To get a sense of how the behavior of the quantum walk differs from the classical one, we first discuss the example of the continuous time quantum walk on the line, before moving on to the discrete case. A general quantum walk does not necessarily have to be defined by the Laplacian it can be defined by any operator which “respects the structure of the graph,” that is, only allows transitions to between neighboring vertices in the graph or remain stationary. Which is, determines the behavior of the quantum analog of the continuous random walk defined previously. Then, the solution to the differential equation: ![]() One can see that the Laplacian preserves the normalization of the state of the system. Recalling the Schrödinger equation, one can see that by inserting a factor of on the left hand side of the equation for p(t) above, the Laplacian can be treated as a Hamiltonian. p(t) is given by the following differential equation: The th entry of p(t) represents the probability of being at vertex at time. The Laplacian determines the behavior of the classical continuous random walk, which is described by a length vector of probabilities, p(t). The adjacency matrix of is defined as follows: First, we review the behavior of the classical continuous random walk in order to develop the definition of the continuous quantum walk. We begin our discussion of quantum walks by introducing the quantum analog of the continuous random walk. Quantum walks can be viewed as a model for quantum computation, providing an advantage over classical and other non-quantum walks based algorithms for certain applications. ![]() In this blog post, we give a broad overview of quantum walks and some quantum walks-based algorithms, including traversal of the glued trees graph, search, and element distinctness.
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