On non-adiabatic holonomic quantum computer

Abstract

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

Application of Geometric Phase in Quantum Computations

“…In this section we give an example of both non-Abelian and non-adiabatic phase for a concrete 4-level quantum system driven by external magnetic field [76]. Let us consider a spin-3/2 system with quadrupole interaction.…”

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].…”