Geometric phases and quantum computations

Abstract

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

Berry and Pancharatnam topological phases of atomic and optical systems

“…The above equations for Principal Bundles are especially important for the use of non-abelian geometric phases in holonomic quantum computation [22][23][24][25][26].…”

“…Following Wilczek and Zee article [12] many investigations have been made on non-abelian topological phases [15][16][17][18][19][20][21] and this area became of spe-cial interest due to the possibility to use non-abelian topological phases in quantum computation [22][23][24][25][26].…”

“…There is much interest in the use of Berry's phases to quantum computation [22][23][24][25][26]. There are different problems which have been treated for the abelian phases, including: a) topological phases for entangled states [66][67][68].…”

Scalar Aharonov‐Bohm effect in the presence of a topological defect

“…The appearance of geometric phases in quantum systems becomes a fundamental feature of the Holonomic Quantum Computation proposed by Zanardy and Raseti in [45]. The implementation of the Holonomic Quantum Computation is based on the using of non-Abelian geometric phases [46][47][48] in such a way that it can achieve great stability [49][50][51][52]. With this motivation, in this work we study the appearance of relativistic and non-relativistic geometric phases in the wave function of a neutral particle with a constant magnetic dipole moment interacting with an external magnetic field in the cosmic string background.…”

Automatic generation of nonadiabatic conditional geometric phase shift with a noncoplanar fibre system

“…Recently, much attention has been attracted to the area of the topological quantum computation both theoretically and experimentally [1][2][3][4][5]. Several basic ideas for realizing the adiabatic geometric quantum computation by using nuclear magnetic resonance (NMR) [3], superconducting nanocircuits [6] and trapped ions [7] were suggested.…”