On the explicit constructions of certain unitary $t$-designs

TL;DR

This paper presents explicit constructions of certain unitary t-designs in quantum information science, including new constructions of 3-designs in U(3) and 4-designs in U(4), addressing open problems in the field.

Contribution

The authors establish a method to construct higher-order unitary t-designs from specific unitary t-groups, providing explicit examples in U(3) and U(4).

Findings

Constructed exact 3-designs in U(3) from the unitary 2-group SL(3,2).

Constructed exact 4-designs in U(4) from the unitary 3-group Sp(4,3).

Provided numerical evidence and discussed related open problems.

Abstract

Unitary $t$-designs are `good' finite subsets of the unitary group $U(d)$ that approximate the whole unitary group $U(d)$ well. Unitary $t$-designs have been applied in randomized benchmarking, tomography, quantum cryptography and many other areas of quantum information science. If a unitary $t$-design itself is a group then it is called a unitary $t$-group. Although it is known that unitary $t$-designs in $U(d)$ exist for any $t$ and $d$, the unitary $t$-groups do not exist for $t\geq 4$ if $d\geq 3$, as it is shown by Guralnick-Tiep (2005) and Bannai-Navarro-Rizo-Tiep (BNRT, 2018). Explicit constructions of exact unitary $t$-designs in $U(d)$ are not easy in general. In particular, explicit constructions of unitary $4$-designs in $U(4)$ have been an open problem in quantum information theory. We prove that some exact unitary $(t+1)$-designs in the unitary group $U(d)$ are constructed from unitary $t$-groups in $U(d)$ that satisfy certain specific conditions. Based on this result, we specifically construct exact unitary $3$-designs in $U(3)$ from the unitary $2$-group $SL(3,2)$ in $U(3),$ and also unitary $4$-designs in $U(4)$ from the unitary $3$-group $Sp(4,3)$ in $U(4)$ numerically. We also discuss some related problems.

Equations

$$({\chi }^{t+1},{\chi }^{t+1}{)}_G=({\chi }^{t+1},{\chi }^{t+1}{)}_{U(d)}+1.$$

$${\int }_{U(d)}f(U)\mathrm{d}U=\frac{1}{|X|}\sum _{U\in X}f(U)$$

$$(1,f{)}_{U(d)}=(1,f{)}_X,$$

$$(1,f{)}_{U(d)}={\int }_{U(d)}\overline{f}\mathrm{d}U$$

$$(1,f{)}_X=\frac{1}{|X|}\sum _{x\in X}\overline{f(x)}.$$

$$\frac{1}{|G|}\sum _{g\in G}|\mathrm{Tr}\chi (g){|}^{2t}={\int }_{U\in U(d)}|\mathrm{Tr}\chi (U){|}^{2t}\mathrm{d}U.$$

$$M_{2t}(G,V)=M_{2t}(U(d),V).$$

$$M_{2t}(G,V):=({\chi }^t,{\chi }^t{)}_G=\frac{1}{|G|}\sum _{g\in G}{\chi }^t(g)\overline{{\chi }^t(g)}=\frac{1}{|G|}\sum _{g\in G}|\mathrm{Tr}\chi (g){|}^{2t}$$

$$M_{2t}(U(d),V):=({\chi }^t,{\chi }^t{)}_{U(d)}={\int }_{U\in U(d)}|\mathrm{Tr}\chi (U){|}^{2t}\mathrm{d}U.$$

$$\frac{1}{|X|}\sum _{U\in X}\rho (U)={\int }_{U(d)}\rho (U)\mathrm{d}U.$$

$$\frac{1}{|X{|}^2}\sum _{U,V\in X}\mathrm{Tr}\rho (U^{†}V)\ge {\int }_{U(d)}\mathrm{Tr}\rho (U)\mathrm{d}U.$$

$$0\le \mathrm{Tr}(S^{†}S)$$

$$+{\int }_{U(d)}{\int }_{U(d)}\mathrm{Tr}\rho (U^{†}V)\mathrm{d}U\mathrm{d}V$$

$$=\frac{1}{|X{|}^2}\sum _{U,V\in X}\mathrm{Tr}\rho (U^{†}V)-{\int }_{U(d)}\mathrm{Tr}\rho (U)\mathrm{d}U.\text{qed}$$

$$M_1=\left[\begin{array}{ccc}1& & \\ & & 1\\ & 1& \end{array}\right],M_2=\left[\begin{array}{ccc}1& & \\ & 1& \\ & & -1\end{array}\right],M_3=\left[\begin{array}{ccc}1/2& -1/2& -a/2\\ -1/2& 1/2& -a/2\\ -\overline{a}/2& -\overline{a}/2& 0\end{array}\right].$$

$$M_1=\left[\begin{array}{cccc}1& & & \\ & 1& & \\ & & {\omega }^2& \\ & & & 1\end{array}\right],$$

$$M_3=\left[\begin{array}{cccc}1& & & \\ & {\omega }^2& & \\ & & 1& \\ & & & 1\end{array}\right],$$

$$f(U)=\sum _{(g_1,g_2)\in G\times G}{\chi }_{\mu }(g_1^{†}U{g}_2).$$

$${\chi }_{\mu }(\Lambda )=s_{\mu }({\lambda }_1,\dots ,{\lambda }_d),$$

$${\chi }_{\mu }(\Lambda )=\det (\Lambda {)}^{{\mu }_d}{\chi }_{{\mu }^{\prime }}(\Lambda ),$$

$${\chi }_{(3,0,-3)}(\Lambda )=\det (\Lambda {)}^{-3}{s}_{(6,3,0)}({\lambda }_1,{\lambda }_2,{\lambda }_3)$$

$$\displaystyle=\frac{1}{(\lambda_{1}\lambda_{2}\lambda_{3})^{3}}\bigg{(}-2\lambda_{1}\lambda_{2}\lambda_{3}\left(\lambda_{1}\lambda_{2}+\lambda_{3}\lambda_{2}+\lambda_{1}\lambda_{3}\right)\left(\lambda_{1}+\lambda_{2}+\lambda_{3}\right){}^{4}$$

