Classical shadows for sample-efficient measurements of gauge-invariant observables

Jacob Bringewatt; Henry Froland; Andreas Elben; Niklas Mueller

doi:10.22331/q-2026-06-08-2127

Classical shadows for sample-efficient measurements of gauge-invariant observables

Jacob Bringewatt^1,2, Henry Froland³, Andreas Elben^4,5, and Niklas Mueller^6,7

¹Volgenau Department of Physics, United States Naval Academy, Annapolis, MD 21402, USA
²Department of Physics, Harvard University, Cambridge, MA 02138, USA
³InQubator for Quantum Simulation (IQuS), Department of Physics,University of Washington, Seattle, WA 98195, USA
⁴PSI Center for Scientific Computing, Theory and Data, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland
⁵ETH Zürich - PSI Quantum Computing Hub, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland
⁶Center for Quantum Information and Control, University of New Mexico, Albuquerque, NM 87106, USA
⁷Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87106, USA

Published:	2026-06-08, volume 10, page 2127
Editor:	Dax Enshan Koh
Eprint:	arXiv:2511.02904v2
Doi:	https://doi.org/10.22331/q-2026-06-08-2127
Citation:	Quantum 10, 2127 (2026).

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Abstract

Classical shadows provide a versatile framework for estimating many properties of quantum states from repeated, randomly chosen measurements without requiring full quantum state tomography. When prior information is available, such as knowledge of symmetries of states and operators, this knowledge can be exploited to significantly improve sample efficiency. In this work, we develop three classical shadow protocols for $\mathbb{Z}_2$ lattice gauge theory, where a dual formulation enables a rigorous analysis of resource requirements, including both circuit depth and sample complexity. Our approaches can offer exponential improvements in sample complexity over symmetry-agnostic methods, albeit at the cost of increased circuit complexity. While our analysis is restricted to $\mathbb{Z}_2$ lattice gauge theory, our approach offers a blueprint for similar protocols for more general lattice gauge theory models which are currently at the forefront of quantum simulation efforts.

Featured image: 1) A quantum state satisfying Gauss' law is prepared on a quantum device. 2) Unitary operations randomizing over the symmetric subspace are applied and the subsequent state is measured. 3) Gauge-invariant observables can be estimated using the collected data. The advantages of using this approach are that for many observables of interest, there is a provably exponential reduction in sample complexity required when compared to standard local randomization schemes. However, the circuit complexity of the randomizing operations can grow, leading to a tradeoff between circuit and sample complexity.

Popular summary

Fully learning an unknown quantum state requires an exponential number of measurements. Nonetheless, using well-chosen random measurements on copies of an unknown quantum state, one can estimate specific properties of interest much more cleverly. In addition, one often knows some properties of the unknown state of interest. For instance, if the state under consideration is the output of a quantum simulation of a physical theory, we know general features, such as symmetries, of the physically-allowed states.

In this paper, we develop random measurement protocol for a class of models, gauge theories, which posses a particular intricate structure and symmetries that one can exploit to drastically improve such learning protocols. Such systems, especially lattice gauge theories, have been at the forefront of advancing quantum simulation capabilities.

We introduce three new measurement protocols. Each protocol uses prior knowledge of the symmetry to restrict the randomization, thereby exponentially reducing the number of measurement samples needed compared to symmetry- agnostic approaches. However, this reduction in sample complexity comes at a cost: the randomizing operations on the quantum computer become more complex. For near-to-intermediate term quantum simulators, leveraging symmetry via classical shadows offers a promising route to efficient characterization of gauge-invariant observables, provided the hardware can handle the more demanding randomization circuits required.

