Geometric manipulation of a quantum system offers a method for fast, universal and robust quantum information processing. Here, we propose a scheme for universal all-geometric quantum computation using non-adiabatic quantum holonomies. We propose three different realizations of the scheme based on an unconventional use of quantum dot and single-molecule magnet devices, which offer promising scalability and robust efficiency.

A common strategy to measure the Abelian geometric phase for a qubit is to let it evolve along an 'orange slice' shaped path connecting two antipodal points on the Bloch sphere by two different semi- great circles. Since the dynamical phases vanish for such paths, this allows for direct measurement of the geometric phase. Here, we generalize the orange slice setting to the non-Abelian case. The proposed method to measure the non-Abelian geometric phase can be implemented in a cyclic chain of four qubits with controllable interactions.Comment: New title, minor revision, journal ref adde

Geometric quantum computation is the idea that geometric phases can be used to implement quantum gates, i.e., the basic elements of the Boolean network that forms a quantum computer. Although originally thought to be limited to adiabatic evolution, controlled by slowly changing parameters, this form of quantum computation can as well be realized at high speed by using nonadiabatic schemes. Recent advances in quantum gate technology have allowed for experimental demonstrations of different types of geometric gates in adiabatic and nonadiabatic evolution. Here, we address some conceptual issues that arise in the realizations of geometric gates. We examine the appearance of dynamical phases in quantum evolution and point out that not all dynamical phases need to be compensated for in geometric quantum computation. We delineate the relation between Abelian and non-Abelian geometric gates and find an explicit physical example where the two types of gates coincide. We identify differences and similarities between adiabatic and nonadiabatic realizations of quantum computation based on non-Abelian geometric phases.

Producing and maintaining entanglement reside at the heart of the optimal construction of quantum operations and are fundamental issues in the realization of universal quantum computation. We here introduce a setup of spin qubits that allows for geometric implementation of entangling gates between the register qubits with any arbitrary entangling power. We show this by demonstrating a circuit through a spin chain, which performs universal nonadiabatic holonomic two-qubit entanglers. The proposed gates are all electric and geometric, which would help to realize fast and robust entangling gates on spin qubits. This family of entangling gates contains gates that are as efficient as the CNOT gate in quantum algorithms. We examine the robustness of the circuit to some extent.

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