Quantum computing with Bianchi groups

TL;DR

This paper explores the use of torsion-free subgroups of Bianchi groups to derive quantum gate generators for universal quantum computing, extending previous work on modular and fundamental groups of 3-manifolds.

Contribution

It introduces a novel approach using Bianchi groups for constructing quantum gates, broadening the mathematical framework for quantum computing.

Findings

Bianchi groups can generate universal quantum gates.

A chain of Bianchi congruence n-cusped links is used in the construction.

The approach generalizes previous modular group methods.

Abstract

It has been shown 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 of the modular group $\Gamma=PSL(2,\mathbb{Z})$ [M. Planat, Entropy 20, 16 (2018)] or more generally from subgroups of fundamental groups of $3$-manifolds [M. Planat, R. Aschheim, M.~M. Amaral and K. Irwin, arXiv 1802.04196(quant-ph)]. In this paper, previous work is encompassed by the use of torsion-free subgroups of Bianchi groups for deriving the quantum gate generators of uqc. A special role is played by a chain of Bianchi congruence $n$-cusped links starting with Thurston's link.