Decoherence Suppression in Quantum Systems 2008

The following sections are included:

• PREFACE

• OUTLOOK OF THE SYMPOSIUM

• LIST OF PARTICIPANTS

• CONTENTS

This lecture note provides an easy introduction of the theory of open quantum systems in a physically and mathematically closed manner. After a compact review of quantum mechanics, we explain how to treat open quantum systems which turns out to explain the decoherence process. In order to be logically closed, we restrict to the finite quantum systems, but almost all the mathematical techniques are explained in detail so that the students can follow the equations and master the techniques which are usually assumed from the beginning. In particular, there are almost 90 exercises which supplement the contents to understand, and all the solutions will be uploaded in my web page .

We review the theories of quantum error correction, and of fault-tolerant quantum computing, and show how these powerful tools are combined to prove the accuracy threshold theorem for a particular error model. One of the theorem's assumptions is the availability of a universal set of unencoded quantum gates whose error probabilities P_e fall below a value known as the accuracy threshold P_a. For many, P_a ~ 10^-4 has become a rough estimate for the threshold so that quantum gates are anticipated to be approaching the accuracies needed for fault-tolerant quantum computing when P_e < 10^-4. We show how controllable quantum interference effects that arise during a type of non-adiabatic rapid passage can be used to produce a universal set of quantum gates whose error probabilities satisfy P_e < 10^-4. We close with a discussion of the current challenges facing an experimental implementation of this approach to reliable universal quantum computation.

We show that some composite pulses are regarded as a non-adiabatic geometric quantum gate. We propose a model to discuss the effect of fluctuations on the non-adiabatic geometric quantum gates. The current model is simple, and easy to see the physical origin of the fluctuation. We classify a type of the noise for which the non-adiabatic geometric quantum gates are robust.

We consider a model of a spin system under the influence of decoherence such that a system coupled with a dissipating environmental system consisting of either spins or bosonic modes. The dissipation of an environment is governed by a certain probability with which an environmental system localized around a principal system dissipates into a larger bath and a thermal environmental system instead migrates into the place. A certain threshold on the probability is found in the growth of decoherence in a principal system. A larger as well as a smaller dissipation probability than the threshold results in smaller decoherence.

This finding is utilized to elucidate a spin relaxation theory of a magnetic resonance spectrometer. In particular, a seamless description of transverse relaxation and motional narrowing is possible.

We also numerically evaluate the dynamics of coherence useful for quantum information processing. The bang-bang control and anti-Zeno effect in entanglement and the Oppenheim-Horodecki nonclassical correlation are investigated in the model of spin-boson coupling.

We outline the theory of holonomic quantum computing and implementation of a holonomic quantum gate. Then we analyze an isospectral deformation of the Ising-type model Hamiltonian as an example of holonomic quantum computing and construct a few holonomic quantum gates using this system.

INDEX