We investigate the action of the first three levels of the Clifford hierarchy
on sets of mutually unbiased bases comprising the Ivanovic MUB and the Alltop
MUBs. Vectors in the Alltop MUBs exhibit additional symmetries when the
dimension is a prime number equal to 1 modulo 3 and thus the set of all Alltop
vectors splits into three Clifford orbits. These vectors form configurations
with so-called Zauner subspaces, eigenspaces of order 3 elements of the
Clifford group highly relevant to the SIC problem. We identify Alltop vectors
as the magic states that appear in the context of fault-tolerant universal
quantum computing, wherein the appearance of distinct Clifford orbits implies a
surprising inequivalence between some magic states.Comment: 20 pages, 2 figures. Published versio

Bengtsson, Ingemar

Blanchfield, Kate

Campbell, Earl

Howard, Mark

English

arXiv

We investigate the action of the first three levels of the Clifford hierarchy
on sets of mutually unbiased bases comprising the Ivanovic mutually unbiased
base (MUB) and the Alltop MUBs. Vectors in the Alltop MUBs exhibit additional
symmetries when the dimension is a prime number equal to 1 modulo 3 and thus
the set of all Alltop vectors splits into three Clifford orbits. These vectors
form configurations with so-called Zauner subspaces, eigenspaces of order 3
elements of the Clifford group highly relevant to the SIC problem. We identify
Alltop vectors as the magic states that appear in the context of fault-
tolerant universal quantum computing, wherein the appearance of distinct
Clifford orbits implies a surprising inequivalence between some magic states

Campbell, Earl Terence

Institutional Repository of the Freie Universität Berlin

Order 3 symmetry in the Clifford hierarchy

We investigate the action of Weyl-Heisenberg and Clifford groups on sets of vectors that comprise mutually unbiased bases (MUBs). We consider two distinct MUB constructions, the standard and Alltop constructions, in Hilbert spaces of prime dimension. We show how the standard set of MUBs turns into the Alltop set under the action of an element at the third level of the Clifford hierarchy. We prove that when the dimension is a prime number equal to one modulo three each Alltop vector is invariant under an element of the Clifford group of order 3. The set of all Alltop vectors splits into three different orbits of the Clifford group, and forms a configuration together with the set of all subspaces invariant under an order 3 element of the Clifford group. There is a well-known conjecture that SIC vectors can be found in the eigenspace of order 3 Cliffords. This, combined with a connection between MUB and SIC vectors, suggests our work may provide a clue to the SIC existence problem in these dimensions. We identify Alltop vectors as so-called magic states which appear in the context of fault-tolerant quantum computing. The appearance of distinct Clifford orbits implies an inequivalence between some magic states

Order 3 Symmetry in the Clifford Hierarchy

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