Abstract
Constructing a discrete model like a cellular automaton is a powerful method for understanding various dynamical systems. However, the relationship between the discrete model and its continuous analogue is, in general, nontrivial. As a quantum-mechanical cellular automaton, a discrete-time quantum walk is defined to include various quantum dynamical behavior. Here we generalize 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. We show that our proposed model has an anomalous slow diffusion characterized by the porous-medium equation, while the conventional discrete-time quantum walk model shows ballistic transport.
Highlights
Constructing a discrete model like a cellular automaton is a powerful method for understanding various dynamical systems
We generalize 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
We show that our proposed model has an anomalous slow diffusion characterized by the porous-medium equation, while the conventional discrete-time quantum walk model shows ballistic transport
Summary
Correspondence and requests for materials should be addressed to Y.S. To simulate quantum mechanical phenomena, Feynman[2] proposed a quantum cellular automaton (the Feynman checkerboard) This model, defined in the general case by Meyer[3], is known as the discrete-time quantum walk (DTQW). Ãbtj ð4Þ with the site-dependent rate function gjt ~atj{1 zibtjz[1 ], ð5Þ by sq50.5(t) , t0.4 These dynamics are consistent with the time evolution of the self-similar solution[35] of the PME, which is known to describe well the anomalous diffusion of an isotropic gas through a porous medium. Since this quantum coin depends on the probability distribution of the coin states on the nearest-neighbor sites at the previous step, this model is called a feed-forward DTQW. We analyze a specific feed-forward DTQW with an experimental proposal using the polarized state and optical mode
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