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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Conditions for independent decoherence

L.-M. Duan, Guang‐Can Guo

Published Aug 1, 1998• 11 citations
Publisher

Abstract

In quantum error correction schemes, it is usually assumed that the qubits are decohered independently. Starting from a practical decoherence model of the quantum register consisting of many qubits, we derive the explicit conditions for independent decoherence.
DOI
10.1088/1355-5111/10/4/004

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We consider decoherence of quantum registers, which consist of the qubits sited approximately periodically in space. The sites of the qubits are permitted to have a small random variance. We derive the explicit conditions under which the qubits can be assumed decohering independently. In other circumstances, the qubits are decohered cooperatively. We describe two kinds of collective decoherence. In each case, a scheme is proposed for reducing the collective decoherence. The schemes operate by encoding the input states of the qubits into some ''subdecoherent'' states.
The dynamical evolution of a quantum register of arbitrary length coupled to an environment of arbitrary coherence length is predicted within a relevant model of decoherence. The results are reported for quantum bits (qubits) coupling individually to different environments (``independent decoherence'') and qubits interacting collectively with the same reservoir (``collective decoherence''). In both cases, explicit decoherence functions are derived for any number of qubits. The decay of the coherences of the register is shown to strongly depend on the input states: We show that this sensitivity is a characteristic of both types of coupling (collective and independent) and not only of the collective coupling, as has been reported previously. A nontrivial behavior (``recoherence'') is found in the decay of the off-diagonal elements of the reduced density matrix in the specific situation of independent decoherence. Our results lead to the identification of decoherence-free states in the collective decoherence limit. These states belong to subspaces of the system's Hilbert space that do not get entangled with the environment, making them ideal elements for the engineering of ``noiseless'' quantum codes. We also discuss the relations between decoherence of the quantum register and computational complexity based on the dynamical results obtained for the register density matrix.
Certain physical aspects of quantum error correction are discussed for a quantum computer (n-qubit register) in contact with a decohering environment. Under rather plausible assumptions upon the form of the computer-environment interaction, the efficiency of a general correcting procedure is evaluated as a function of the spontaneous-decay duration and the rank of errors covered by the procedure. It is proved that the probability of errors can be made arbitrarily small by enhancing the correction method, provided the decohering interaction is represented by a bounded operator.
We show that errors are not generated correlatedly provided that quantum bits do not directly interact with (or couple to) each other. Generally, this no-qubit-interaction condition is assumed except for the case where two-qubit gate operation is being performed. In particular, the no-qubit-interaction condition is satisfied in the collective decoherence models. Thus, errors are not correlated in the collective decoherence. Consequently, we can say that current quantum error correcting codes that correct single-qubit errors will work in most cases including the collective decoherence.