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.

Ämnesord

NATURVETENSKAP  -- Fysik (hsv//swe)

NATURAL SCIENCES  -- Physical Sciences (hsv//eng)

Nyckelord

Mutually unbiased bases

magic state distillation

Clifford hierarchy

Theoretical Physics

teoretisk fysik

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