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arXiv:0708.3775 · Physical Review A

Decoherence of encoded quantum registers

Stefan Borghoff, Rochus Klesse

Published Jan 8, 2008• 2 citations
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Abstract

In order to eliminate disturbing effects of decoherence, encoding of quantum information in decoherence-free subspaces has been suggested. We analyze the benefits of this concept for a quantum register that is realized in a spin chain in contact with a common bosonic bath. Within a dissipationless model, we provide explicit analytical results for the average fidelity of plain and encoded quantum registers. For the investigation of dissipative spin-boson couplings, we employ a master equation of Bloch-Redfield type.

DOI
10.1103/physreva.77.012306

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In order to eliminate disturbing effects of decoherence, encoding of quantum information in decoherence-free subspaces has been suggested. We analyze the benefits of this concept for a quantum register that is realized in a spin chain in contact with a common bosonic bath. Within a dissipationless model, we provide explicit analytical results for the average fidelity of plain and encoded quantum registers. For the investigation of dissipative spin-boson couplings, we employ a master equation of Bloch-Redfield type.
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A model for a quantum register dissipatively coupled with a bosonic thermal bath is studied. The register consists of NNN quantum bits (qubits) the bath is described by Nb{N}_{b}Nb​ bosonic modes. The register-bath coupling is chosen in such a way that the total number of excitations is conserved. The Hilbert space splits, allowing the study of the dynamics separately in each sector. Assuming that the coupling with the bath is the same for all qubits, the excitation sectors have a further decomposition according the irreducible representations of the su(2) spin algebra. The stability against environment-generated noise of the information encoded in a quantum state of the register depends on its SU(2) symmetry content. At zero temperature we find that states belonging to the vacuum symmetry sector have, for long time, vanishing fidelity, whereas each lowest spin vector is decoupled from the bath and therefore is decoherence free. Numerical results are shown in the one-excitation space in the case of qubit-dependent bath-system coupling.
In the scope of this work the coherence of quantum information, which is encoded into a qubit register, is analysed. The qubit register is modelled by a spin chain with finite inter-spin distance. In most physically relevant realisations this spin chain irreversibly interacts with a surrounding environment, such that a spin-boson model is used to describe the setting. Due to the interaction decoherence occurs among the qubits register and quantum information gets lost. Mechanisms to slow down this decoherence process are investigated. For that purpose, the techniques of encoding qubits into decoherence-reduced subspaces and quantum error correction are used. In both cases only a linear subspace of the complete available Hilbert space of the spin chain is used as quantum code. The stability of such a code against decoherence has to be evaluated. This evaluation is performed on average over all states within the code by a code fidelity.
We investigate the decoherence process for a quantum register composed of NNN qubits coupled to an environment. We consider an environment composed of one common phonon bath and several electronic baths. This environment is relevant to the implementation of a charge based solid-state quantum computer. We explicitly compute the time evolution of all off-diagonal terms of the register's reduced density matrix. We find that in realistic configurations, ``superdecoherence'' and ``decoherence-free subspaces'' do not exist for an NNN-qubit system. This means that all off-diagonal terms decay, but not faster than ${e}^{\ensuremath{-}q(t)N}$, where q(t)q(t)q(t) is of the same order as the decay function of a single qubit.
We propose and analyze a robust quantum-state transfer protocol by the use of a combination of coherent quantum coupling and decoherence-free subspaces in a coupled quantum spin chain. Under decoherence, an arbitrary unknown quantum state embedded in a decoherence-free subspace can be perfectly transferred through a noisy channel being weakly coupled to two end registers. The method protects quantum information from both the channel noise and the environmental decoherence. A special case of utilizing two physical qubits to encode one logical qubit is considered as an example and the robustness is confirmed by numerical performances.
In idealized models of a quantum register and its environment, quantum information can be stored indefinitely by encoding it into a decoherence-free subspace (DFS). Nevertheless, perturbations to the idealized register-environment coupling will cause decoherence in any realistic setting. Expanding a measure for state preservation, the dynamical fidelity, in powers of the strength of the perturbations, we prove stability to linear order is a generic property of quantum state evolution. The effect of noise perturbation is quantified by a concise expression for the strength of the quadratic leading order, which we define as the dynamical fidelity susceptibility of DFSs. Under the physical restriction that noise acts on the register kkk-locally, this susceptibility is bounded from above by a polynomial in the system size. These general results are illustrated by two physically relevant examples. Knowledge of the susceptibility can be used to increase coherence times of future quantum computers.