Dephasing of entangled qubits

We investigate the influence of nearby two-level systems on the dynamics of a qubit. The intrinsic decoherence is given by a coupling of both the qubit and the two-level systems to a heat bath. Assuming weak interactions between the qubit and the two-level systems we show that the dephasing of the qubit follows a multi-exponential decay. For a flat distribution of energy splittings of the two-level systems the multi-exponential behavior simplifies to a good approximation to a single exponential decay. The rate is given by the sum of the dephasing rate of the isolated qubit plus all relaxation rates for the two-level systems out of their present state. This latter contribution sums up to a rate approximately given by the mean coupling between the qubit and the two-level systems times the number of thermally active neighbors. In recent years, extensive studies have been done on quantum computing. One of the major problems to overcome in that field is the decoherence of the qubit dynamics. Besides the dephasing due to experimental imperfections the interactions with the environment yields an intrinsic source of decoherence [1, 2, 3, 4]. The so-called quality factor of the qubit, the number of quantum operations performed during the qubit coherence time, should be at least 10 to allow for quantum error correction[5]. We will not discuss the various physical sources of decoherence. Instead we investigate a model where the qubit is weakly coupled to defects described as two-level systems (TLSs). Since the qubits are also described as TLSs, we do formally not distinguish between qubits and defects. All TLSs/qubits are coupled to a heat bath represented by a set of harmonic oscillators providing the basic source of relaxation and decoherence. Both result from the energy exchange with the heat bath like in the spin-boson model [6] which is different from the model proposed by Unruh [1]. Our investigation focuses on the question how neighboring TLSs influence the dephasing behavior of a given qubit. In the following we use qubit to address the TLS of interest and TLS for all the systems which form a background for the qubit. Qubits which are not used for an actual calculation contributes in that picture as a background. We restrict our investigation to longitudinally coupled TLSs leading to the Hamiltonian