$$+\left({\lambda }_1{\lambda }_2+{\lambda }_3{\lambda }_2+{\lambda }_1{\lambda }_3\right){}^3\left({\lambda }_1+{\lambda }_2+{\lambda }_3\right){}^3+2{\lambda }_1^2{\lambda }_2^2{\lambda }_3^2\left({\lambda }_1+{\lambda }_2+{\lambda }_3\right){}^3$$

$$+3{\lambda }_1{\lambda }_2{\lambda }_3\left({\lambda }_1{\lambda }_2+{\lambda }_3{\lambda }_2+{\lambda }_1{\lambda }_3\right){}^2\left({\lambda }_1+{\lambda }_2+{\lambda }_3\right){}^2-2\left({\lambda }_1{\lambda }_2+{\lambda }_3{\lambda }_2+{\lambda }_1{\lambda }_3\right){}^4\left({\lambda }_1+{\lambda }_2+{\lambda }_3\right)$$

$$\displaystyle-5\lambda_{1}^{2}\lambda_{2}^{2}\lambda_{3}^{2}\left(\lambda_{1}\lambda_{2}+\lambda_{3}\lambda_{2}+\lambda_{1}\lambda_{3}\right)\left(\lambda_{1}+\lambda_{2}+\lambda_{3}\right)+\lambda_{1}^{3}\lambda_{2}^{3}\lambda_{3}^{3}+2\lambda_{1}\lambda_{2}\lambda_{3}\left(\lambda_{1}\lambda_{2}+\lambda_{3}\lambda_{2}+\lambda_{1}\lambda_{3}\right){}^{3}\bigg{)}$$

$${\chi }_{(3,0,-3)}(\Lambda )=\mathrm{Tr}({\Lambda }^2){\overline{\mathrm{Tr}(\Lambda )}}^2+\mathrm{Tr}({\Lambda }^3)\overline{\mathrm{Tr}(\Lambda )\mathrm{Tr}({\Lambda }^2)}+\mathrm{Tr}(\Lambda {)}^2\overline{\mathrm{Tr}({\Lambda }^2)}$$

$$-2\mathrm{Tr}({\Lambda }^2)\overline{\mathrm{Tr}({\Lambda }^2)}+\mathrm{Tr}(\Lambda )\mathrm{Tr}({\Lambda }^2)\overline{\mathrm{Tr}({\Lambda }^3)}-\mathrm{Tr}({\Lambda }^3)\overline{\mathrm{Tr}({\Lambda }^3)}-3\mathrm{Tr}(\Lambda )\overline{\mathrm{Tr}(\Lambda )}+10$$

$$\displaystyle\chi_{(4,0,0,-4)}(\Lambda)=\bigg{(}\left|18\operatorname{Tr}(\Lambda^{4})-12\operatorname{Tr}(\Lambda)\operatorname{Tr}(\Lambda^{3})-6\operatorname{Tr}(\Lambda^{2})^{2}+4\operatorname{Tr}(\Lambda^{2})\operatorname{Tr}(\Lambda)^{2}\right|^{2}$$

$$+48\Re \left[(2\mathrm{Tr}(\Lambda )\mathrm{Tr}({\Lambda }^3)+\mathrm{Tr}({\Lambda }^2{)}^2)\overline{\mathrm{Tr}({\Lambda }^2)\mathrm{Tr}(\Lambda {)}^2}\right]$$

$$-16{\left|\mathrm{Tr}({\Lambda }^2)\mathrm{Tr}(\Lambda {)}^2\right|}^2+{\left|24\mathrm{Tr}({\Lambda }^3)-27\mathrm{Tr}({\Lambda }^2)\mathrm{Tr}(\Lambda )+3\mathrm{Tr}(\Lambda {)}^3\right|}^2$$

$$-{\left|3\mathrm{Tr}(\Lambda {)}^3-27\mathrm{Tr}(\Lambda )\mathrm{Tr}({\Lambda }^2)\right|}^2+360{\left|\mathrm{Tr}({\Lambda }^2)\right|}^2+216{\left|\mathrm{Tr}(\Lambda {)}^2\right|}^2$$

$$\displaystyle-1296\Re\left[\overline{\operatorname{Tr}(\Lambda^{2})}\operatorname{Tr}(\Lambda)^{2}\right]+432\left|\operatorname{Tr}(\Lambda)\operatorname{Tr}(\Lambda^{2})\right|^{2}-720\left|\operatorname{Tr}(\Lambda)\right|^{2}-5040\bigg{)}/144$$

$$\cos (u)[281838\cos (v)-156\cos (2v)-158]+\sqrt{7}\sin (u)\left[24\cos (v)+6\cos (2v)+2\right]$$

$$+[28125-181\cos (v)+140901\cos (2v)-65\cos (3v)]=0.$$

Full text

I Introduction

The basic idea of “design theory” is to approximate a given space $M$ by a good finite subset $X$ of $M$. The spherical $t$-designs are those finite subsets $X$ of the unit sphere $M=S^{n-1}$ such that for any polynomial $f$ of degree up to $t$, the spherical integral of $f$ on the sphere is given by the average value of $f$ at the finitely many points of $X$ of $S^{n-1}$ MR0485471 . So are the concept of combinatorial $t$-designs ($t$-$(v,k,\lambda)$ designs) of the $M={\binom{V}{k}}$, the set of all the $k$-element subsets of a set $V$ of cardinality $v$. The space $M$ has the structure of an association scheme called Johnson association scheme $J(v,k)$. This concept of $t$-design was generalized further to the concept of $t$-designs in $Q$-polynomial association schemes by Delsarte MR0384310 . There are many different kinds of design theories and there is vast literature on various design theories. We would like to refer the readers, in particular, to the following two papers MR2535394 ; MR3594369 for the review of the developments of design theory including many generalizations of the concept of $t$-designs, from viewpoint of algebraic combinatorics.

