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Physics World

Feynman's ‘quantum walk’ observed

Published Aug 1, 2009• 1 citation
Publisher

Abstract

The concept of a "random walk" has been widely used to describe many real-world systems from the Brownian motion of smoke particles to the fluctuation of prices on the stock market. The mathematics is usually explained though the analogy of a person dictating their movements by the toss of a coin: get heads, for example, and they move one step to the right; get tails and they shuffle one step to the left. After many coin tosses, the person's position is random – though, statistically, it is likely to be near the starting point.

DOI
10.1088/2058-7058/22/08/5

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The concept of a "random walk" has been widely used to describe many real-world systems from the Brownian motion of smoke particles to the fluctuation of prices on the stock market. The mathematics is usually explained though the analogy of a person dictating their movements by the toss of a coin: get heads, for example, and they move one step to the right; get tails and they shuffle one step to the left. After many coin tosses, the person's position is random – though, statistically, it is likely to be near the starting point.
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Imagine that a pair of coins are tossed in a black box. The box reports only one of the following three results at random: (1) the outcome of the first coin (heads or tails), (2) the outcome of the second coin (heads or tails), or (3) whether the outcomes of the two coins matched or were different. Our task is to construct a joint probability distribution of the four possible outcomes of the coins (HH, TT, HT, TH) based on many observations of the black box outputs. Now suppose that after many trials, the black box reports that each coin comes up heads two-thirds of the time when measured individually, yet the coins never match when they are compared. (Clearly the results of the coin tosses have been correlated—perhaps a joker in the black box flips the coins and then changes the outcomes appropriately). We seek a distribution that both reflects this correlation and obeys the marginal distributions of each coin as two-thirds heads, one-third tails (see the three tables on page 23). The only way to satisfy both requirements is to force the joint probability P(TT) of getting two tails to be negative! Mathematically, this is because P(HH)+P(TT) is observed to be zero, yet we expect P(HH)to be greater than P(TT), because the individual coins are weighted toward heads.
The evolution of a walker in standard "Discrete-time Quantum Walk (DTQW)" is determined by coin and shift unitary operators. The conditional shift operator shifts the position of the walker to right or left by unit step size while the direction of motion is specified by the coin operator. This scenario can be generalized by choosing the step size randomly at each step in some specific interval. For example, the value of the roll of a dice can be used to specify the step size after throwing the coin. Let us call such a quantum walk "Discrete-time Random Step Quantum Walk (DTRSQW)". A completely random probability distribution is obtained whenever the walker follows the DTRSQW. We have also analyzed two more types of quantum walks, the "Discrete-time Un-biased Quantum Walk (DTUBQW)" and the "Discrete-time Biased Quantum Walk (DTBQW)". In the first type, the step size is kept different than unit size but the same for left and right shifts, whereas in the second type left and right shifts can also be different. The probability distribution in DTUBQW is found to follow a certain rule. The standard deviation (σσσ) of DTRSQW is higher than DTQW and hence DTRSQW spreads faster. The σσσ of DTUBQW shows sawtooth behavior with faster spread than DTQW for some specific values of rotation angles and steps.