Generalized group designs: constructing novel unitary 2-, 3- and 4-designs
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AbstractUnitary 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 summaryIn 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} }► References [1] Christoph Dankert. ``Efficient Simulation of Random Quantum States and Operators'' (2005). arXiv:quant-ph/0512217. arXiv:quant-ph/0512217 [2] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. ``Exact and approximate unitary 2-designs and their application to fidelity estimation''. Phys. Rev. A 80 (2009). https://doi.org/10.1103/physreva.80.012304 [3] A J Scott. ``Optimizing quantum process tomography with unitary 2-designs''. J. Phys. A 41, 055308 (2008). https://doi.org/10.1088/1751-8113/41/5/055308 [4] Aidan Roy and A. J. Scott. ``Unitary designs and codes''. Des. Codes Cryptogr. 53, 13–31 (2009). https://doi.org/10.1007/s10623-009-9290-2 [5] D. Gross, F. Krahmer, and R. Kueng. ``A Partial Derandomization of PhaseLift Using Spherical Designs''. J. Fourier Anal. Appl. 21, 229–266 (2014). https://doi.org/10.1007/s00041-014-9361-2 [6] Joonwoo Bae, Beatrix C Hiesmayr, and Daniel McNulty. ``Linking entanglement detection and state tomography via quantum 2-designs''. New J. Phys. 21, 013012 (2019). https://doi.org/10.1088/1367-2630/aaf8cf [7] Daniel A Roberts and Beni Yoshida. ``Chaos and complexity by design''. J.
High Energy Phys. 2017, 1–64 (2017). https://doi.org/10.1007/JHEP04(2017)121 [8] Lorenzo Leone, Salvatore FE Oliviero, You Zhou, and Alioscia Hamma. ``Quantum chaos is quantum''. Quantum 5, 453 (2021). https://doi.org/10.22331/q-2021-05-04-453 [9] Joel J Wallman and Steven T Flammia. ``Randomized benchmarking with confidence''. New J. Phys. 16, 103032 (2014). https://doi.org/10.1088/1367-2630/16/10/103032 [10] Mahnaz Jafarzadeh, Ya-Dong Wu, Yuval R Sanders, and Barry C Sanders. ``Randomized benchmarking for qudit Clifford gates''. New J. Phys. 22, 063014 (2020). https://doi.org/10.1088/1367-2630/ab8ab1 [11] Hsin-Yuan Huang, Richard Kueng, and John Preskill. ``Predicting many properties of a quantum system from very few measurements''. Nat. Phys. 16, 1050–1057 (2020). https://doi.org/10.1038/s41567-020-0932-7 [12] Senrui Chen, Wenjun Yu, Pei Zeng, and Steven T Flammia. ``Robust shadow estimation''. PRX Quantum 2, 030348 (2021). https://doi.org/10.1103/PRXQuantum.2.030348 [13] Mirko Arienzo, Markus Heinrich, Ingo Roth, and Martin Kliesch. ``Closed-form analytic expressions for shadow estimation with brickwork circuits''. Quantum Inf. Comput. 