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Generalized group designs: constructing novel unitary 2-, 3- and 4-designs

Ágoston Kaposi, Zoltán Kolarovszki, Adrián Solymos, Zoltán Zimborás

TL;DR

The paper addresses the limitation that exact group-based unitary $t$-designs are generally limited to $t\le 3$, proposing generalized group designs constructed from products of finite subgroups to surpass the $4$-design barrier. It develops a representation-theoretic framework and proves a theorem enabling higher-degree designs from suitably chosen subgroups, and provides explicit $2$-designs in arbitrary dimension via the monomial reflection group $G(3,1,d)$. Additionally, it establishes a practical route to convert orthogonal $t$-designs into unitary designs through consecutive twirls with rotated orthogonal groups, supported by concrete examples and GAP data. These constructions broaden the toolkit for Haar-averaging in quantum information tasks such as tomography, benchmarking, and shadow estimation, across general dimensions.

Abstract

Unitary designs are essential tools in several quantum information protocols. Similarly to other design concepts, unitary designs are mainly used to facilitate averaging over a relevant space, in this case, the unitary group $\mathrm{U}(d)$. While it is known that exact unitary $t$-designs exist for any degree $t$ and dimension $d$, the most appealing type of designs, group designs (in which the elements of the design form a group), can provide at most $3$-designs. Moreover, even group $2$-designs can exist only in limited dimensions. In this paper, we present novel construction methods for creating exact generalized group designs based on the representation theory of the unitary group and its finite subgroups that overcome the $4$-design-barrier of unitary group designs. Furthermore, a construction is presented for creating generalized group $2$-designs in arbitrary dimensions.

Generalized group designs: constructing novel unitary 2-, 3- and 4-designs

TL;DR

The paper addresses the limitation that exact group-based unitary -designs are generally limited to , proposing generalized group designs constructed from products of finite subgroups to surpass the -design barrier. It develops a representation-theoretic framework and proves a theorem enabling higher-degree designs from suitably chosen subgroups, and provides explicit -designs in arbitrary dimension via the monomial reflection group . Additionally, it establishes a practical route to convert orthogonal -designs into unitary designs through consecutive twirls with rotated orthogonal groups, supported by concrete examples and GAP data. These constructions broaden the toolkit for Haar-averaging in quantum information tasks such as tomography, benchmarking, and shadow estimation, across general dimensions.

Abstract

Unitary designs are essential tools in several quantum information protocols. Similarly to other design concepts, unitary designs are mainly used to facilitate averaging over a relevant space, in this case, the unitary group . While it is known that exact unitary -designs exist for any degree and dimension , the most appealing type of designs, group designs (in which the elements of the design form a group), can provide at most -designs. Moreover, even group -designs can exist only in limited dimensions. In this paper, we present novel construction methods for creating exact generalized group designs based on the representation theory of the unitary group and its finite subgroups that overcome the -design-barrier of unitary group designs. Furthermore, a construction is presented for creating generalized group -designs in arbitrary dimensions.
Paper Structure (14 sections, 18 theorems, 109 equations, 1 table)

This paper contains 14 sections, 18 theorems, 109 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constructing higher-degree generalized group designs
  4. Constructing 2-designs in arbitrary dimension
  5. Monomial reflection group
  6. Constructing generalized unitary 2-design
  7. Constructing unitary designs from orthogonal designs
  8. Orthogonal designs
  9. Representation theory of the orthogonal group
  10. Construction
  11. Summary and Outlook
  12. Rotation of the d-dimensional irreducible representation of the monomial reflection group G(3,1,d)
  13. Recap of representation theory
  14. Recap of character theory

Key Result

Proposition 1

A finite set $\mathcal{V}\subset \mathrm{U}(d)$ and a weight function $w: \mathcal{V} \to [0, 1]$ forms a unitary $t$-design if and only if the following equation is true for all linear transformations $M$ on $(\mathbb{C}^d)^{\otimes t}$: where we used the notation as before, and $\Tr_{\mathcal{H}_{\gamma}}$ is the partial trace over $\mathcal{H}_{\gamma}$ of operators supported on the subspace $

Theorems & Definitions (45)

  • Definition 1: Unitary $t$-design
  • Definition 2: Generalized group $t$-design
  • Remark 1
  • Proposition 1
  • Theorem 1
  • proof
  • Example 1: A $6$-dimensional $4$-design
  • Example 2: A $12$-dimensional $4$-design
  • Example 3: A $10$-dimensional $3$-design
  • Example 4: A $13$-dimensional $3$-design
  • ...and 35 more