► BibTeX data

@article{Bringewatt2026classicalshadows,
  doi = {10.22331/q-2026-06-08-2127},
  url = {https://doi.org/10.22331/q-2026-06-08-2127},
  title = {Classical shadows for sample-efficient measurements of gauge-invariant observables},
  author = {Bringewatt, Jacob and Froland, Henry and Elben, Andreas and Mueller, Niklas},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2127},
  month = jun,
  year = {2026}
}

► References

[1] J. Eisert, M. Friesdorf, and C. Gogolin, Nat. Phys. 11, 124 (2015).
https://doi.org/10.1038/nphys3215

[2] J. Schachenmayer, L. Pollet, M. Troyer, and A. J. Daley, EPJ Quantum Technology 2, 1 (2015).
https://doi.org/10.1140/epjqt15

[3] R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
https://doi.org/10.1146/annurev-conmatphys-031214-014726

[4] A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Science 353, 794 (2016).
https://doi.org/10.1126/science.aaf6725

[5] A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Nature 607, 667 (2022).
https://doi.org/10.1038/s41586-022-04940-6

[6] N. Mueller, T. Wang, O. Katz, Z. Davoudi, and M. Cetina, Nat. Commun. 16, 5492 (2025).
https://doi.org/10.1038/s41467-025-60177-7

[7] A. De, A. Lerose, D. Luo, F. M. Surace, A. Schuckert, E. R. Bennewitz, B. Ware, W. Morong, K. S. Collins, Z. Davoudi, et al., arXiv preprint arXiv:2410.13815 (2024).
https://doi.org/10.48550/arXiv.2410.13815
arXiv:2410.13815

[8] T. A. Cochran, B. Jobst, E. Rosenberg, Y. D. Lensky, G. Gyawali, N. Eassa, M. Will, D. Abanin, R. Acharya, L. A. Beni, et al., Nature , 315–320 (2025).
https://doi.org/10.1038/s41586-025-08999-9

[9] D. González-Cuadra, M. Hamdan, T. V. Zache, B. Braverman, M. Kornjača, A. Lukin, S. H. Cantú, F. Liu, S.-T. Wang, A. Keesling, et al., Nature , 1 (2025).
https://doi.org/10.1038/s41586-025-09051-6

[10] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Nature 549, 242 (2017).
https://doi.org/10.1038/nature23879

[11] P. Scholl, M. Schuler, H. J. Williams, A. A. Eberharter, D. Barredo, K.-N. Schymik, V. Lienhard, L.-P. Henry, T. C. Lang, T. Lahaye, et al., Nature 595, 233 (2021).
https://doi.org/10.1038/s41586-021-03585-1

[12] G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Samajdar, et al., Science 374, 1242 (2021).
https://doi.org/10.1126/science.abi8794

[13] K. Satzinger, Y.-J. Liu, A. Smith, C. Knapp, M. Newman, C. Jones, Z. Chen, C. Quintana, X. Mi, A. Dunsworth, et al., Science 374, 1237 (2021).
https://doi.org/10.1126/science.abi8378

[14] S. Aaronson, in Proceedings of the 50th annual ACM SIGACT symposium on theory of computing (2018) pp. 325–338.
https://doi.org/10.1145/3188745.3188802

[15] H.-Y. Huang, R. Kueng, and J. Preskill, Nat. Phys. 16, 1050 (2020).
https://doi.org/10.1038/s41567-020-0932-7

[16] J. Emerson, R. Alicki, and K. Życzkowski, J. Opt. B: Quantum Semiclassical Opt. 7, S347 (2005).
https://doi.org/10.1088/1464-4266/7/10/021

[17] E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, Phys. Rev. A 77, 012307 (2008).
https://doi.org/10.1103/PhysRevA.77.012307

[18] M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, Nat Commun. 1, 149 (2010).
https://doi.org/10.1038/ncomms1147

[19] E. Magesan, J. M. Gambetta, and J. Emerson, Phys. Rev. Lett. 106, 180504 (2011).
https://doi.org/10.1103/PhysRevLett.106.180504

[20] S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, J. M. Martinis, and H. Neven, Nat. Phys. 14, 595 (2018).
https://doi.org/10.1038/s41567-018-0124-x

