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Advances in Mathematical Physics

Quantum Computing in Decoherence‐Free Subspace Constructed by Triangulation

Bi Qiao, LIU JIAN GUO, Harry E. Ruda

Published Jan 1, 2010• 5 citations
Publisher

Abstract

A formalism for quantum computing in decoherence‐free subspaces is presented. The constructed subspaces are partial triangulated to an index related to environment. The quantum states in the subspaces are just projected states which are ruled by a subdynamic kinetic equation. These projected states can be used to perform ideal quantum logical operations without decoherence.

DOI
10.1155/2010/365653

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A formalism for quantum computing in decoherence‐free subspaces is presented. The constructed subspaces are partial triangulated to an index related to environment. The quantum states in the subspaces are just projected states which are ruled by a subdynamic kinetic equation. These projected states can be used to perform ideal quantum logical operations without decoherence.
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.
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
The idea of coding the information in decoherence-free subspaces (DFS) to avoid decoherence in quantum computations has been receiving a lot of attention in circuit-based quantum computations, since first proposed by Zanardi & Rasetti [1] (Phys. Rev. Lett., 1997). Following [2] (Altaisky & Kaputkina, JISST, 2025), we study the possibility to harness DFS to quantum machine learning, where no explicit algorithm is present, but the quantum system learns to classify data. We have shown that a simple classification problem of distinguishing between 2-qubit product states and 2-qubit entangled states can be solved by 4-qubit quantum reservoir [3] (Ghosh, et al., npj Quantum Inf., 2019), the output of which we fed to decoherence-free subspaces.