Hamilton's turns as a visual tool kit for designing single-qubit unitary gates
TLDR
In this article, it was shown 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).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${}^{2}\ensuremath{\subset}{\mathbb{R}}^{3}$, 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 $\ensuremath{\pi}/2$ and $\ensuremath{\pi}$ pulses respectively for QWPs and HWPs, the phases of the pulses playing the role of the orientation of fast axes of these plates.read more
Citations
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Demonstrating Quantum Coherence and Metrology that is Resilient to Transversal Noise.
Chao Zhang,Thomas R. Bromley,Yun-Feng Huang,Huan Cao,Wei-Min Lv,Bi-Heng Liu,Chuan-Feng Li,Guang-Can Guo,Marco Cianciaruso,Gerardo Adesso +9 more
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Realization of arbitrary discrete unitary transformations using spatial and internal modes of light
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TL;DR: In this article, an algorithm to realize an arbitrary discrete unitary transformation on the combined spatial and internal degrees of freedom of light was proposed. But the number of beam splitters required to realize a unitary transform was increased by a factor of 2.
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