The microscopic world is described by quantum physics, where the time-evolution of a closed system is expressed by a unitary transformation. Accordingly, study of unitary transformations, or unitary matrices if the system is finite-dimensional, is essential to understand the quantum world. Needless to say, unitary transformations play central roles in quantum computing and quantum information theory. So, it is natural for us to approximate the whole unitary group $\operatorname{U}(d)$ by a finite subset $X$ of $M=\operatorname{U}(d)$. This lead physicists and mathematicians to formulate the concept of unitary $t$-designs MR2326329 ; MR2433437 . A systematic study of unitary $t$-designs from a mathematical viewpoint is given by Roy-Scott MR2529619 and we use their paper as a basic reference on unitary $t$-designs. There are many further developments on the theory of unitary $t$-designs, including those so called approximate unitary $t$-designs. Those unitary $t$-designs which satisfy Equation 1 in Definition 3 in Section II is called exact unitary $t$-designs. Approximate unitary $t$-designs have also been considered and studied mainly in physics.

A unitary $t$-design $X$ of $\operatorname{U}(d)$ is called a unitary $t$-group if $X$ is a subgroup of $\operatorname{U}(d)$ as well. In physics, cf Zhu_2017 ; Webb:2016:CGF:3179439.3179447 ; 1609.08172 , some unitary 3-groups have been known, say Clifford groups and some sporadic examples, but the difficulty of finding unitary 4-groups (except for the case of $d=2$, cf. 1810.02507 ) has been noticed. Actually, the non-existence of unitary 4-groups was known for $d\geq 5$ in a disguised form in finite group theory, in a very deep paper of Guralnick-Tiep MR2123127 ) that uses the classification of finite simple groups. This was recently pointed out by BNRT 1810.02507 and the complete classification of unitary $t$-groups on $\operatorname{U}(d)$ for all $t\geq 2$ and $d\geq 2$ was obtained therein.

Although unitary $4$-groups on $\operatorname{U}(d)$ do not exist for $d\geq 3$ at all, unitary $t$-designs exist for all $t$ and $d$ as was proved in Seymour-Zaslavsky MR744857 . However, the explicit constructions of unitary $t$-designs are challenging in general, similarly as in the case for the explicit constructions of spherical $t$-designs. In particular, while the existence of unitary $4$-designs in $\operatorname{U}(4)$ have been known, their explicit constructions were not obtained so far to our knowledge Nakata . Explicit constructions of unitary $t$-designs are essential in many areas of quantum information processing such as efficient randomized benchmarking of quantum channels PhysRevA.77.012307 ; PhysRevLett.106.180504 ; PhysRevA.85.042311 ; wallman2014randomized ; 1510.02767 ; 1711.08098 , quantum process tomographyMR2433437 ; 2016APS..MARB44006L , quantum state tomographyPhysRevA.72.032325 ; scott2006tight ; PhysRevA.84.022327 ; PhysRevA.90.012115 , decoupling szehr2013decoupling ; nakata2017decoupling , quantum cryptographyambainis2009nonmalleable and data hiding985948 , among others. Their efficient implementation in terms of the number of local gates have been actively studied Cleve:2016:NCE:3179473.3179474 ; MR2551028 ; MR3535891 ; PhysRevX.7.021006 .

The main purpose of this paper is to give explicit constructions of unitary $3$-designs in $\operatorname{U}(3)$ and unitary $4$-designs in $\operatorname{U}(4)$ numerically. In order to do that, we first obtain the following purely mathematical theorem that explains how we can construct unitary $(t+1)$-designs from certain unitary $t$-group $G$ explicitly. Namely, we obtain the following Theorem:

Theorem 1**.**

Let ${G}$ be a finite subgroup of $\operatorname{U}(d)$, and let $\chi:\operatorname{U}(d)\to\operatorname{U}(d)$ be the natural (fundamental) unitary representation of $\operatorname{U}(d)$. We abuse the notation by considering $\chi:G\hookrightarrow\operatorname{U}(d)$ as the natural embedding of $G$. Suppose that ${G}$ is a unitary $t$-group in ${\operatorname{U}(d)}$. Let $\chi^{t+1}$ be the $(t+1)$ times tensor product of the fundamental representation $\chi$. Suppose

[TABLE]

Then there exists a non-zero $G\times G$-invariant homogeneous polynomial $f\in\operatorname{Hom}(\operatorname{U}(d),t+1,t+1)$, unique up to scalar multiplication, such that $\int_{\operatorname{U}(d)}f(U)\,\mathrm{d}U=0$. Let ${U_{0}\in\operatorname{U}(d)}$ be a zero of ${f.}$ Then the orbit ${X}=GU_{0}G$ of ${U_{0}}$ under the action ${G\times G}$ on ${\operatorname{U}(d)}$ becomes a unitary $(t+1)$-design in ${\operatorname{U}(d)}$.

Here we defined the inner product of two representations $\rho_{1}$ and $\rho_{2}$ of a group $\operatorname{U}(d)$ by $(\rho_{1},\rho_{2})_{\operatorname{U}(d)}=\int_{\operatorname{U}(d)}\operatorname{Tr}\rho_{1}(U)\overline{\operatorname{Tr}\rho_{2}(U)}\,\mathrm{d}U$ and $(\rho_{1},\rho_{2})_{G}=\frac{1}{|G|}\sum_{x\in G}\operatorname{Tr}\rho_{1}(x)\overline{\operatorname{Tr}\rho_{2}(x)}$ for a finite subgroup $G\subset\operatorname{U}(d)$. The Haar measure is normalized as $\int_{\operatorname{U}(d)}\,\mathrm{d}U=1$.

