Quantum computing with Bianchi groups
Michel Planat
Université de Bourgogne/Franche Comté, Institut FEMTOST Besançon
Michel Planat currently works as a research scientist at FEMTOST institute in Besançon, France. Since his hiring at CNRS in 1982, he did research in several areas including nonlinear waves in piezoelectric structures, frequency metrology, number theory, signal processing, discrete mathematics and quantum information theory. He is now investigating topological quantum computing and quantum gravity.He wrote about 140 peer reviewed papers in journals, book chapters and books, had 26 invited talks and organized 3 workshops and one international conference.Being retired from his institution by the end of 2018, he keeps ready for new international actions and cooperations.
Abstract
\begin{document}{\it Introduction}We found that nonstabilizer eigenstates of permutation gates are appropriate for allowing $d$dimensional universal quantum computing (uqc) based on minimal informationally complete POVMs.... [ view full abstract ]
\begin{document}
{\it Introduction}
We found that nonstabilizer eigenstates of permutation gates are appropriate for allowing $d$dimensional universal quantum computing (uqc) based on minimal informationally complete POVMs. The relevant quantum gates may be built from subgroups of finite index $d$ of the modular group $\Gamma=PSL(2,\mathbb{Z})$ [1] or more generally from subgroups of fundamental groups $\pi$ of $3$manifolds [2]. Here previous work is encompassed by the use of torsionfree subgroups of Bianchi groups as quantum gate generators of uqc.
\vspace{5mm}{\it Methods and results}
A Bianchi group $\Gamma_k=PSL(2,\mathcal{O}_k)
The calculations are performed thanks to the softwares Magma and SnaPy.
One specializes on uqc based on the qutrit Hesse SIC and on the twoqubit geometry of the generalized quadrangle $GQ(2,2)$, these geometries are illustrated in [1], fig 1c and 2c. With the \lq magic' manifold $M$, the qutrit uqc follows from a link called 'L14n63788' with Poincar\'e polyhedron shown in the attached caption (a) and for the twoqubit uqc it follows from a $3$manifold with Poincar\'e polyhedron shown in caption (b). The former case corresponds to the 3fold irregular covering of $\pi(M)$ whose $3$manifold has first homology $\mathbb{Z}^{\oplus 5}$, $5$ cusps and volume $\approx 16.00$. The later case corresponds to the 4fold irregular covering whose $3$manifold has first homology $\mathbb{Z}/2\oplus\mathbb{Z}^{\oplus 4}$, $4$ cusps and volume $\approx 21.34$. \newline
Further congruence links encompass $M$ culminating in the $8$cusped Thurston's congruence link [with $\Gamma_{3}$ and ideal $(5+\sqrt{3})/2]$ [3], their possible role in uqc is under investigation.
\vspace{5mm}
[1] M. Planat, Entropy, 20, 16 (2018).
[2] M. Planat, R. Aschheim, M.~M. Amaral and K. Irwin, arXiv: 1802.04196 (quantph).
[3] M.~D. Baker et al, arXiv: 1802.01275.
\end{document}
Authors

Michel Planat
(Université de Bourgogne/Franche Comté, Institut FEMTOST Besançon)
Topic Area
Quantum information processing and computing
Session
OS1bR236 » Quantum information processing and computing (16:40  Wednesday, 5th September, Room 236)
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