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arXiv:1407.2713 · Journal of Physics A Mathematical and Theoretical

Order 3 symmetry in the Clifford hierarchy

Ingemar Bengtsson, Kate Blanchfield, Earl T. Campbell, Mark Howard

Published Oct 23, 2014• 16 citations
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Abstract

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.

DOI
10.1088/1751-8113/47/45/455302

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Journal of Physics A Mathematical and Theoretical•Jan 18, 2019•4 citations
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.
We express Alltop's construction of mutually unbiased bases as orbits under the Weyl-Heisenberg group in prime dimensions and find a related construction in dimensions 2 and 4. We reproduce Alltop's mutually unbiased bases using abelian subgroups of the Clifford group in prime dimensions, in direct analogy to the well-known construction of mutually unbiased bases using abelian subgroups of the Weyl-Heisenberg group.Finally, we prove three theorems relating to the distances and linear dependencies among different sets of mutually unbiased bases.
We express Alltop's construction of mutually unbiased bases as orbits under the Weyl-Heisenberg group in prime dimensions and find a related construction in dimensions 2 and 4. We reproduce Alltop's mutually unbiased bases using abelian subgroups of the Clifford group in prime dimensions, in direct analogy to the well-known construction of mutually unbiased bases using abelian subgroups of the Weyl-Heisenberg group. Finally, we prove three theorems relating to the distances and linear dependencies among different sets of mutually unbiased bases.
Abstract In quantum mechanics, mutually unbiased bases (MUBs) represent orthonormal bases (ONBs) that are as ‘far apart’ as possible, and their classification reveals rich underlying geometric structure. Given a complex inner product space, we construct the space of its ONBs as a discrete quotient of the complete flag manifold. We introduce a metric on this space, which corresponds to the ‘MUBness’ distance. This allows us to describe equivalence between sets of MUBs in terms of the geometry of this space. The subspace of bases that are unbiased with respect to the standard basis decomposes into orbits under a certain group action, and this decomposition corresponds to the classification of complex Hadamard matrices. More generally, we consider a list of k MUBs, that one wishes to extend. The candidates are points in the subspace comprising all bases which are unbiased with respect to the entire list. This space also decomposes into orbits under a group action, and we prove that points in distinct orbits yield inequivalent MUB lists. Thus, we generalize the relation between complex Hadamard matrices and MUBs. As an application, we identify new symmetries that reduce the parameter space of MUB triples in dimension 4 by a factor of 4.
Mutually unbiased bases (MUBs) are interesting for various reasons. The most attractive example of (a complete set of) MUBs is the one constructed by Ivanovi\ifmmode \acute{c}\else \'{c}\fi{} as well as Wootters and Fields, which is referred to as the canonical MUB. Nevertheless, little is known about anything that is unique to this MUB. We show that the canonical MUB in any prime power dimension is uniquely determined by an extremal orbit of the (restricted) Clifford group except in dimension 3, in which case the orbit defines a special symmetric informationally complete measurement (SIC), known as the Hesse SIC. Here the extremal orbit is the orbit with the smallest number of pure states. Quite surprisingly, this characterization does not rely on any concept that is related to bases or unbiasedness. As a corollary, the canonical MUB is the unique minimal 2-design covariant with respect to the Clifford group except in dimension 3. In addition, these MUBs provide an infinite family of highly symmetric frames and positive-operator-valued measures (POVMs), which are of independent interest.
We consider a version of the extended Clifford Group which is defined in terms of a finite Galois field in odd prime power dimension. We show that Neuhauser's result, that with the appropriate choice of phases the standard (or metaplectic) representation of the discrete symplectic group is faithful also holds for the anti-unitary operators of the extended group. We also improve on Neuhauser's result by giving explicit formulae for the (anti-)unitary corresponding to an arbitrary (anti-)symplectic matrix. We then go on to find the eigenvalues and the order of an arbitrary (anti-)symplectic matrix. The fact that in prime power dimension the matrix elements belong to a field means that this can be done using the same techniques which are used to find the eigenvalues of a matrix defined over the reals-including the use of an extension field (the analogue of the complex numbers) when the eigenvalues are not in the base field. We then give an application of these results to SIC-POVMs (symmetric informationally complete positive operator valued measures). We show that in prime dimension our results can be used to find a natural basis for the eigenspace of the Zauner unitary in which SIC-fiducials are expected to lie. Finally, we apply our results to the MUB cycling problem. We show that in odd prime power dimension d, although there is no Clifford unitary, there is a Clifford anti-unitary which cycles through the full set of Wootters-Fields MUBs if d=3 (mod 4). Also, irrespective of whether d=1 or 3 (mod 4), the Wootters-Fields MUBs split into two groups of (d+1)/2 bases in such a way that there is a single Clifford unitary which cycles through each group separately.
Abstract For over a decade, there has been intensive work on the numerical and analytic construction of SICs ( d 2 equiangular lines in ) as an orbit of the Heisenberg group. The Clifford group, which consists of the unitary matrices which normalise the Heisenberg group, plays a key role in these constructions. All of the known fiducial (generating) vectors for such SICs are eigenvectors of symplectic operations in the Clifford group with canonical order 3. Here we describe the Clifford group and the subgroup of symplectic operations in terms of a natural set of generators. From this, we classify all its elements of canonical order three. In particular, we show (contrary to prior claims) that there are symplectic operations of canonical order 3 for , which are not conjugate to the Zauner matrix. It is as yet unknown whether these give rise to SICs.