# On the Relationship Between Continuous- and Discrete-Time Quantum Walk

@article{Childs2010OnTR, title={On the Relationship Between Continuous- and Discrete-Time Quantum Walk}, author={Andrew M. Childs}, journal={Communications in Mathematical Physics}, year={2010}, volume={294}, pages={581-603} }

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…

## 277 Citations

Continuous-time quantum walks: simulation and application

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This thesis introduces the classical and quantum walks, as well as key graph theory concepts that will underpin later results, and proposes a centrality measure based on the continuous-time quantum walk that is highly correlated with the classical eigenvector centrality and suggests that it provides an extension of the eigen vector centrality to the quantum realm.

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This article provides a new continuous-time quantum walk search algorithm that can find a marked node in any graph with any number of marked nodes, in a time that is quadratically faster than classical random walks.

Quantum walks: a comprehensive review

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This paper has reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.

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Exact simulation of coined quantum walks with the continuous-time model

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This work produces the evolution of a coined quantum walk on a generic graph using a continuous-time quantumWalk on a larger graph to accommodate the alternation between the shift and coin operators from the coined model.

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Crossovers induced by discrete-time quantum walks

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This weak convergence theorem gives a phase diagram which maps sufficiently long-time behaviors of the discrete- and continuous-time quantum and random walks.

An effective Hamiltonian approach to quantum random walk

- Physics
- 2015

In this article we present an effective Hamiltonian approach for discrete time quantum random walk. A form of the Hamiltonian for one-dimensional quantum walk has been prescribed, utilizing the fact…

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This work generalizes a discrete-time quantum walk on a line into the feed-forward quantum coin model, which depends on the coin state of the previous step, and shows that the proposed model has an anomalous slow diffusion characterized by the porous-medium equation, while the conventional discrete- time quantum walk model shows ballistic transport.

Discretization of continuous-time quantum walks via the staggered model with Hamiltonians

- PhysicsNatural Computing
- 2018

We characterize a close connection between the continuous-time quantum-walk model and a discrete-time quantum-walk version, based on the staggered model with Hamiltonians in a class of Cayley graphs,…

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