibility of a quantum walk algorithm yielding such an exponential speedup over possible classical algorithms, without the use of an oracle. We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum walks can be constructed and discuss potential applications to quantum search for marked vertices along these graphs.