On the relationship between continuous- and discrete-time quantum walk
Andrew M. Childs
TL;DR
The paper builds a formal bridge between continuous- and discrete-time quantum walks on arbitrary graphs via Szegedy's framework, showing how a discrete-time walk can emulate a given Hamiltonian and, in turn, how continuous-time dynamics emerge as a limit of discrete-time dynamics with lazy augmentation. It then leverages this correspondence to achieve linear-time Hamiltonian simulation using phase estimation on the discrete-time walk, and applies the method to obtain a near-optimal continuous-time algorithm for element distinctness. The work also connects these techniques to the continuous-time query model, analyzes limitations due to the sign problem, and outlines open problems in simulating non-sparse Hamiltonians and in applying these ideas to exponential sums and association schemes. Overall, it provides a versatile toolkit for converting between quantum-walk models, enabling efficient Hamiltonian simulation beyond sparse cases and enabling new quantum algorithms.
Abstract
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.