Decoherence, Control, and Symmetry in Quantum Computers

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 the method discussed in the first part of this thesis is to use a symmetry of the decoherence mechanism to find robust encodings of the quantum information. Stability, control, and methods for using decoherence-free information in a quantum computer are presented with a specific emphasis 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. Rigorous definitions of control and the use of encoded universality in quantum computers are addressed. Explicit gate constructions for encoded universality on ion trap and exchange based quantum computers are given. In the second part of the thesis we examine physical systems with error correcting properties. We examine systems that can store quantum information in their ground state such that decoherence processes are prohibited via energetics. We present the theory of supercoherent systems whose ground states are quantum error detecting codes and describe a spin ladder whose ground state has both the error detecting and correcting properties. We conclude by discussing naturally fault-tolerant quantum computation.