Quantum computing thanks to Bianchi groups

Université de Bourgogne/Franche Comté, Institut FEMTO-ST UMR 6174, 15 B Avenue des Montboucons, F-25044 Besançon, France

^* e-mail: michel.planat@femto-st.fr

Published online: 15 January 2019

Abstract

It has been shown that the concept of a magic state (in universal quantum computing: uqc) and that of a minimal informationally complete positive operator valued measure: MIC-POVMs (in quantum measurements) are in good agreement when such a magic state is selected in the set of non-stabilizer eigenstates of permutation gates with the Pauli group acting on it [1]. Further work observed that most found low-dimensional MICs may be built from subgroups of the modular group PS L(2, Z) [2] and that this can be understood from the picture of the trefoil knot and related 3-manifolds [3]. Here one concentrates on Bianchi groups PS L(2, O₁₀) (with O₁₀ the integer ring over the imaginary quadratic field) whose torsion-free subgroups define the appropriate knots and links leading to MICs and the related uqc. One finds a chain of Bianchi congruence n-cusped links playing a significant role [4].