The quantum Fourier transform and quantum chaos

Abstract

In this thesis I study control of quantum systems while implementing complex quantum operations. Through experimental implementations of such operations, I test the accuracy of control and provide methods for identifying the type and strength of experimental errors. The centerpiece of this work is the quantum Fourier transform (QFT), an essential gate for quantum algorithms and quantum simulations. Experiments are performed on a three qubit liquid-state nuclear magnetic resonance quantum information processor, and demonstrate salient features of the QFT in both of these venues. The first experiment exhibits the ability of the QFT to extract periodicity, a necessary process for many quantum algorithms. As an example of a quantum simulation, I implement a three qubit quantum baker's map, which is composed of QFTs, and discuss how various conjectures of quantum chaos could be experimentally realized on a quantum computer. Another example of complex quantum operations are 'pseudo-random' maps. These are operators which pass statistical tests of randomness but can be efficiently implemented on a quantum computer. I explore the importance of pseudo-random maps for the study of quantum chaos and a host of quantum information processing protocols. I also implement a set of such maps experimentally. In order to determine the type and strength of the errors effecting our implemetations, quantum process tomography is done on the QFT.

(cont.) From the constructed QFT superoperator and Kraus forms I show how best to analyze the data in order to extract information about coherent, incoherent, and decoherent errors. Finally, I explore fidelity decay as a signature of quantum chaos. The simulations performed concentrate on the exact determination of fidelity decay behavior for quantum chaotic systems, and attempt to identify properties of the evolution operator that cause the observed fidelity decay behavior.