5
$\begingroup$

The paper Real Randomized Benchmarking

https://quantum-journal.org/papers/q-2018-08-22-85/

https://arxiv.org/abs/1801.06121

makes use of the fact that the real Clifford group is an orthogonal 2-design on $ n $ qubits in order to do randomized benchmarking (in other words, it uses that fact that the real Clifford group is a 2-design on $ O(2^n) $).

Are there other well known orthogonal 2-designs?

$\endgroup$
6
  • $\begingroup$ groups or general sets? $\endgroup$ Oct 10, 2022 at 8:25
  • $\begingroup$ @MarkusHeinrich oh good question. It would be cool to hear about some general sets but mostly I'm interested in groups $\endgroup$ Oct 10, 2022 at 13:32
  • $\begingroup$ I thought the answer for groups would be more or less the same as in the unitary case (i.e. there's not much beyond the Clifford group), but turns out that I don't know enough about the rep theory of the orthogonal group. $\endgroup$ Oct 11, 2022 at 10:12
  • $\begingroup$ @MarkusHeinrich what makes you say that? $\endgroup$ Oct 16, 2022 at 16:12
  • $\begingroup$ Well, the fact that I don't know enough about the rep theory of $O(n)$ ;) The classification of unitary group designs relies on [Guralnick-Tiep, 2005] and that $\mathfrak{sl}_n$ is the complexification of $\mathfrak{su}_n$. I couldn't see how it would work in the orthogonal case, but it seems that you managed to show something, would you care to elaborate? $\endgroup$ Oct 17, 2022 at 18:13

1 Answer 1

2
$\begingroup$

The paper https://arxiv.org/abs/1810.02507 does the following

"Relying on the main results of [GT], we classify all unitary t-groups for t ≥ 2 in any dimension d ≥ 2."

In this answer I'm roughly trying to do the same for orthogonal t-groups (finite subgroups of orthogonal groups that are also 2-designs). As a side note, it turns out some of these 2-designs are also 3-designs, such as the real qubit Clifford groups for $ d=2^n $ and some exceptional cases for $ d=8,8,24,52 $. These are the rows of [GT] Table I which list an orthogonal group in the 4th column and have a $ * $ in the 2nd column, or equivalently have a 6 in the 5th column.

This answer is taken from Thm 1.5 page 3 of [GT].

Part (A) is not of interest to us because we only want finite subgroups (the commutator is not interesting it is just $ SO_n $ for $ O_n $, $ SU_n $ for $ U_n $ basically it is positive dimensional and really is just the determinant 1 subgroup)

Part (B) we can interpret with the help of Table II

Part (C) we can interpret using Lemma 5.1 I think it is basically that the real $ n $ qubit Clifford group is a 2-design for $ d=2^n $. And moreover that we can think of the real $ n $ qubit Clifford group as being (approximately/exactly?) isomorphic to $ \mathbb{F}_2^n \rtimes O_{2n}^+(\mathbb{F}_2) $ and for any subgroups $ \Gamma $ of $ O_{2n}^+(\mathbb{F}_2) $ that act transitively on the nonzero elements of $ \mathbb{F}_2^n $ then $ \mathbb{F}_2^n \rtimes \Gamma $ will also be a 2-design for $ d=2^n $. Point is, the real $ n $ qubit Clifford group is an orthogonal 2-design (indeed $ 3 $-design) for $ d=2^n $ and so are certain of its large subgroups (although I would imagine they are rarely 3-designs)(although it is possible, for example drawing from my experience with the 2 qubit (complex) Clifford group, there are 2 proper subgroups that just 2 designs and one proper subgroup that is a 3-design). Apparently these subgroups can be found in M. W. Liebeck, The affine permutation groups of rank three, Proc. London Math. Soc. 54 (1987), 477 − 516

Part (D) just look at table I all the rows with an orthogonal group in the 4th column.

In addition to the real $ n $ qubit Clifford group being an orthogonal 2-design in dimensions $ d=2^n $ it is also the case that the Weil module for $$ SU(2n,2) $$ is an orthogonal 2-design for dimensions $ d=\frac{4^n+2}{3}=6,22,86, \dots $, $ n \geq 2 $, and the Weil module $$ PSp(2n,5) $$ is an orthogonal 2-design for dimensions $ d=\frac{5^n+1}{2}=3,13,63, \dots $, $ n \geq 1 $.

In addition to the these two infinite families of non-Clifford orthogonal 2-designs there are also a finite number of sporadic/low dimensional examples. The full list of these exceptional cases is:

The ultra low dimension $ d=3,d=4 $ are not discussed in [GT]. For $ d=3 $ I'm confident that the only 2-designs are $ PSp(2,5)=PSL(2,5)=A_5 $, the $ n=1 $ case for the second Weyl module above, together with

  • $ S_4 $ and $ A_4 $ for dimension $ d=3 $.
  • For $ d=4 $ I'm less sure. $ Lift(S_4 \times S_4 ) $ should be a 2-design since its the real 2-qubit Clifford group. Probably given any $ H_1,H_2 $ 2-designs for $ d=3 $ then the lift through the double cover $ SO_4 \to SO_3 \times SO_3 $ will give a 2-design $$ Lift(H_1 \times H_2) $$

Then some small finite groups of Lie type/alternating groups for small dimensions $ d \geq 5 $

  • $ Sp(6,2) $ for dimension $ d=7 $.
  • $ 2.A_9 $ (Schur cover of $ A_9 $) for dimension $ d=8 $.
  • $ 2\Omega^+(8,2) $ for dimension $ d=8 $.
  • $ G(2,3) $ for dimension $ d=14 $.
  • $ (2 × Sp(4,4)) · 4 $ for dimension $ d=18 $.

And finally some of the sporadic finite simple groups (or associated quasisimple/almost simple groups)

  • $ McL $ for dimension $ d=22 $.
  • $ Co_2 $ for dimension $ d=23 $.
  • a much larger Conway group $ Co_3 $ for dimension $ d=23 $.
  • quasi-simple double cover $ 2.Co_1 $ of Conway 1 for dimension $ d=24 $.
  • $ 2F_4(2) · 2 $ for dimension $ d=52 $ (see Tits group https://en.wikipedia.org/wiki/Tits_group sometimes the Tits group is considered of Lie type other times it is listed as the "27th" sporadic group).
  • $ Fi_{22} $ for dimension $ d=78 $.
  • $ HN $ for dimension $ d=133 $.
  • $ Th $ for dimension $ d=248 $.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.