Continuous-time quantum walks with long-range steps ${R}^{\ensuremath{-}\ensuremath{\gamma}}$ (
R being the distance between sites) on a discrete line behave in similar ways for all $\ensuremath{\gamma}\ensuremath{\ge}2$. This is in contrast to classical random walks, which for $\ensuremath{\gamma}>3$ belong to a different universality class than for $\ensuremath{\gamma}\ensuremath{\le}3$. We show that the average probabilities to be at the initial site after time
t as well as the mean square displacements are of the same functional form for quantum walks with $\ensuremath{\gamma}=2$,
4, and with nearest neighbor steps. We interpolate this result to arbitrary $\ensuremath{\gamma}\ensuremath{\ge}2$.
We propose a scheme to implement the simplest and best-studied version of the quantum random walk, the discrete Hadamard walk, in one dimension using a coherent macroscopic sample of ultracold atoms, Bose-Einstein condensate (BEC). Implementation of the quantum walk using a BEC gives access to the familiar quantum phenomena on a macroscopic scale. This paper uses a rf pulse to implement the Hadamard operation (rotation) and stimulated Raman transition technique as a unitary shift operator. The scheme suggests the implementation of the Hadamard operation and unitary shift operator while the BEC is trapped in a long Rayleigh range optical dipole trap. The Hadamard rotation and a unitary shift operator on a BEC prepared in one of the internal states followed by a bit-flip operation, implements one step of the Hadamard walk. To realize a sizable number of steps, the process is iterated without resorting to intermediate measurement. With current dipole trap technology, it should be possible to implement enough steps to experimentally highlight the discrete quantum random walk using a BEC leading to further exploration of quantum random walks and its applications.
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 (
σ) of DTRSQW is higher than DTQW and hence DTRSQW spreads faster. The
σ of DTUBQW shows sawtooth behavior with faster spread than DTQW for some specific values of rotation angles and steps.