Master Thesis 2014

This thesis is about the Quantum Walk on the line, the integers representing the position space of the particle. The walk is the quantum analogue of the classical random walk. In the quantum case, however, an extra qubit degree of freedom must be added to the position space due to a no-go theorem that states that otherwise only trivial walks are possible. The walks of a qubit are then determined by a unitary transformation of the qubit freedom. In the literature one mostly considers a symmetric case corresponding to the so-called Hadamard matrix.

The main techniques used in the analysis of the long time limiting probability density of the position are discussed. One well-known result from the analysis is Konno’s limit theorem, which states the probability density of a quantum walker under the weak convergence limit.

One of the standard techniques, the Fourier method, is examined more thoroughly as it is used to search for limit theorems with general unitary transformations of the internal degree of freedom and general quantum initial conditions, i.e., superpositions of position eigenstates.

The probability density at time t for a general unitary transformation of the qubit space has been derived here, using the stationary phase approximation. This result reduces to a previously known result for the Hadamard case. Building upon this, a theorem was obtained for initial conditions that are superpositions of two position eigenstates. This leads to oscillations in the probability density.