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=π1(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).
arXiv:1906.06068v1 [math.GR] 14 Jun 2019GROUP GEOMETRICAL AXIOMS FOR MAGIC STATES
OF QUANTUM COMPUTING
MICHEL PLANAT†, RAYMOND ASCHHEIM‡,
MARCELO M. AMARAL‡ AND KLEE IRWIN‡
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 sub-
group 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 measure-
ments. 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 defin-
ing the low dimensional MIC states from subgroups of the fundamental
group G = π1(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).
MSC codes: 81P68, 51E12, 57M05, 81P50, 57M25, 57R65, 14H30
1. Introduction
Interpreting quantum theory is a long standing effort and not a single
approach can exhaust all facets of this fascinating subject. Quantum infor-
mation owes much to the concept of a (generalized) Pauli group for under-
standing quantum observables, their commutation, entanglement, contextu-
ality and many other aspects, e.g. quantum computing. Quite recently, it
has been shown that quantum commutation relies on some finite geometries
such as generalized polygons and polar spaces [1]. Finite geometries connect
to the classification of simple groups as understood by prominent researchers
as Jacques Tits, Cohen Thas and many others [2, 3].
In the Atla