This theorem guarantees that if there is such $G$ satisfying the conditions of Theorem 1, then there is a non-trivial homogeneous polynomial $f$ in $\operatorname{Hom}(\operatorname{U}(d),t+1,t+1)$ that is invariant under the action of $G\times G.$ Take any zero $U_{0}$ of $f$ on $\operatorname{U}(d)$, then the orbit of $U_{0}$ under the action of $G\times G$, say $X=GU_{0}G$, gives a unitary $(t+1)$-design on $\operatorname{U}(d).$ In Section V, we apply this Theorem in particular for the two cases

$d=3,G=\operatorname{SL}(3,2),t=2$ 2. 2.

$d=4,G=\operatorname{Sp}(4,3),t=3$

to construct the explicit unitary $(t+1)$-designs in $\operatorname{U}(d)$ numerically.

This technique also works for other $G$ satisfying the conditions of Theorem 1, but the large order of the group so far prevented us from getting the explicit examples for other cases. They should be manageable if we have more computational resources.

Theorem 1 claims $X$ is a unitary $(t+1)$-design, although it does not rule out the possibility that $X$ is also a unitary $(t+2)$-design. We have the following theorem to bound the strength of the design.

Theorem 2**.**

Let $G$ be a finite subgroup of $\operatorname{U}(d)$. Let $X_{1}=GU_{1}G$ and $X_{2}=GU_{2}G$ be two orbits of the natural action $G\times G$ on $\operatorname{U}(d)$. Suppose $X_{i}$ is a unitary $t_{i}$-design but not a unitary $(t_{i}+1)$-design where $i=1,2$. Then $t_{1}\leq 2t_{2}+1$ and $t_{2}\leq 2t_{1}+1$.

This theorem is motivated by MR740321 which proves a similar result for spherical designs.

We will conclude our paper by giving some discussions.

II Unitary $t$-designs and unitary $t$-groups

Let us recall the definition of unitary $t$-designs in $\operatorname{U}(d).$

Definition 3**.**

A finite subset ${X}$ of the unitary group ${\operatorname{U}(d)}$ is called a unitary ${t}$-design, if

[TABLE]

for any ${f(U)\in\operatorname{Hom}(\operatorname{U}(d),t,t).}$ Here ${\operatorname{Hom}(\operatorname{U}(d),r,s)}$ is the space of polynomials that are homogeneous of degree ${r}$ in the matrix entries of ${U}$, and homogeneous of degree ${s}$ in the matrix entries of the Hermitian conjugate ${U^{\dagger}}$ of $U$.

Those satisfying the condition (1) above are called exact unitary $t$-designs in some literature. In this paper, we consider only these unitary $t$-designs. While those with the condition (1) replaced by the condition that the difference of both sides is very small, are called approximate unitary $t$-designs. Of course exact unitary $t$-designs are approximate unitary $t$-designs, and both types of unitary $t$-designs are studied extensively in physics MR2551028 ; MR3535891 ; PhysRevX.7.021006 ; dankert2009exact ; harrow2009random .

It is known that there are many equivalent characterizations of unitary $t$-design in $\operatorname{U}(d)$. (cf. Roy-Scott MR2529619 , Zhu-Kueng-Grassl-Gross 1609.08172 .) Here, we will use some of the equivalent conditions later in our paper. One equivalent definition is as follows (MR2529619, , p.14):

A finite subset $X$ in $\operatorname{U}(d)$ is a unitary $t$-design, if and only if for any $f\in\operatorname{Hom}(\operatorname{U}(d),t,t)$,

[TABLE]

where

[TABLE]

and

[TABLE]

There are several different characterization of unitary $t$-groups (MR2529619, , Corollary 8) and (1609.08172, , Proposition 3). Let ${\chi}$ be the natural (fundamental) representation of ${G\hookrightarrow\operatorname{U}(d)}$ as well as the natural representation of $\operatorname{U}(d)$, $\chi:\operatorname{U}(d)\hookrightarrow\operatorname{U}(d)$. The notation $\chi^{t}$ is the shorthand for $t$ times tensor product $\chi\otimes\cdots\otimes\chi$.

A finite subgroup ${G}$ is a unitary ${t}$-group in ${\operatorname{U}(d)}$, if and only if

[TABLE] 2. 2.

A finite subgroup $G$ is a unitary $t$-group if and only if the decomposition of $\operatorname{U}(d)^{\otimes t}$ into the irreducible representations of ${\operatorname{U}(d)}$ is the same as the decomposition of $G^{\otimes t}$ into the irreducible representations of ${G}$ in the sense of both dimension and multiplicity. 3. 3.

A finite subgroup ${G\subset\operatorname{U}(d)}$ is a unitary ${t}$-group, if and only if

[TABLE]

where the LHS

[TABLE]

and the RHS ${M_{2t}(\operatorname{U}(d),V)}$ is the corresponding inner product

[TABLE]

Let us recall that unitary $t$-groups in $\operatorname{U}(d)$ are completely classified for all $t\geq 2$ and $d\geq 2.$ (Cf. Guralnick-Tiep MR2123127 and BNRT 1810.02507 .) The main purpose of this paper is to prove Theorem 1 given in Section II and construct new unitary designs accordingly.

III Proofs

It is known that the irreducible representations of $\operatorname{U}(d)$ appearing in $\chi^{t+1}\otimes\overline{\chi}^{t+1}$ are parametrized by the non-increasing integer sequence $\mu=(\mu_{1},\mu_{2},\ldots,\mu_{d}).$ The irreducible representation of $\operatorname{U}(d)$ corresponding the sequence $\mu$ is denoted by $\rho_{\mu}.$ (cf. MR2529619 ). Let $\Phi$ be the set of $\mu$ with $\rho_{\mu}$ in the representation $\chi^{t+1}\otimes\overline{\chi}^{t+1}$. Such $\mu$ is characterized by $\mu_{+}=-\mu_{-}\leq t+1$. Here, $\mu_{+}$ is the sum of all positive $\mu_{i}$’s and $\mu_{-}$ is the sum of all negative $\mu_{i}$’s.

