Quantum computing with Bianchi groups

\begin{document}{\it Introduction}We found that non-stabilizer 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}

We found that non-stabilizer 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 torsion-free 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 two-qubit 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 two-qubit uqc it follows from a $3$-manifold with Poincar\'e polyhedron shown in caption (b). The former case corresponds to the 3-fold 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 4-fold 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 (quant-ph).
[3] M.~D. Baker et al, arXiv: 1802.01275.

\end{document}