23, 961–993 (2023). https://doi.org/10.26421/qic23.11-12-5 [14] Jonas Helsen and Michael Walter. ``Thrifty Shadow Estimation: Reusing Quantum Circuits and Bounding Tails''. Phys. Rev. Lett. 131 (2023). https://doi.org/10.1103/physrevlett.131.240602 [15] D. Gross, K. Audenaert, and J. Eisert. ``Evenly distributed unitaries: On the structure of unitary designs''. J. Math. Phys. 48, 052104 (2007). https://doi.org/10.1063/1.2716992 [16] Maxwell West, Diego García-Martín, N. L. Diaz, M. Cerezo, and Martin Larocca. ``No-go theorems for sublinear-depth group designs'' (2025). arXiv:2506.16005. arXiv:2506.16005 [17] Lorenzo Grevink, Jonas Haferkamp, Markus Heinrich, Jonas Helsen, Marcel Hinsche, Thomas Schuster, and Zoltán Zimborás. ``Will it glue? on short-depth designs beyond the unitary group'' (2025). arXiv:2506.23925. arXiv:2506.23925 [18] Hoi Fung Chau. ``Unconditionally secure key distribution in higher dimensions by depolarization''. IEEE Trans. Inf. Theory 51, 1451–1468 (2005). https://doi.org/10.1109/TIT.2005.844076 [19] Markus Heinrich, Martin Kliesch, and Ingo Roth. ``Randomized benchmarking with random quantum circuits'' (2023). arXiv:2212.06181. arXiv:2212.06181 [20] Zak Webb. ``The Clifford group forms a unitary 3-design''. Quantum Inf. Comput. 16, 1379–1400 (2015). https://doi.org/10.26421/QIC16.15-16-8 [21] Huangjun Zhu. ``Multiqubit Clifford groups are unitary 3-designs''. Phys. Rev. A 96, 062336 (2017). https://doi.org/10.1103/PhysRevA.96.062336 [22] Richard Kueng and David Gross. ``Qubit stabilizer states are complex projective 3-designs'' (2015). arXiv:1510.02767. arXiv:1510.02767 [23] Eiichi Bannai, Gabriel Navarro, Noelia Rizo, and Pham Huu Tiep. ``Unitary $t$-groups''. J. Math. Soc. Jpn. 72, 909 – 921 (2020). https://doi.org/10.2969/jmsj/82228222 [24] Matthew A Graydon, Joshua Skanes-Norman, and Joel J Wallman. ``Clifford groups are not always 2-designs'' (2021). arXiv:2108.04200. arXiv:2108.04200 [25] Fernando G. S. L. Brandão, Aram W. Harrow, and Michał Horodecki. ``Local Random Quantum Circuits are Approximate Polynomial-Designs''. Commun. Math. Phys. 346, 397–434 (2016). https://doi.org/10.1007/s00220-016-2706-8 [26] Michał Oszmaniec, Adam Sawicki, and Michał Horodecki. ``Epsilon-Nets, Unitary Designs, and Random Quantum Circuits''. IEEE Trans. Inf. Theory 68, 989–1015 (2022). https://doi.org/10.1109/TIT.2021.3128110 [27] Jonas Haferkamp. ``Random quantum circuits are approximate unitary $t$-designs in depth $O \left(nt^{5+o (1)} \right) $''. Quantum 6, 795 (2022). https://doi.org/10.22331/q-2022-09-08-795 [28] P.D Seymour and Thomas Zaslavsky. ``Averaging sets: A generalization of mean values and spherical designs''. Adv. Math. 52, 213–240 (1984). https://doi.org/10.1016/0001-8708(84)90022-7 [29] Ingemar Bengtsson and Karol Życzkowski. ``Geometry of Quantum States: An Introduction to Quantum Entanglement''.
Cambridge University Press. Cambridge, UK (2017). 2 edition. https://doi.org/10.1017/9781139207010 [30] Eiichi Bannai, Yoshifumi Nakata, Takayuki Okuda, and Da Zhao. ``Explicit construction of exact unitary designs''. Adv. Math. 405, 108457 (2022). https://doi.org/10.1016/j.aim.2022.108457 [31] W. Fulton and J. Harris. ``Representation Theory: A First Course''. Graduate Texts in Mathematics.
Springer New York.