Let $G$ be a subgroup of $\operatorname{U}(d)$, and let $\chi$ be the natural embedding of $G$ into $\operatorname{U}(d)$. Suppose that $d\geq t+1$. Let $(\chi^{t+1}\overline{\chi}^{t+1},1)_{G}=(t+1)!+1$. First we prove the following proposition:

Proposition 4**.**

With the notation given above, there is a unique non-trivial irreducible representation $\rho_{\widetilde{\mu}}$ such that $(\rho_{\widetilde{\mu}},1)_{G}=1$, where $\widetilde{\mu}\in\Phi$.

Proof.

We write $\displaystyle\chi^{t+1}=\bigoplus_{\lambda}H_{\lambda}=\bigoplus_{\lambda}W_{\lambda}\otimes S_{\lambda}$ as given in (1609.08172, , p.12). Here $W_{\lambda}$ is the Weyl module carrying the irreducible representation of $\operatorname{U}(d)$ associated with the partition $\lambda$ while $S_{\lambda}$ is the Specht module of which the symmetric group $S_{t+1}$ acts irreducibly.

By our assumption $(\chi^{t+1}\overline{\chi}^{t+1},1)_{G}=\sum_{\lambda,\rho}d_{\lambda}d_{\tau}(W_{\lambda}\overline{W_{\tau}},1)_{G}=(t+1)!+1$. Here $\lambda$ and $\tau$ are non-increasing partitions of $t+1$ into no more than $d$ parts and $d_{\lambda}$ is the degree of the Specht module $S_{\lambda}$. Note that $(\chi^{t+1}\overline{\chi}^{t+1},1)_{\operatorname{U}(d)}=\sum_{\lambda,\tau}d_{\lambda}d_{\tau}(W_{\lambda}\overline{W_{\tau}},1)_{\operatorname{U}(d)}=(t+1)!=\sum{d_{\lambda}}^{2}$.

On the other hand, the irreducible representations $\rho_{\mu}$ of $\operatorname{U}(d)$ appearing in $\chi^{t+1}\overline{\chi}^{t+1}$ are characterized in (MR2529619, , Theorem 4). We do not know the exact multiplicities in which each irreducible representation $\rho_{\mu}$ (of $\operatorname{U}(d)$) appearing in $\chi^{t+1}\overline{\chi}^{t+1}$. However, we know that the multiplicity of $\rho_{(0,...,0)}$ is $(t+1)!$. Since $(\rho_{\mu},1)_{\operatorname{U}(d)}=0$ for $\mu\not={(0,...,0)}$, we conclude that there is exactly one $\widetilde{\mu}\neq{(0,...,0)}$ such that $(\rho_{\tilde{\mu}},1)_{G}=1$. ∎

Next we introduce the concept of unitary $\rho$-design for later proof.

Definition 5**.**

Let $(\rho,V)$ be a unitary representation of $\operatorname{U}(d)$. Let $X$ be a finite subset of $\operatorname{U}(d)$. Then $X$ is called a unitary $\rho$-design if

[TABLE]

Obviously we have another characterization of unitary $t$-design.

Theorem 6**.**

$X$* is a unitary $t$-design if and only if $X$ is a unitary $\rho$-design for every irreducible representation $\rho$ appearing in $U^{\otimes t}\otimes(U^{\dagger})^{\otimes t}$.*

We mimic the proof in (MR2433437, , Theorem 5.4) to get an equivalent definition of unitary $\rho$-design, whose condition is easy to confirm.

Theorem 7**.**

For any finite $X\subset\operatorname{U}(d)$,

[TABLE]

with equality if and only if $X$ is a unitary $\rho$-design.

Corollary 8**.**

If $X$ is a unitary $\rho$-design, then $UX$ is also a unitary $\rho$-design for every $U\in\operatorname{U}(d)$.

Proof of Theorem 7.

Let $S:=\frac{1}{|X|}\sum_{U\in X}\rho(U)-\int_{\operatorname{U}(d)}\rho(U)\,\mathrm{d}U$. Then

[TABLE]

Now we are able to prove Theorem 1.

Proof of Theorem 1.

First we recall the fact that all the matrix coefficient functions of $\rho_{\mu}$ where $\mu\in\Phi$ form a basis of $\operatorname{Hom}(\operatorname{U}(d),t+1,t+1)$ (MR2529619, , Theorem 7). By Proposition 4, there is a unique non-trivial irreducible representation $\rho_{\widetilde{\mu}}$ such that $(\rho_{\widetilde{\mu}},1)_{G}=1$. By symmetry $\widetilde{\mu}=-\widetilde{\mu}$. Here $-\widetilde{\mu}$ is obtained by negating entries of $\widetilde{\mu}$ and put them in reverse order.

For every non-trivial $\mu\neq\widetilde{\mu}$, we have shown that $(\rho_{\mu},1)_{G}=0$. Therefore $\frac{1}{|G|^{2}}\sum_{U,V\in G}\operatorname{Tr}\rho_{\mu}(U^{\dagger}V)=0$. By Theorem 7, $G$ is a unitary $\rho_{\mu}$-design. It implies that $\frac{1}{|G|}\sum_{U\in G}\rho_{\mu}(U)=0$. Hence every matrix coefficient function of $\rho_{\mu}$ becomes [math] after $G\times G$ averaging.

