1110.3127 (B. Neethi Simon et al.)

Hamilton's turns as visual tool-kit for designing of single-qubit
unitary gates    [PDF]

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

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 as addition of directed great circle arcs on the unit
sphere S$^2 \subset \mathbb{R}^3$, 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.

View original: http://arxiv.org/abs/1110.3127