The paper by D. A. Lidar, I. L. Chuang, and K. B. Whaley discusses the formulation of decoherence in quantum computers using the Semigroup approach. They identify error generators with the generators of a Lie algebra, providing a comprehensive description that includes the spin-boson model as a special case. The authors present a generic condition for error-less quantum computation: decoherence-free subspaces are spanned by states that are annihilated by all the generators. These subspaces are shown to be stable to perturbations and support universal quantum computation. The paper categorizes decoherence models into four types: total decoherence, independent qubit decoherence, collective decoherence, and cluster decoherence. It derives the necessary and sufficient conditions for the existence of a generic decoherence-free subspace, which involves states being degenerate eigenstates of all error generators or being annihilated by all generators if the Lie algebra is semisimple. The authors also analyze the effect of the system Hamiltonian on decoherence-free dynamics, showing that while the absence of decoherence may be spoiled by the Hamiltonian, the full dynamics in the decoherence-free subspace is decoherence-free to first order. They further demonstrate that the decoherence-free subspace is stable to symmetry-breaking perturbations, making it suitable for quantum error correction. Finally, the paper discusses the dimension of decoherence-free subspaces and their potential for universal quantum computation, showing that universal quantum computation can be performed within these subspaces. The authors conclude by highlighting the experimental challenges and the potential advantages of decoherence-free quantum computing.The paper by D. A. Lidar, I. L. Chuang, and K. B. Whaley discusses the formulation of decoherence in quantum computers using the Semigroup approach. They identify error generators with the generators of a Lie algebra, providing a comprehensive description that includes the spin-boson model as a special case. The authors present a generic condition for error-less quantum computation: decoherence-free subspaces are spanned by states that are annihilated by all the generators. These subspaces are shown to be stable to perturbations and support universal quantum computation. The paper categorizes decoherence models into four types: total decoherence, independent qubit decoherence, collective decoherence, and cluster decoherence. It derives the necessary and sufficient conditions for the existence of a generic decoherence-free subspace, which involves states being degenerate eigenstates of all error generators or being annihilated by all generators if the Lie algebra is semisimple. The authors also analyze the effect of the system Hamiltonian on decoherence-free dynamics, showing that while the absence of decoherence may be spoiled by the Hamiltonian, the full dynamics in the decoherence-free subspace is decoherence-free to first order. They further demonstrate that the decoherence-free subspace is stable to symmetry-breaking perturbations, making it suitable for quantum error correction. Finally, the paper discusses the dimension of decoherence-free subspaces and their potential for universal quantum computation, showing that universal quantum computation can be performed within these subspaces. The authors conclude by highlighting the experimental challenges and the potential advantages of decoherence-free quantum computing.