For $\mu=\widetilde{\mu}$, let us consider its matrix coefficient functions. Let $A=\frac{1}{|G|}\sum_{U\in G}\rho_{\widetilde{\mu}}(U)$ and $M=M(U)=\frac{1}{|G|^{2}}\sum_{g_{1},g_{2}\in G}\rho_{\widetilde{\mu}}(g_{1}^{\dagger}Ug_{2})$. Note that $\rho_{\widetilde{\mu}}(g)M=M\rho_{\widetilde{\mu}}(g)=M$ for every $g\in G$ and hence $AM=MA=M$. Suppose $\rho_{\widetilde{\mu}}$ decomposes into irreducible representations $(\rho_{\eta},V_{\eta})$ of $G$ by $\rho_{\widetilde{\mu}}=\bigoplus_{\eta\in\Gamma}m_{\eta}\rho_{\eta}$. Then $M$ is a block diagonal matrix with blocks corresponding to these $\rho_{\eta}$. By Proposition 4, one of the $\rho_{\eta}$’s is the trivial representation and its multiplicity is one. For every other $\eta$, since $(\rho_{\eta},1)_{G}=0$, we have $A|_{V_{\eta}}=0_{V_{\eta}}$. Therefore $M|_{V_{\eta}}=0_{V_{\eta}}$. Hence the matrix coefficient functions in the block corresponding to $\rho_{\eta}$ becomes [math] as well after $G\times G$ averaging. Note that the trivial representation in $\rho_{\widetilde{\mu}}$ is of dimension 1 and of multiplicity 1, therefore besides the trivial constant function only one matrix coefficient is non-zero after $G\times G$ averaging. In fact, its $G\times G$ averaging is equal to the polynomial $f(U)=\frac{1}{|G|^{2}}\sum_{g_{1},g_{2}\in G}\chi_{\widetilde{\mu}}(g_{1}^{\dagger}Ug_{2})$. Note that $(\rho_{\widetilde{\mu}},1)_{\operatorname{U}(d)}=0$ implies that $\int_{\operatorname{U}(d)}f(U)\,\mathrm{d}U=0$. So far, we have shown the existence and uniqueness of the non-zero $G\times G$-invariant homogeneous polynomial $f\in\operatorname{Hom}(\operatorname{U}(d),t+1,t+1)$ such that $\int_{\operatorname{U}(d)}f(U)\,\mathrm{d}U=0$.

Now we take a zero $U_{0}$ of the polynomial $f(U)=\frac{1}{|G|^{2}}\sum_{g_{1},g_{2}\in G}\chi_{\widetilde{\mu}}(g_{1}^{\dagger}Ug_{2})$. Let $X$ be the orbit of $U_{0}$ under the action of $G\times G$ on $\operatorname{U}(d)$. For every non-trivial $\mu\neq\widetilde{\mu}$, we have shown that $G$ is a unitary $\rho_{\mu}$-design. By Corollary 8 and the additivity of unitary $\rho$-design, $X=GU_{0}G$ is a unitary $\rho_{\mu}$-design. For $\mu=\widetilde{\mu}$, since $U_{0}$ is a zero of $f$, we get $\operatorname{Tr}M(U_{0})=0$. Combined with the argument in the last paragraph, $M(U_{0})$ is indeed the zero matrix. Hence $X$ is a unitary $\rho_{\widetilde{\mu}}$-design.

Finally by Theorem 6, we conclude that $X$ is a unitary $(t+1)$-design. ∎

Proof of Theorem 2.

Without loss of generality, let us assume that $t_{1}\leq t_{2}$. Since $X_{1}=GU_{1}G$ is not a unitary $(t_{1}+1)$-design, there exists a $G\times G$-invariant homogeneous polynomial $h\in\operatorname{Hom}(\operatorname{U}(d),t_{1}+1,t_{1}+1)$ such that $h(U_{1})\neq 0$ and $\int_{\operatorname{U}(d)}h(U)\,\mathrm{d}U=0$. Now let us consider the $G\times G$-invariant homogeneous polynomial $h\overline{h}\in\operatorname{Hom}(\operatorname{U}(d),2t_{1}+2,2t_{1}+2)$. Note that $c:=\int_{\operatorname{U}(d)}(h\overline{h})(U)\,\mathrm{d}U>0$. Let $f:=h\overline{h}-c\in\operatorname{Hom}(\operatorname{U}(d),2t_{1}+2,2t_{1}+2)$, then $\int_{\operatorname{U}(d)}f(U)\,\mathrm{d}U=0$. Suppose $X_{2}$ is a unitary $(2t_{1}+2)$-design, then we must have $f(U_{2})=0$. Note that $2t_{1}+2>t_{1}+1$, so $h(U_{2})=0$. Therefore $f(U_{2})=h(U_{2})\overline{h}(U_{2})-c=-c<0$, contradiction. Hence $t_{2}\leq 2t_{1}+1$. ∎

IV Examples of unitary $t$-groups $G$ in $\operatorname{U}(d)$ satisfying the conditions of Theorem 1

The followings are some examples of ${G\subset\operatorname{U}(d)}$ that satisfy the conditions in Theorem 1. Here, we basically use the notation of An Atlas of Finite Groups MR827219 . Also, see Guralnick-Tiep MR2123127 and BNRT 1810.02507 .

For ${t=3,}$ (We assume ${d\geq 3.}$)

1. (a)

${d=4,G=\operatorname{Sp}(4,3),}$ 2. (b)

${d=6,G=6_{1}.\operatorname{U}_{4}(3),}$ 3. (c)

${d=12,G=6\operatorname{Suz}.}$ 2. 2.

For ${t=2,}$ (We assume ${d\geq 3.}$)

1. (a)

${d=3,G=\operatorname{SL}(3,2)=\operatorname{PSL}(2,7),}$ 2. (b)

${d=10,G=\operatorname{M}_{22},}$ 3. (c)

${d=28,G=\operatorname{Rd},}$ 4. (d)

${d=(3^{m}\pm 1)/2,G=\operatorname{PSp}(2m,3),\operatorname{Sp}(2m,3)}$. (see (1810.02507, , Section 4) for the details of Weil representations in this case.)

The above list might exhaust all such examples, although we will not try to give a rigorous proof of this claim.

V Computation

V.1 The unitary representation of $\operatorname{SL}(3,2)$ and $\operatorname{Sp}(4,3)$

We aim to construct some unitary $(t+1)$-designs based on certain unitary $t$-groups. This urges us to find the unitary representations of these groups first.

