  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 ($\sigma$) of DTRSQW is higher
than DTQW and hence DTRSQW spreads faster. The $\sigma$ of DTUBQW shows
sawtooth behavior with faster spread than DTQW for some specific values of
rotation angles and steps.

Quantum Physics