Explicit construction of exact unitary designs Eiichi Bannai ∗ Yoshifumi Nakata † Takayuki Okuda ‡ Da Zhao § September 24, 2020 Abstract The purpose of this paper is to give explicit constructions of unitary t- designs in the unitary group U(d) for all t and d. It seems that the explicit constructions were so far known only for very special cases. Here explicit construction means that the entries of the unitary matrices are given by the values of elementary functions at the root of some given polynomials. We will discuss what are the best such unitary 4-designs in U(4) obtained by these methods. Indeed we give an inductive construction of designs on compact groups by using Gelfand pairs (G, K). Note that (U(n), U(m) × U(n − m)) is a Gelfand pair. By using the zonal spherical functions for (G, K), we can construct designs on G from designs on K. We remark that our proofs use the representation theory of compact groups crucially. We also remark that this method can be applied to the orthogonal groups O(d), and thus provides another explicit construction d 1 of spherical t-designs on the d dimensional sphere S − by the induction on d. 1 Introduction The aim of design theory is to approximate a space M by a good finite subset X. arXiv:2009.11170v1 [math.CO] 23 Sep 2020 There have been numerous studies on spherical designs[17] and combinatorial designs[14]. The sphere is a canonical continuous space while the combinatorial V design is in the discrete space M = k of k-subsets of V . The concept of ∗Professor Emeritus of Kyushu University, Fukuoka, Japan. Postal Address: Asagaya- minami 3-2-33, Suginami-ku, Tokyo 166-0004, Japan. [email protected] †Photon Science Center, Graduate School of Engineering, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan. JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan. [email protected] ‡Department of Mathematics, Hiroshima University, 1-3-1 Kagamiyama, Higashihiroshima, 739-8526, Japan. [email protected] §School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang District, Shanghai 200240, China. [email protected] 1 combinatorial design was generalized to designs on Q-polynomial association schemes[16]. Other continuous spaces such as projective space, Grassmannian space[1, 2, 38] have been considered as well. In this paper we focus on the construction of unitary designs, which is designs on the unitary group. There is an increasing demand for unitary designs in quantum information science that aims to realize information processing based on quantum mechan- ics, where protocols are described by unitary transformations. Physically im- plementing unitary transformations is the key to realize information processing in quantum information science. A unitary transformation chosen uniformly at random from the whole unitary group is of particular importance since it is used in many protocols, such as benchmarking quantum devices [18, 30], characteriz- ing quantum systems [31, 36, 40], improving computational complexity [11], and transmitting information [27]. However, the resources available in experiments are quite limited and so, it is in general hard to experimentally implement the uniformly random unitary transformation. Thus, mimicking the whole unitary group by unitary designs is attracting much attention. While numerous constructions of unitary 2-designs are known [37, 13, 28, 21, 25, 34], less is known about unitary t-designs for t 3 [10, 23, 24, 26, 33]. Most of them are approximate ones. Hence, finding explicit≥ constructions of exact unitary t-designs on U(d) is highly desired, which will lead to more accurate realizations of quantum information protocols. Some finite subgroups of the unitary group are unitary designs of small strength[7, 22, 39]. The Clifford groups on U(2n) for any n N are exact unitary 3-designs[44, 48, 49]. An exact unitary 4-design on U(4) is∈ constructed in [6]. The main purpose of this paper is to provide inductive constructions of exact unitary t-design on U(d) for arbitrary strength t and dimension d. The method can be applied to orthogonal groups as well. As a by-product we have constructions for complex spherical designs and real spherical designs. The existence of spherical designs were proved by Seymour-Zaslavsky[41]. Bondarenko-Radchenko-Viazovsk[9] showed that there exist spherical t-designs d d on S of size at least cdt for some fixed constant cd with t tending