Quantum walk is one of the main tools for quantum algorithms. Defined by

analogy to classical random walk, a quantum walk is a time-homogeneous quantum

process on a graph. Both random and quantum walks can be defined either in

continuous or discrete time. But whereas a continuous-time random walk can be

obtained as the limit of a sequence of discrete-time random walks, the two

types of quantum walk appear fundamentally different, owing to the need for

extra degrees of freedom in the discrete-time case.

In this article, I describe a precise correspondence between continuous- and

discrete-time quantum walks on arbitrary graphs. Using this correspondence, I

show that continuous-time quantum walk can be obtained as an appropriate limit

of discrete-time quantum walks. The correspondence also leads to a new

technique for simulating Hamiltonian dynamics, giving efficient simulations

even in cases where the Hamiltonian is not sparse. The complexity of the

simulation is linear in the total evolution time, an improvement over

simulations based on high-order approximations of the Lie product formula. As

applications, I describe a continuous-time quantum walk algorithm for element

distinctness and show how to optimally simulate continuous-time query

algorithms of a certain form in the conventional quantum query model. Finally,

I discuss limitations of the method for simulating Hamiltonians with negative

matrix elements, and present two problems that motivate attempting to

circumvent these limitations.