Springer New York, NY (1991). https://doi.org/10.1007/978-1-4612-0979-9 [32] Ágoston Kaposi, Zoltán Kolarovszki, Adrian Solymos, Tamás Kozsik, and Zoltán Zimborás. ``Constructing Generalized Unitary Group Designs''. Pages 233–245. Springer. (2023). https://doi.org/10.1007/978-3-031-36030-5_19 [33] ``GAP – Groups, Algorithms, and Programming, Version 4.14.0''. https://www.gap-system.org (2024). https://www.gap-system.org [34] David Amaro-Alcalá, Barry C Sanders, and Hubert de Guise. ``Randomised benchmarking for universal qudit gates''. New J. Phys. 26, 073052 (2024). https://doi.org/10.1088/1367-2630/ad6635 [35] Giulio Chiribella, Giacomo Mauro D’Ariano, and Martin Roetteler. ``Identification of a reversible quantum gate: assessing the resources''. New J. Phys. 15, 103019 (2013). https://doi.org/10.1088/1367-2630/15/10/103019 [36] Fernando GSL Brandao, Aram W Harrow, and Michał Horodecki. ``Local random quantum circuits are approximate polynomial-designs''. Comm. Math. Phys. 346, 397–434 (2016). https://doi.org/10.1007/s00220-016-2706-8 [37] Thomas Schuster, Jonas Haferkamp, and Hsin-Yuan Huang. ``Random unitaries in extremely low depth''. Science 389, 92–96 (2025). https://doi.org/10.1126/science.adv8590 [38] Adrian Solymos, Dávid Jakab, and Zoltán Zimborás. ``Extendibility of Brauer states'' (2024). arXiv:2411.04597. arXiv:2411.04597 [39] A. K. Hashagen, S. T. Flammia, D. Gross, and J. J. Wallman. ``Real Randomized Benchmarking''. Quantum 2, 85 (2018). https://doi.org/10.22331/q-2018-08-22-85 [40] Bruce C Berndt, Ronald J Evans, and Kenneth S Williams. ``Gauss and Jacobi sums''. Wiley. (1998). https://doi.org/10.2307/3619097 [41] Aleksandr A. Kirillov. ``Elements of the Theory of Representation''. Springer. (1976). https://doi.org/10.1007/978-3-642-66243-0 [42] Roe Goodman and Nolan R. Wallach. ``Symmetry, Representations, and Invariants''. Springer. (2009). https://doi.org/10.1007/978-0-387-79852-3 [43] Matthias Christandl. ``The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography''. PhD thesis. University of Cambridge. (2006). arXiv:quant-ph/0604183. arXiv:quant-ph/0604183 [44] K. Lux and H. Pahlings. ``Representations of Groups: A Computational Approach''. Cambridge Stud. Adv. Math.
Cambridge University Press. Cambridge, UK (2010). https://doi.org/10.1017/CBO9780511750915Cited byCould not fetch Crossref cited-by data during last attempt 2026-02-23 13:17:58: Could not fetch cited-by data for 10.22331/q-2026-02-23-2008 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-02-23 13:17:59: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractUnitary 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 summaryIn 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} }► References [1] Christoph Dankert. ``Efficient Simulation of Random Quantum States and Operators'' (2005). arXiv:quant-ph/0512217. arXiv:quant-ph/0512217 [2] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. ``Exact and approximate unitary 2-designs and their application to fidelity estimation''. Phys. Rev. A 80 (2009). https://doi.org/10.1103/physreva.80.012304 [3] A J Scott. ``Optimizing quantum process tomography with unitary 2-designs''. J. Phys. A 41, 055308 (2008). https://doi.org/10.1088/1751-8113/41/5/055308 [4] Aidan Roy and A. J. Scott. ``Unitary designs and codes''. Des. Codes Cryptogr. 53, 13–31 (2009). https://doi.org/10.1007/s10623-009-9290-2 [5] D. Gross, F. Krahmer, and R. Kueng. ``A Partial Derandomization of PhaseLift Using Spherical Designs''. J. Fourier Anal. Appl. 21, 229–266 (2014). https://doi.org/10.1007/s00041-014-9361-2 [6] Joonwoo Bae, Beatrix C Hiesmayr, and Daniel McNulty. ``Linking entanglement detection and state tomography via quantum 2-designs''. New J. Phys. 21, 013012 (2019). https://doi.org/10.1088/1367-2630/aaf8cf [7] Daniel A Roberts and Beni Yoshida. ``Chaos and complexity by design''. J.