to infin- ity. This lower bound is asymptotically best possible order of magnitude. The asymptotically best lower bound for fixed strength t with d tending to infinity is yet unknown. The explicit construction of spherical designs is relatively in- volved. Rabau-Bajnok[35] and Wagner[43] constructed spherical t-designs from interval t-designs with Gegenbauer weight. Cui-Xia-Xiang[15] showed the ex- istence of spherical designs over Q(√p) where p is a prime number. Later Xiang[47] obtained explicit spherical designs. The readers can find surveys on spherical designs in [3, 4]. Let us now define unitary designs. There are several equivalent definitions of unitary t-designs. Definition 1.1 ([39, pp. 14-15]). Let X be a finite subset X of U(d). The following are equivalent. 1. X is a unitary t-design. 2 2. 1 U ⊗t (U †)⊗t = U ⊗t (U †)⊗t dU. |X| U∈X ⊗ U(d) ⊗ 3. 1 P f(U)= f(UR) dU for every f Hom(U(d),t,t), the space |X| U∈X U(d) ∈ of polynomials of homogeneous degree t in entries of U and of homogeneous P R † degree t in the entries of U . Representation theory is used extensively in our construction. There is an- other equivalent definition of unitary t-designs by irreducible representations of U(d), which is quite useful for our purpose. The irreducible representations of unitary group are characterized by the highest weight. Theorem 1.1 ([12, Theorem 25.5]). The irreducible representations of unitary group U(n) are indexed by non-increasing integer sequence λ = (λ1, λ2,...,λn) of length n. + − We denote by λ the sum of positive terms in λ1, λ2,...,λn and by λ the absolute value of sum of negative terms in λ1, λ2,...,λn. And we define λ = λ+ λ−. The following two collections of irreducible representations are used| | to characterize− unitary design. s,t := λ : λ λ λ , λ+ s, λ− t . n { 1 ≥ 2 ≥···≥ n ≤ ≤ } t := λ : λ λ λ , λ+ = λ− t t,t. n { 1 ≥ 2 ≥···≥ n ≤ }⊂ n Theorem 1.2 ([6, Theorem 6]). A finite subset X U(d) is a unitary t-design if and only if ⊂ 1 ρ (U)= ρ (U) dU. (1) X λ λ U X U(d) | | X∈ Z for every irreducible representation ρ where λ t . λ ∈ n t t,t Due to the inductive nature of our construction, we will replace n by n . We will also adopt the relaxation from set to multi-set for technical reasons. Definition 1.2. Let X be a finite multi-set on U(d). The following are equiv- alent. 1. X is a strong unitary t-design on U(d). 1 ⊗r † ⊗s ⊗r † ⊗s 2. |X| U∈X U (U ) = U(d) U (U ) dU for every integers 0 r, s t. ⊗ ⊗ ≤ P≤ R 3. 1 f(U)= f(U) dU for every f Hom(U(d),r,s), the space |X| U∈X U(d) ∈ of polynomials of homogeneous degree r in entries of U and of homoge- P R neous degree s in the entries of U †, for every integers 0 r, s t. ≤ ≤ 4. 1 ρ (U)= ρ (U) dU for every λ t,t. |X| U∈X λ U(d) λ ∈ n This paperP shows the followingR theorem. 3 Theorem. Strong unitary t-designs on U(n) can be constructed from strong unitary t-designs on U(m) and strong unitary t-designs on U(n m) using the zeroes of zonal spherical functions of the complex Grassmannian− . Gm,n The paper is organized as follows. In Sections 2 to 4 the notation for multi- sets, representations and Haar measure are set up. In Section 5 we introduce Gelfand pairs and zonal spherical functions. The central object in this paper, designs on compact groups, are given in Section 6. We explain our inductive construction in Section 7. In Section 9 we compare different constructions of unitary 4-designs on U(4). We briefly mention how our method gives spherical designs as a by-product in Section 10. And finally in Section 11 we discuss the relation between designs in this paper and those in classic design theory. The Appendix include the zonal polynomials together with their zeroes. 2 finite multi-sets on groups Let G be a group. We use the terminology of “non-empty finite multi-sets X on G” in the following sense: Let N Z . Each S -orbit in GN is ∈ ≥1 N said to be an N-point multi-set on G, where GN denotes the direct product N of N-copies of G and SN the symmetric group of order N acting on G as permutations of coordinates. For the simplicity, the S -orbit of (x ,...,x ) N 1 N ∈ GN will be denoted by x ,...,x . Note that x , x ,...,x = { 1 N }mult { 1 2 N }mult 6 x2,...,xN mult even if x1 = x2. { We also} use the following notation: For each N-point multi-set X = x1,...,xN mult on G, we put X := N • and { } | | N ρ(x) := ρ(xi) x X i=1 X∈ X for each map ρ : G W where W is an Abelian group.