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

Ágoston Kaposi; Zoltán Kolarovszki; Adrian Solymos; Zoltán Zimborás

doi:10.22331/q-2026-02-23-2008

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

Ágoston Kaposi^1,2, Zoltán Kolarovszki^1,2, Adrian Solymos^1,3, and Zoltán Zimborás^1,2,4,5

¹Quantum Computing and Quantum Information Research Group, HUN-REN Wigner Research Centre for Physics, Konkoly–Thege Miklós út 29-33, Budapest, H-1525, Hungary
²Department of Programming Languages and Compilers, Eötvös Loránd University, Pázmány Péter sétány 1/C, Budapest, H-1117, Hungary
³Department of Physics of Complex Systems, Eötvös Loránd University, Pázmány Péter sétány 1/A, Budapest, H-1117, Hungary
⁴Algorithmiq Ltd, Kanavakatu 3C, Helsinki, 00160, Finland
⁵University of Helsinki, Yliopistonkatu 4, Helsinki, 00100, Finland

Published:	2026-02-23, volume 10, page 2008
Editor:	Álvaro Alhambra
Eprint:	arXiv:2405.00919v4
Doi:	https://doi.org/10.22331/q-2026-02-23-2008
Citation:	Quantum 10, 2008 (2026).

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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.

Featured image: Simple illustration of the general procedure for obtaining generalized unitary group $t$-designs in the case of $t=2$. Starting with an arbitrary operator $M$, we first perform a twirling operation with a finite group $H$, where $H$ acts irreducibly on the symmetric subspace. Secondly, we apply twirling by the finite group $K$, which acts irreducibly on the antisymmetric subspace. Applying the two operations together yields a unitary $2$-design. The generalization of this procedure can be used to obtain higher-degree unitary designs.

Presentation Generalised group designs overcoming the 3 design barrier and constructing novel 2 At QIP2024

Popular summary

In this paper, we introduce a new framework for constructing exact unitary designs, which are finite sets of unitaries used to mimic random quantum operations. Although these designs are essential for testing quantum hardware, a fundamental constraint known as the $4$-design barrier for group designs limits the availability of easy-to-construct designs. (The $4$-design barrier states that in dimensions greater than $2$, a single group cannot form an exact $4$-design.) To overcome this, we introduce generalized group designs, defined as the set formed by the products of multiple finite subgroups of the unitary group. This new definition allows for a more flexible structure.

The core of our approach is a “recipe” based on representation theory that determines how to combine different finite groups so that their collective average matches the average over the entire unitary group, which has infinite elements. This technique focuses on building designs that are mathematically exact. By applying this method, we discovered specific new examples of exact designs, including a $4$-design in $6$ dimensions and another in $12$ dimensions, as well as various exact $3$-designs in dimensions $10$, $13$, and $18$.

Beyond these high-degree designs, we provide a significant advancement for universal applications by presenting the first non-inductive construction for exact $2$-design in any arbitrary dimension. This is achieved by taking the product of a certain monomial reflection group and a “rotated” version of itself. Furthermore, we give a procedure that yields exact unitary $2$- and $3$-designs from their orthogonal counterparts.

► BibTeX data

@article{Kaposi2026generalizedgroup,
  doi = {10.22331/q-2026-02-23-2008},
  url = {https://doi.org/10.22331/q-2026-02-23-2008},
  title = {Generalized group designs: constructing novel unitary 2-, 3- and 4-designs},
  author = {Kaposi, {\'{A}}goston and Kolarovszki, Zolt{\'{a}}n and Solymos, Adrian and Zimbor{\'{a}}s, Zolt{\'{a}}n},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2008},
  month = feb,
  year = {2026}
}

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Cited by

[1] Ryotaro Suzuki, Hosho Katsura, Yosuke Mitsuhashi, Tomohiro Soejima, Jens Eisert, and Nobuyuki Yoshioka, "More global randomness from less-random local gates", Physical Review A 114 2, 022424 (2026).

[2] Ryotaro Suzuki, Hosho Katsura, Yosuke Mitsuhashi, Tomohiro Soejima, Jens Eisert, and Nobuyuki Yoshioka, "More global randomness from less random local gates", arXiv:2410.24127, (2024).

[3] Juntai Zhou, Stefano Chessa, Eric Chitambar, and Felix Leditzky, "On the distinguishability of geometrically uniform quantum states", Journal of Physics A Mathematical General 58 41, 415303 (2025).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-17 16:00:14) and SAO/NASA ADS (last updated successfully 2026-08-17 16:00:15). The list may be incomplete as not all publishers provide suitable and complete citation data.

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