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arXiv:quant-ph/0110002

Quantum Computing Using an Open System and Projected Subspace

Bi Qiao, Harry E. Ruda, Zhen, X. H.

Published Sep 28, 2001• 0 citations
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Abstract

Using the subdynamical kinetic equation for an open quantum system, a formulation is presented for performing decoherence-free (DF) quantum computing in Rigged Liouville Space (RLS). Three types of interactions were considered, and in each case, stationary and evolutionary states were evaluated for DF behavior in both the total space and the projected subspace. Projected subspaces were found using the subdynamics kinetic equation. It was shown that although the total space may be decoherent, the subspace can be DF. In the projected subspace, the evolution of the density operator may be time asymmetric. Hence, a formulation for performing quantum computing in RLS or rigged Hilbert space (RHS) was proposed, and a quantum Controlled-Not Logical gate with corresponding operations in RLS (RHS) was constructed. A generalized quantum Turing machine in RHS was also discussed. Key Words: Quantum Computing, Subdynamics, Rigged Liouvile Space, Decoherence, Open System PACS: 05.30.-d+85.30+82.20.Db+84.35.+i

DOI
10.48550/arxiv.quant-ph/0110002

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Using the subdynamical kinetic equation for an open quantum system, a formulation is presented for performing decoherence-free (DF) quantum computing in Rigged Liouville Space (RLS). Three types of interactions were considered, and in each case, stationary and evolutionary states were evaluated for DF behavior in both the total space and the projected subspace. Projected subspaces were found using the subdynamics kinetic equation. It was shown that although the total space may be decoherent, the subspace can be DF. In the projected subspace, the evolution of the density operator may be time asymmetric. Hence, a formulation for performing quantum computing in RLS or rigged Hilbert space (RHS) was proposed, and a quantum Controlled-Not Logical gate with corresponding operations in RLS (RHS) was constructed. A generalized quantum Turing machine in RHS was also discussed. Key Words: Quantum Computing, Subdynamics, Rigged Liouvile Space, Decoherence, Open System PACS: 05.30.-d+85.30+82.20.Db+84.35.+i
A formulation for performing quantum computing in a projected subspace is presented, based on the subdynamical kinetic equation (SKE) for an open quantum system. The eigenvectors of the kinetic equation are shown to remain invariant before and after interaction with the environment. However, the eigenvalues in the projected subspace exhibit a type of phase shift to the evolutionary states. This phase shift does not destroy the decoherence-free (DF) property of the subspace because the associated fidelity is 1.1.1. This permits a universal formalism to be presented---the eigenprojectors of the free part of the Hamiltonian for the system and bath may be used to construct a DF projected subspace based on the SKE. To eliminate possible phase or unitary errors induced by the change in the eigenvalues, a cancellation technique is proposed, using the adjustment of the coupling time, and applied to a two-qubit computing system. A general criteria for constructing a DF-projected subspace from the SKE is discussed. Finally, a proposal for using triangulation to realize a decoherence-free subsystem based on SKE is presented. The concrete formulation for a two-qubit model is given exactly. Our approach is general and appears to be applicable to any type of decoherence.
We discuss a super bra-ket formalism for quasi-Hermitian Liouvillian operators within the framework of rigged Hilbert spaces (RHS). An RHS in terms of the Liouville space, referred to as a rigged Liouville space (RLS), is reconstructed by exploiting the mathematical fact that the space of Hilbert-Schmidt operators is unitarily equivalent to the tensor product of Hilbert spaces. The obtained RLS endows a rigorous foundation of the construction for the super bra-ket and for the spectral decompositions of both Hermitian and quasi-Hermitian Liouville operators, which are characterized by the generalized eigenvectors in the dual spaces. Furthermore, within this framework, the non-Hermitian Liouvillian operator and its adjoint can be constructed symmetrically, with their symmetric structure preserved. As an application of our RLS methodology, we examine the Liouville operators corresponding to Hermitian and non-Hermitian harmonic oscillators and elucidate the differences between their spectral decomposition forms.
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