Abstract
Let H be a nontrivial subgroup of index d of a free group G and N be 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 states associated to minimal informationally complete measurements, called MIC states. It is shown that, in most cases, the existence of a MIC state entails 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 = π 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).
Highlights
Interpreting quantum theory is a long-standing effort, and no single approach can exhaust all facets of this fascinating subject
It has been shown that quantum commutation relies on some finite geometries such as generalized polygons and polar spaces [1]
Let H be a subgroup of index d of a free group G with generators and relations
Summary
Interpreting quantum theory is a long-standing effort, and no single approach can exhaust all facets of this fascinating subject. It has been shown that quantum commutation relies on some finite geometries such as generalized polygons and polar spaces [1]. The straightforward relationship of quantum commutation to the appropriate symmetries and finite groups was made possible thanks to techniques developed by the first author (and coauthors) that we briefly summarize. This will be useful at a later stage of the paper with the topic of magic state quantum computing (For a relation of finite groups to anyons and universal quantum computation, see, for instance, [5]).
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