Quantum walk on the line as an interference phenomenon

We show that the coined quantum walk on a line can be understood as an interference phenomenon, can be classically implemented, and indeed already has been. The walk is essentially two independent walks associated with the different coin sides, coupled only at initiation. There is a simple analogy between the evolution of walker positions and the propagation of light in a dispersive optical fiber. The concept of a quantum random walk (QW for short) was first proposed ten years ago by Aharonov et al. [1] as the quantum analog of the classical random walk (RW). In the last few years, QWs have received much attention [2, 3, 4, 5, 6, 7, 8]. As some problems are best solved in classical computation with algorithms based on RWs, it is expected that these type of problems could be solved even faster in a quantum computer. Preliminary investigations focused on the nature of the QWs themselves. For example, Kempe [4] has shown that the hitting time of the discrete QW from one corner of anN -bit hypercube to the opposite corner is polynomial in the number of steps, n, whilst it is exponential in n the classical case. Subsequently Shenvi et al. [5] showed that a QW can perform the same tasks as the Grover’s search algorithm, and Childs et al. [6] introduced an algorithm for crossing a special graph exponentially faster that can be done with a classical random walk. Kempe [8] has recently reviewed the field. In a common version of the (classical) RW, the “walker” (the particle or system performing the RW) randomly takes one step to the right or to the left (in the line version) depending on the result of tossing a coin. After n steps, the probability of finding the walker at a distance m from the origin is given E-mail: [email protected] Permanent address: Departament d’Òptica, Universitat de València, Dr. Moliner 50, 46100–Burjassot, Spain. E-mail: [email protected] Permanent address: Department of Physics, University of Toronto, Toronto M5S 1A7, Canada. E-mail: [email protected]