Quantum computing thanks to Bianchi groups

D. Mogilevtsev,Michel Planat

doi:10.1051/epjconf/201919800012

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].

Highlights

A Bianchi group Γk = PS L(2, Ok) < PS L(2, C) acts as a subset of orientation-preserving isometries of 3-dimens√ional hyperbolic space H3 with Ok the ring of integers of the imaginary quadratic field I = Q( −k)
There exists a connection between the Poincaré conjecture -it states that every connected closed 3-manifold is homeomorphic to the 3-sphere- and the Bloch sphere S 3 that houses the qubits
Thurston and led a proof of the Poincaré conjecture) many of them corresponding to minimal informationally complete POVMs (MICs) and the related uqc

Summary

Introduction

One starts by upgrading these models of uqc by using other torsion-free subgroups of Bianchi groups and their corresponding 3-manifold such as the Bergé manifold that comes from the Bergé link L6a2 [with Γ−3(24)] or the so-called magic manifold that comes from the link L6a5 [with Γ−7(6)]. The latter link is a small congruence link and belongs to a chain of√eight links starting with Thurston’s eight-cusped congruence link [with Γ−3 and ideal (5 + −3)/2 [8, 9] and ending with the Whitehead link and the figureeight knot. Their possible role for uqc and the relevant 3-manifolds is discussed

Minimal informationally complete POVMs with permutations

A Bianchi factory for quantum computing

Universal quantum computing from the Bianchi factory

Congruence links in the Bianchi factory

Conclusion