Next Article in Journal
Autocorrelation Values of Generalized Cyclotomic Sequences with Period pn+1
Previous Article in Journal
Connectedness and Path Connectedness of Weak Efficient Solution Sets of Vector Optimization Problems via Nonlinear Scalarization Methods
Previous Article in Special Issue
Quiver Gauge Theories: Finitude and Trichotomoty
Open AccessArticle

Group Geometrical Axioms for Magic States of Quantum Computing

1
Institut FEMTO-ST CNRS UMR 6174, Université de Bourgogne/Franche-Comté, 15 B Avenue des Montboucons, F-25044 Besançon, France
2
Quantum Gravity Research, Los Angeles, CA 90290, USA
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(10), 948; https://doi.org/10.3390/math7100948
Received: 20 June 2019 / Revised: 18 September 2019 / Accepted: 7 October 2019 / Published: 11 October 2019
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). View Full-Text
Keywords: quantum computing; free group theory; Coxeter-Todd algorithm; magic states; informationally complete quantum measurementds; 3- and 4-manifolds; finite geometries quantum computing; free group theory; Coxeter-Todd algorithm; magic states; informationally complete quantum measurementds; 3- and 4-manifolds; finite geometries
Show Figures

Figure 1

MDPI and ACS Style

Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Group Geometrical Axioms for Magic States of Quantum Computing. Mathematics 2019, 7, 948.

Show more citation formats Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Back to TopTop