High Energy Phys. 2017, 1–64 (2017). https://doi.org/10.1007/JHEP04(2017)121 [8] Lorenzo Leone, Salvatore FE Oliviero, You Zhou, and Alioscia Hamma. ``Quantum chaos is quantum''. Quantum 5, 453 (2021). https://doi.org/10.22331/q-2021-05-04-453 [9] Joel J Wallman and Steven T Flammia. ``Randomized benchmarking with confidence''. New J. Phys. 16, 103032 (2014). https://doi.org/10.1088/1367-2630/16/10/103032 [10] Mahnaz Jafarzadeh, Ya-Dong Wu, Yuval R Sanders, and Barry C Sanders. ``Randomized benchmarking for qudit Clifford gates''. New J. Phys. 22, 063014 (2020). https://doi.org/10.1088/1367-2630/ab8ab1 [11] Hsin-Yuan Huang, Richard Kueng, and John Preskill. ``Predicting many properties of a quantum system from very few measurements''. Nat. Phys. 16, 1050–1057 (2020). https://doi.org/10.1038/s41567-020-0932-7 [12] Senrui Chen, Wenjun Yu, Pei Zeng, and Steven T Flammia. ``Robust shadow estimation''. PRX Quantum 2, 030348 (2021). https://doi.org/10.1103/PRXQuantum.2.030348 [13] Mirko Arienzo, Markus Heinrich, Ingo Roth, and Martin Kliesch. ``Closed-form analytic expressions for shadow estimation with brickwork circuits''. Quantum Inf. Comput. 23, 961–993 (2023). https://doi.org/10.26421/qic23.11-12-5 [14] Jonas Helsen and Michael Walter. ``Thrifty Shadow Estimation: Reusing Quantum Circuits and Bounding Tails''. Phys. Rev. Lett. 131 (2023). https://doi.org/10.1103/physrevlett.131.240602 [15] D. Gross, K. Audenaert, and J. Eisert. ``Evenly distributed unitaries: On the structure of unitary designs''. J. Math. Phys. 48, 052104 (2007). https://doi.org/10.1063/1.2716992 [16] Maxwell West, Diego García-Martín, N. L. Diaz, M. Cerezo, and Martin Larocca. ``No-go theorems for sublinear-depth group designs'' (2025). arXiv:2506.16005. arXiv:2506.16005 [17] Lorenzo Grevink, Jonas Haferkamp, Markus Heinrich, Jonas Helsen, Marcel Hinsche, Thomas Schuster, and Zoltán Zimborás. ``Will it glue? on short-depth designs beyond the unitary group'' (2025). arXiv:2506.23925. arXiv:2506.23925 [18] Hoi Fung Chau. ``Unconditionally secure key distribution in higher dimensions by depolarization''. IEEE Trans. Inf. Theory 51, 1451–1468 (2005). https://doi.org/10.1109/TIT.2005.844076 [19] Markus Heinrich, Martin Kliesch, and Ingo Roth. ``Randomized benchmarking with random quantum circuits'' (2023). arXiv:2212.06181. arXiv:2212.06181 [20] Zak Webb. ``The Clifford group forms a unitary 3-design''. Quantum Inf. Comput. 16, 1379–1400 (2015). https://doi.org/10.26421/QIC16.15-16-8 [21] Huangjun Zhu. ``Multiqubit Clifford groups are unitary 3-designs''. Phys. Rev. A 96, 062336 (2017). https://doi.org/10.1103/PhysRevA.96.062336 [22] Richard Kueng and David Gross. ``Qubit stabilizer states are complex projective 3-designs'' (2015). arXiv:1510.02767. arXiv:1510.02767 [23] Eiichi Bannai, Gabriel Navarro, Noelia Rizo, and Pham Huu Tiep. ``Unitary $t$-groups''. J. Math. Soc. Jpn. 72, 909 – 921 (2020). https://doi.org/10.2969/jmsj/82228222 [24] Matthew A Graydon, Joshua Skanes-Norman, and Joel J Wallman. ``Clifford groups are not always 2-designs'' (2021). arXiv:2108.04200. arXiv:2108.04200 [25] Fernando G. S. L. Brandão, Aram W. Harrow, and Michał Horodecki. ``Local Random Quantum Circuits are Approximate Polynomial-Designs''. Commun. Math. Phys. 346, 397–434 (2016). https://doi.org/10.1007/s00220-016-2706-8 [26] Michał Oszmaniec, Adam Sawicki, and Michał Horodecki. ``Epsilon-Nets, Unitary Designs, and Random Quantum Circuits''. IEEE Trans. Inf. Theory 68, 989–1015 (2022). https://doi.org/10.1109/TIT.2021.3128110 [27] Jonas Haferkamp. ``Random quantum circuits are approximate unitary $t$-designs in depth $O \left(nt^{5+o (1)} \right) $''. Quantum 6, 795 (2022). https://doi.org/10.22331/q-2022-09-08-795 [28] P.D Seymour and Thomas Zaslavsky. ``Averaging sets: A generalization of mean values and spherical designs''. Adv. Math. 52, 213–240 (1984). https://doi.org/10.1016/0001-8708(84)90022-7 [29] Ingemar Bengtsson and Karol Życzkowski. ``Geometry of Quantum States: An Introduction to Quantum Entanglement''.
