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

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?

• groups or general sets? Oct 10, 2022 at 8:25
• @MarkusHeinrich oh good question. It would be cool to hear about some general sets but mostly I'm interested in groups Oct 10, 2022 at 13:32
• 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. Oct 11, 2022 at 10:12
• @MarkusHeinrich what makes you say that? Oct 16, 2022 at 16:12
• 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? Oct 17, 2022 at 18:13

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 $$d=248$$.