What are well-known orthogonal 2-designs, other than the real Clifford group?

Question

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?

Answer 1

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 $.