Order 3 Symmetry in the Clifford Hierarchy

TL;DR

This paper explores the symmetries and structures of certain quantum states within the Clifford hierarchy, revealing new orbit classifications and their implications for quantum computing and SIC problems.

Contribution

It uncovers the orbit structure of Alltop vectors under Clifford group actions and links these to magic states and Zauner subspaces in prime dimensions.

Findings

Alltop vectors split into three Clifford orbits in prime dimensions ≡ 1 mod 3.

Identifies Alltop vectors as magic states relevant for fault-tolerant quantum computing.

Reveals inequivalence between certain magic states based on orbit structure.

Abstract

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.