Calculation aspects of holonomic quantum computer (HQC) are considered. Wilczek-Zee potential defining the set of quantum calculations for HQC is explicitly evaluated. Principal possibility of realization of the logical gates for this case is discussed.The conception of quantum computer (QC) and quantum calculation developed in 80-th [1]-[2] were found to be fruitful both for computer science and mathematics as well as for physics [3]. Although a device being able to perform quantum calculations is far away from practical realization, there is a number of theoretical proposals of such a construct [4]- [16]. Recently holonomic realization of QC was proposed [17], [18]. It is based on the notion of non-Abelian Berry phase [19].To perform quantum computations in this approach one needs a parametrically driven quantum system described by the Hamiltonian H(λ) with adiabatically evolving parameters λ(t) = (λ 1 (t), . . . , λ N (t)). Adiabaticity means that Ω ≪ ω min where Ω is the characteristic frequency in the Fourier spectrum of λ(t) and ω min is the minimal transition frequency in the spectrum of H(λ). If the spectrum is degenerate then the cyclic evolution of N parameters λ A (t) results in a unitary transformation of each eigenspace of H. Such transformations can be treated as computations on the eigenspace representing a part of the qubit space of HQC. These computations are shown to form a universal set of quantum gates for a specific relevant model [20]. For instance, one should choose of independent 2-loops C in the control manifold M = {all possible λ} and then the non-Abelian Berry phases U(C) corresponding to each loop represent all basic gates. To produce such gates one should have a possibility to control i.e. for each closed adiabatic curve C in M one is to know the explicit form of U(C). To know that it is necessary to be aware of an explicit expression of non-Abelian Wilczek-Zee potential A which determines U(C) completely. A more or less universal method of explicit calculation of A is to our knowledge up to now absent. For an early overview on this subject see [21]. A variant of Berry phase evaluation for a finite level system relevant to HQC was proposed in [18] where a specific parametrization of SU(n) is used. In this paper we present some less general but to our mind some more effective method of evaluation of non-Abelian Berry phase which can appear to be useful for HQC. For this purpose we restrict ourselves with systems

The Halting problem of a quantum computer is considered. It is shown that if halting of a quantum computer takes place the associated dynamics is described by an irreversible operator.

Abstract. Three massless limits of the Dirac-Kähler theory are considered. It is shown that the Dirac-Kähler equation for massive particles can be represented as a result of the gauge-invariant mixture (topological interaction) of the above massless fields.

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Non-adiabatic non-Abelian geometric phase of spin-3/2 system in the rotating magnetic field is considered. Explicit expression for the corresponding effective non-Abelian gauge potential is obtained. This formula can be used for construction of quantum gates in quantum computations. QC and in [19] for the ion-trap model. In this paper we show a realization of quantum gates for a concrete 4-level quantum system driven by external magnetic field. Let us consider a spin-3/2 system with quadrupole interaction. Physically it can be thought of as a single nucleus with the spin above. A coherent ensemble of such nuclei manifest geometric phase when placed in rotating magnetic field. This phase is non-Abelian due to degenerate energy levels with respect to the sign of the spin projection. Depending on the experiment setup the phase can be both adiabatic as in Rb experiment by Tycko [15] and non-adiabatic as in Xe experiment by Appelt et al [16]. This non-Abelian phase results in mixing of ±1/2 states in one subspace and ±3/2 in another one and thus can be regarded as a 2-qubit gate. The gate is generated by a non-Abelian effective gauge potential A that is the subject of * Phone (+375) 172 283438; e-mail alexm@hep.by † Phone (+375) 172 283438; e-mail a.tregub@open.by PACS