Avoiding quantum chaos in quantum computation

Abstract

We study a one-dimensional chain of nuclear 1/2−spins in an external time-dependent magnetic field. This model is considered as a possible candidate for experimental realization of quantum computation. According to general theory of interacting particles, one of the most dangerous effects is quantum chaos which can destroy the stability of quantum operations. The standard viewpoint is that the threshold for the onset of quantum chaos due to an interaction between spins (qubits) strongly decreases with an increase of the number of qubits. Contrary to this opinion, we show that the presence of a non-homogeneous magnetic field can strongly reduce quantum chaos effects. We give analytical estimates which explain this effect, together with numerical data supporting our analysis.PACS numbers: 05.45Pq, 05.45Mt, 03.67,Lx Much attention is paid in recent years to the idea of quantum computation (see, for example, [1][2][3] and references therein). The burst of interest to this subject is caused by the discovery of fast quantum algorithm for the factorization of integers [4] demonstrating the effectiveness of quantum computers in comparison to the classical ones. Nowadays, there are different projects for the experimental realization of quantum computers, based on interacting two-level systems (qubits). One of the most important problems widely discussed in the literature, is the problem of decoherence which arises in many-qubit systems due to the influence of an environment [5]. However, even in the absence of the environment, the interaction between qubits may lead to the "internal decoherence" related to the onset of quantum chaos [6].The latter subject of quantum chaos in closed systems of interacting particles has been developed recently in application to nuclear, atomic and solid state physics (see, e.g., [7] and references therein). When the (two-body) interaction between particles exceeds the critical value, fast transition to chaos occurs in the Hilbert space of manyparticle states [8]. Different aspects of this transition are now well understood, such as statistical description of eigenstates and the onset of thermalization in finite systems (see, e.g., [9] and references therein).Direct application of the quantum chaos theory to a simple model of quantum computer [6] has shown that for a strong enough interaction between qubits the onset of quantum chaos is unavoidable. Although for L = 14 − 16 qubits the critical value J cr for quantum chaos threshold is quite large, with an increase of L it decreases as J cr ∼ 1/L. From the viewpoint of the standard approach for closed systems of interacting particles, the decrease of the chaos threshold with an increase of qubits looks generic. However, in this Letter we demonstrate that this conclusion is not universal and the quantum chaos can be avoided, for example, with a proper choice of the external magnetic field.Our consideration is based on the one-dimensional model of L nuclear 1/2−spins subjected to the timedependent magnetic field of the following form [10],w...

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