Schematic Diagram Of This Study (IMAGE)
Caption
The random walk is one of the models of the stochastic process. (A) represents the basic operations composed of coin tosses and shifts in the random walk. The walker with a coin shifts to the right or left by one step depending on the result of the coin toss, and then repeats to do that. The probability distribution of the walker's position after many steps has a peak at the starting point as shown in (B). The quantum walk is the counterpart in the quantum world of the random walk. In the quantum walk, the walker behaves like waves as well as particles due to the wave-particle duality. Therefore, the probability distribution spreads on both sides indicated in (C) because of the superposition of states and the Interference effect unlike that of the random walk. In our study, we use the time-dependent coin to explore the controllability of the quantum walk. We have revealed the walking mechanism i.e. the physical quantity corresponding to the coin by both of the wave nature and the particle nature (see the flow diagram around the central diagram (D)). In the wave nature, we have derived the wave equation and found that the wave velocity corresponds to the coin rate. In the particle nature, we have focused on the equation of the motion of the particle (walker) and obtained the same result. As a result, we have unveiled the walking mechanism that the coin rate of the quantum walk controls the trajectory. In addition, we have found that the quantum walk with the desired trajectory can be realized on demand as shown in (D) by designing the coin.
Credit
Haruna Katayama, Hiroshima University
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