We adopt the notation $E(n)$ being the $n$-th root of unity from the mathematical software GAP GAP4 . The following two constructions are taken from (MR0059914, , Equation 10.1 and Equation 10.5).

Example 9**.**

Let $a:=-(E(7)^{4}+E(7)^{2}+E(7))$. Let $\mathcal{M}$ be the matrix group generated by the following three matrices.

[TABLE]

Then $G=\mathcal{M}^{(1)}$, the commutator subgroup of $\mathcal{M}$, is isomorphic to $\operatorname{SL}(3,2)$ and is embedded in $\operatorname{U}(3)$.

Example 10**.**

Let $\omega:=E(3)$. Let $\mathcal{M}$ be the matrix group generated by the following four matrices.

[TABLE]

Then $G=\mathcal{M}^{(1)}$, the commutator subgroup of $\mathcal{M}$, is isomorphic to $\operatorname{Sp}(4,3)$ and is embedded in $\operatorname{U}(4)$.

V.2 The $G\times G$-invariant polynomial

The construction of the $G\times G$-invariant polynomial $f$ in $\operatorname{Hom}(\operatorname{U}(d),t+1,t+1)$ is based on the irreducible characters of $\operatorname{U}(d)$.

Suppose $\chi_{\mu}$ is the character of an irreducible representation $(\rho_{\mu},V_{\mu})$ of the unitary group $\operatorname{U}(d)$. It naturally induces a $G\times G$-invariant function on $\operatorname{U}(d)$, namely

[TABLE]

A closed form of $\chi_{\mu}(\Lambda)$ can be expressed as a symmetric polynomial with respect to the spectrum of the unitary matrix $\Lambda$. Note that if $\mu=-\mu$, then $\chi_{\mu}$, thus $f$, is a real function.

Theorem 11** ((MR2529619, , Theorem 8) or (MR2062813, , Theorem 38.2 and Proposition 38.2)).**

Let $V_{\mu}$ be the irreducible representation of $\operatorname{U}(d)$ indexed by non-increasing integer sequence $\mu$.

If $\mu_{d}=0$, then the character of $V_{\mu}$ is

[TABLE]

where $s_{\mu}$ is the Schur polynomial, and $\{\lambda_{1},\ldots,\lambda_{d}\}$ are the eigenvalues of $\Lambda$.

If $\mu_{d}\neq 0$, then the character of $V_{\mu}$ is

[TABLE]

where $\mu^{\prime}=(\mu_{1}-\mu_{d},\ldots,\mu_{d-1}-\mu_{d},0)$.

For numerical computation, it takes considerable time to find the eigenvalues of a matrix and meantime it loses accuracy. Therefore we prefer to express $\chi_{\mu}$ by $\operatorname{Tr}(\Lambda^{k})$ and $\overline{\operatorname{Tr}({\Lambda}^{k})}$ where $0\leq k\leq d$. This can be done by Newton–Girard formulae (MR1740388, , §10.12, pp. 278-279).

Example 12**.**

By Theorem 11, we have

[TABLE]

Note that $\frac{1}{\lambda_{i}}=\overline{\lambda_{i}}$ and $\operatorname{Tr}(\Lambda^{k})=\sum_{i}{\lambda_{i}^{k}}$. We can simplify the above expression by Newton-Girard formulae of symmetric polynomials.

[TABLE]

Example 13**.**

[TABLE]

V.3 The approximation algorithm

Now our goal is reduced to the following problem.

Problem 14**.**

Given a continuous real function $f$ defined on a connected Lie group, find a zero of this function (numerically).

In particular, the function $f$ is a non-trivial $G\times G$-invariant polynomial on a unitary group $\operatorname{U}(d)$. The unitary group $\operatorname{U}(d)$ is connected, and the existence of zero is guaranteed because the integration of $f$ on $\operatorname{U}(d)$ is [math].

Suppose $f(L)<0$ and $f(R)>0$ where $L,R$ are two matrices representing the elements of the Lie group. By intermediate value theorem, there exists at least one matrix $Z$ on a path connecting $L$ and $R$ such that $f(Z)=0$. There are infinitely such paths and we will choose some special paths in the following.

It is natural to use bisection method or false position method to approximate the zero in arbitrary precision. The trouble here is that the function is defined on a manifold rather than the Euclidean space. For Lie groups, there is a canonical atlas given by the exponential map from the Lie algebra to the Lie group. We take advantage of this property to define the mid-point and the false position. The mid-point of $L$ and $R$ is defined to be $\exp\left((\log L+\log R)/2\right)$, and the false position between $L$ and $R$ is defined to be $\exp\left(\frac{f(R)\log L-f(L)\log R}{f(R)-f(L)}\right)$. The false position method usually converges faster than the bisection method. Nevertheless we use the bisection method when $L$ and $R$ are far away for the sake of robustness. One may consider other iterative methods to speed up the convergence. We did not use them because evaluation of the function is the heavy part of the computation. The initial value of $L$ and $R$ are obtained by taking unitary matrices randomly until both of them are found.

We are ready to construct the unitary designs, but let us put further constraint on the solution for the moment.

Problem 15**.**

Find a zero $U_{0}$ with good property, namely the size of the orbit $GU_{0}G$ is as small as possible.

Suppose $GU_{0}G$ is an orbit whose size is smaller than $|G|^{2}$, then there must exist $g_{1},g_{2},g_{3},g_{4}\in G$, such that $g_{1}^{\dagger}U_{0}g_{2}=g_{3}^{\dagger}U_{0}g_{4}$. Therefore $U_{0}^{\dagger}g_{i}U_{0}=g_{j}$ where $g_{i}=g_{3}g_{1}^{\dagger}$ and $g_{j}=g_{4}g_{2}^{\dagger}$ are also elements of $G$. This implies that $g_{i}$ and $g_{j}$ have the same spectrum. If $g_{i}$ has distinct eigenvalues, then $U_{0}$ is on a submanifold isomorphic to $\operatorname{U}(1)\times\operatorname{U}(1)\times\cdots\times\operatorname{U}(1)$. If the eigenvalues of $g_{i}$ are not simple, then $U_{0}$ is on a submanifold isomorphic to $\operatorname{U}(m_{1})\times\operatorname{U}(m_{2})\times\cdots\times\operatorname{U}(m_{k})$, where $m_{1},m_{2},\ldots,m_{k}$ are the multiplicities of the eigenvalues. Note that there is no guarantee that a zero exists on the submanifold.