Cambridge University Press. Cambridge, UK (2017). 2 edition. https://doi.org/10.1017/9781139207010 [30] Eiichi Bannai, Yoshifumi Nakata, Takayuki Okuda, and Da Zhao. ``Explicit construction of exact unitary designs''. Adv. Math. 405, 108457 (2022). https://doi.org/10.1016/j.aim.2022.108457 [31] W. Fulton and J. Harris. ``Representation Theory: A First Course''. Graduate Texts in Mathematics.
Springer New York.
Springer New York, NY (1991). https://doi.org/10.1007/978-1-4612-0979-9 [32] Ágoston Kaposi, Zoltán Kolarovszki, Adrian Solymos, Tamás Kozsik, and Zoltán Zimborás. ``Constructing Generalized Unitary Group Designs''. Pages 233–245. Springer. (2023). https://doi.org/10.1007/978-3-031-36030-5_19 [33] ``GAP – Groups, Algorithms, and Programming, Version 4.14.0''. https://www.gap-system.org (2024). https://www.gap-system.org [34] David Amaro-Alcalá, Barry C Sanders, and Hubert de Guise. ``Randomised benchmarking for universal qudit gates''. New J. Phys. 26, 073052 (2024). https://doi.org/10.1088/1367-2630/ad6635 [35] Giulio Chiribella, Giacomo Mauro D’Ariano, and Martin Roetteler. ``Identification of a reversible quantum gate: assessing the resources''. New J. Phys. 15, 103019 (2013). https://doi.org/10.1088/1367-2630/15/10/103019 [36] Fernando GSL Brandao, Aram W Harrow, and Michał Horodecki. ``Local random quantum circuits are approximate polynomial-designs''. Comm. Math. Phys. 346, 397–434 (2016). https://doi.org/10.1007/s00220-016-2706-8 [37] Thomas Schuster, Jonas Haferkamp, and Hsin-Yuan Huang. ``Random unitaries in extremely low depth''. Science 389, 92–96 (2025). https://doi.org/10.1126/science.adv8590 [38] Adrian Solymos, Dávid Jakab, and Zoltán Zimborás. ``Extendibility of Brauer states'' (2024). arXiv:2411.04597. arXiv:2411.04597 [39] A. K. Hashagen, S. T. Flammia, D. Gross, and J. J. Wallman. ``Real Randomized Benchmarking''. Quantum 2, 85 (2018). https://doi.org/10.22331/q-2018-08-22-85 [40] Bruce C Berndt, Ronald J Evans, and Kenneth S Williams. ``Gauss and Jacobi sums''. Wiley. (1998). https://doi.org/10.2307/3619097 [41] Aleksandr A. Kirillov. ``Elements of the Theory of Representation''. Springer. (1976). https://doi.org/10.1007/978-3-642-66243-0 [42] Roe Goodman and Nolan R. Wallach. ``Symmetry, Representations, and Invariants''. Springer. (2009). https://doi.org/10.1007/978-0-387-79852-3 [43] Matthias Christandl. ``The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography''. PhD thesis. University of Cambridge. (2006). arXiv:quant-ph/0604183. arXiv:quant-ph/0604183 [44] K. Lux and H. Pahlings. ``Representations of Groups: A Computational Approach''. Cambridge Stud. Adv. Math.
Cambridge University Press. Cambridge, UK (2010). https://doi.org/10.1017/CBO9780511750915Cited byCould not fetch Crossref cited-by data during last attempt 2026-02-23 13:17:58: Could not fetch cited-by data for 10.22331/q-2026-02-23-2008 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-02-23 13:17:59: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.