Group geometrical axioms for magic states of quantum computing

Abstract

Let $H$ be a non trivial subgroup of index $d$ of a free group $G$ and $N$ the normal closure of $H$ in $G$. The coset organization in a subgroup $H$ of $G$ provides a group $P$ of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states consists of MIC states associated to minimal informationally complete measurements. It is shown that, in most cases, the existence of a MIC state entails that the two conditions (i) $N=G$ and (ii) no geometry (a triple of cosets cannot produce equal pairwise stabilizer subgroups), or that these conditions are both not satisfied. Our claim is verified by defining the low dimensional MIC states from subgroups of the fundamental group $G=\pi_1(M)$ of some manifolds encountered in our recent papers, e.g. the $3$-manifolds attached to the trefoil knot and the figure-eight knot, and the $4$-manifolds defined by $0$-surgery of them. Exceptions to the aforementioned rule are classified in terms of geometric contextuality (which occurs when cosets on a line of the geometry do not all mutually commute).