Hamilton's turns as a visual tool kit for designing single-qubit unitary gates

Simon, B. Neethi; Chandrashekar, C. M.; Simon, Sudhavathani

Hamilton's turns as a visual tool kit for designing single-qubit unitary gates

Files

3554.pdf(1.64 MB)

Published Version

Date

2012

Authors

Simon, B. Neethi

Chandrashekar, C. M.

Simon, Sudhavathani

Publisher

American Physical Society

Published Version

10.1103/PhysRevA.85.022323

Abstract

Unitary evolutions of a qubit are traditionally represented geometrically as rotations of the Bloch sphere, but the composition of such evolutions is handled algebraically through matrix multiplication [of SU(2) or SO(3) matrices]. Hamilton's construct, called turns, provides for handling the latter pictorially through the addition of directed great circle arcs on the unit sphere S-2 subset of R-3, resulting in a non-Abelian version of the parallelogram law of vector addition of the Euclidean translation group. This construct is developed into a visual tool kit for handling the design of single-qubit unitary gates. As an application, it is shown, in the concrete case wherein the qubit is realized as polarization states of light, that all unitary gates can be realized conveniently through a universal gadget consisting of just two quarter-wave plates (QWP) and one half-wave plate (HWP). The analysis and results easily transcribe to other realizations of the qubit: The case of NMR is obtained by simply substituting pi/2 and pi pulses respectively for QWPs and HWPs, the phases of the pulses playing the role of the orientation of fast axes of these plates.

Keywords

Quantum kinematic approach , Geometric phase , Polarization optics , Bargmann invariants , Pancharatnam phase , Lorentz group , SU(2) gadget , Computation , Light , Representation

Citation

Simon, B. N., Chandrashekar, C. M. and Simon, S. (2012) 'Hamilton's turns as a visual tool kit for designing single-qubit unitary gates', Physical Review A, 85(2), 022323 (11pp). doi: 10.1103/PhysRevA.85.022323