Though it does not solve 15 completely, we have the clue to find them.

V.4 Solutions

For $G\cong\operatorname{SL}(3,2)\hookrightarrow\operatorname{U}(3)$, we find a zero on the submanifold $\operatorname{U}(1)\times\operatorname{U}(1)\times\operatorname{U}(1)$, namely the diagonal unitary matrices. The size of the orbit is at most $|G|^{2}/4=7056$.

Example 16**.**

Let $G\cong\operatorname{SL}(3,2)$ be the matrix group in Example 9, and let $f$ be the $G\times G$-invariant polynomial induced by the irreducible character $\chi_{(3,0,-3)}$ in Equation 5. Then $U_{0}=\operatorname{diag}(u_{11},u_{22},u_{33})$ is a zero of $f$, where $u_{11}=1$, $u_{22}\approx 0.6480674529649858-0.7615829412529393i$ and $u_{33}\approx-0.3307476956662597-0.9437192176762438i$. The error bound in Algorithm 1 is $\epsilon=10^{-15}$. Hence the orbit $GU_{0}G$ is a unitary $3$-design on $\operatorname{U}(3)$. The size of this orbit is $7056$.

Moreover, we can characterize all the diagonal unitary matrices in $\operatorname{U}(3)$ which make $GU_{0}G$ a unitary $3$-design. Let $u,v,t$ be real numbers and let $U_{0}=\operatorname{diag}\left(e^{it},e^{i(t+\frac{u+v}{2})},e^{i(t+\frac{u-v}{2})}\right)$. Then $GU_{0}G$ is a unitary $3$-design if and only if $u$ and $v$ satisfy

[TABLE]

The solution of this equation is shown in Figure 1.

For $G\cong\operatorname{Sp}(4,3)\hookrightarrow\operatorname{U}(4)$, we find a zero on the submanifold $\operatorname{U}(2)\times\operatorname{U}(2)$. The size of the orbit is at most $|G|^{2}/6=447897600$.

Example 17**.**

Let $G\cong\operatorname{Sp}(4,3)$ be the matrix group in Example 10, and let $f$ be the $G\times G$-invariant polynomial induced by the irreducible character $\chi_{(4,0,0,-4)}$ in Example 13. Then $U_{0}=\begin{bmatrix}A&0\\ 0&B\end{bmatrix}$ is a zero of $f$, where $A\approx\begin{bmatrix}-0.106632-0.973877i&0.0621677\,-0.190601i\\ 0.197341\,+0.0353545i&-0.807683-0.554486i\end{bmatrix}$ and $B\approx\begin{bmatrix}-0.596879-0.434093i&-0.562033-0.373388i\\ 0.372766\,-0.562445i&-0.381284+0.631921i\end{bmatrix}$. The error bound in Algorithm 1 is $\epsilon=10^{-6}$. Hence the orbit $GU_{0}G$ is a unitary $4$-design on $\operatorname{U}(4)$. The size of this orbit is $447897600$.

Remark 18**.**

The existence of exact $t$-designs are guaranteed by Theorem 1, which is different from finding approximate unitary $t$-designs. Algorithm 1 can approximate such a unitary design with arbitrary precision if one has enough time and computational resources. The time complexity of evaluating Equation 4 is $O(tp(t)d^{3}|G|^{2})$ where $p(n)\sim\frac{1}{4n\sqrt{3}}e^{\pi\sqrt{2n/3}}$, the partition function, is equal to the number of partitions of $n$. The error $\epsilon$ is ideally halved after each iteration. So it takes about $\log_{2}10\approx 3.3$ iterations to get one more significant digit. For Example 17, our program (written in Mathematica) ran on a PC equipped with Core i7-6700 CPU and 8GB RAM, and it took about half a day for each iteration.

VI Discussion

It would be interesting to classify those unitary $t$-groups $G\subset U(d)$ that satisfy $(\chi^{t+1},\chi^{t+1})_{G}=(\chi^{t+1},\chi^{t+1})_{U(d)}+1=(t+1)!+1$, which is the condition of Theorem 1. This should be certainly possible for $t\geq 2$ as such $G$ are among those already classified. The problem would be interesting for $t=1$ as well. We expect the existence of many such examples of unitary 2-designs by our method mentioned in this paper. Such classification may be obtained by extending the method in Guralnick-Tiep MR2123127 , although actually doing so would not be trivial at all. This would lead to explicit constructions of many families of explicit unitary $2$-designs. We believe this is an independently interesting open problem from the viewpoint of finite group theory.

Concerning Examples 16 and 17, it would be interesting to find what are the smallest sizes of unitary 3-designs, respectively 4-designs, that can be obtained by our method. This may be done by discussing the possible submanifolds which contain the orbit. If the function can achieve zero on a submanifold, we can still apply Algorithm 1. On the other hand it is not easy to show the non-existence of zeros on a submanifold.

Acknowledgment

The authors thank TGMRC (Three Gorges Mathematical Research Center) of China Three Gorges University in Yichang, Hubei, China, for supporting the visits of the authors to work on this research project in October 2018. The work is supported in part by NSFC Grant 11671258. The work of M. N. is partly supported by KAKENHI from JSPS Grant-in-Aid for Scientific Research (KAKENHI Grant No. 17K05554). Y. Z. is supported by NSFC Grant No. 11801353 and China Postdoctoral Science Foundation No. 2018M632078.