Dynamic Quantum Walk
A quantum walk is the quantum counterpart of the coin-flip walk. In the classical version you flip a coin at each step and move one place left or right. In the quantum version the walker carries a qubit, and nothing is random until you look. At every position the state of the walk is a pair of arrows (phasors), one for “left” and one for “right”. Each step applies the same fixed rule to every pair—the coin gate—and then shifts one arrow to the left and the other to the right. Where two arrows meet, the next coin gate adds them: they can reinforce each other or cancel. This is interference.
Because arrows are added instead of probabilities, the walk spreads differently. A classical coin-flip walk creeps outward with the square root of the number of steps,
\[\text{classical spread} \sim \sqrt{n},\]and settles into the familiar bell curve. The quantum walk spreads in proportion to the number of steps ($\sim n$) and builds a two-horned shape with most of the probability near the edges.
This visualization shows the probability of finding the walker at each position as time goes on—the sum of the squared lengths of the two arrows there—alongside the classical case. Chance enters only when the position is measured; measured after every step, the walk gives the bell curve again. On a line all of this can be computed classically: the walk shows how interference works, not a speed-up. On other graphs, quantum walks are building blocks of quantum algorithms.