Decoherence, control, and symmetry in quantum computers

Computers built on the physical principles of quantum theory offer the possibility of tremendous computational advantages over conventional computers. To actually realize such quantum computers will require technologies far beyond present day capabilities. One problem which particularly plagues quantum computers is the coupling of the quantum computer to an environment and the subsequent destruction of the quantum information in the computer through the process known as decoherence. In this thesis we describe methods for avoiding the detrimental effects of decoherence while at the same time still allowing for computation of the quantum information. The philosophy of our method is to use a symmetry of the decoherence mechanism to find robust encodings of the quantum information. The theory of such decoherence-free systems is developed in this thesis with a particular emphasis on the manipulation of the decoherence-free information. Stability, control, and methods for using decoherence-free information in a quantum computer are presented. Specific emphasis is put on decoherence due to a collective coupling between the system and its environment. Universal quantum computation on such collective decoherence decoherence-free encodings is demonstrated. Along the way, rigorous definitions of control and the use of encoded universality in the physical implementations of quantum computers are addressed. Explicit gate constructions for encoded universality on ion trap and exchange based quantum computers are given. The second part of the thesis is devoted to methods of reducing the decoherence problem which rely on more classically motivated reasoning for the robust storage of information. We examine quantum systems that can store information in their ground state such that decoherence processes are prohibited via considerations of energetics. We present the theory of supercoherent systems whose ground states are quantum error detecting codes and give examples of supercoherent systems which allow universal quantum computation. We also give examples of a spin ladder whose ground state has both the error detecting properties of supercoherence as well as error correcting properties. We present the first example of a quantum error correcting ground state which is a natural error correcting code under reasonable physical assumptions. We conclude by discussing the radical possibility of a naturally fault-tolerant quantum computer.