[21] A. Erhard, J. J. Wallman, L. Postler, M. Meth, R. Stricker, E. A. Martinez, P. Schindler, T. Monz, J. Emerson, and R. Blatt, Nat. Commun. 10, 5347 (2019).
https://doi.org/10.1038/s41467-019-13068-7

[22] C. Kokail, C. Maier, R. van Bijnen, T. Brydges, M. K. Joshi, P. Jurcevic, C. A. Muschik, P. Silvi, R. Blatt, C. F. Roos, et al., Nature 569, 355 (2019).
https://doi.org/10.1038/s41586-019-1177-4

[23] R. Blume-Kohout, T. Proctor, and K. Young, arXiv preprint arXiv:2503.16383 (2025).
https://doi.org/10.48550/arXiv.2503.16383
arXiv:2503.16383

[24] A. Czerwinski, Optics 3, 268 (2022).
https://doi.org/10.3390/opt3030026

[25] D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, Phys. Rev. Lett. 105, 150401 (2010).
https://doi.org/10.1103/PhysRevLett.105.150401

[26] S. T. Flammia, D. Gross, Y.-K. Liu, and J. Eisert, New J. Phys. 14, 095022 (2012).
https://doi.org/10.1088/1367-2630/14/9/095022

[27] J. Haah, A. W. Harrow, Z. Ji, X. Wu, and N. Yu, in Proceedings of the forty-eighth annual ACM symposium on Theory of Computing (2016) pp. 913–925.
https://doi.org/10.1145/2897518.2897585

[28] R. O'Donnell and J. Wright, in Proceedings of the forty-eighth annual ACM symposium on Theory of Computing (2016) pp. 899–912.
https://doi.org/10.1145/2897518.2897544

[29] G. Torlai, G. Mazzola, J. Carrasquilla, M. Troyer, R. Melko, and G. Carleo, Nat. Phys. 14, 447 (2018).
https://doi.org/10.1038/s41567-018-0048-5

[30] A. Saxena, E. Abbasgholinejad, A. Majumdar, and R. Trivedi, Phys. Rev. Res. 6, 033058 (2024).
https://doi.org/10.1103/PhysRevResearch.6.033058

[31] S. J. van Enk and C. W. Beenakker, Phys. Rev. Lett. 108, 110503 (2012).
https://doi.org/10.1103/PhysRevLett.108.110503

[32] A. Elben, B. Vermersch, M. Dalmonte, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 120, 050406 (2018).
https://doi.org/10.1103/PhysRevLett.120.050406

[33] B. Vermersch, A. Elben, M. Dalmonte, J. I. Cirac, and P. Zoller, Phys. Rev. A 97, 023604 (2018).
https://doi.org/10.1103/PhysRevA.97.023604

[34] A. Elben, B. Vermersch, C. F. Roos, and P. Zoller, Phys. Rev. A 99, 052323 (2019).
https://doi.org/10.1103/PhysRevA.99.052323

[35] T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos, Science 364, 260 (2019).
https://doi.org/10.1126/science.aau4963

[36] Z. Du, Y. Tang, A. Elben, I. Roth, J. Eisert, and Z. Liu, arXiv preprint arXiv:2505.09206 (2025).
https://doi.org/10.48550/arXiv.2505.09206
arXiv:2505.09206

[37] M. Paini and A. Kalev, arXiv preprint arXiv:1910.10543 (2019).
arXiv:1910.10543

[38] H.-Y. R. Huang, M. B. Broughton, J. Cotler, S. Chen, J. Li, M. Mohseni, H. Neven, R. Babbush, R. Kueng, J. Preskill, et al., Science 376, 1182 (2021a).
https://doi.org/10.1126/science.abn7293

[39] K. Van Kirk, J. Cotler, H.-Y. Huang, and M. D. Lukin, arXiv preprint arXiv:2212.06084 (2022).
https://doi.org/10.48550/arXiv.2212.06084
arXiv:2212.06084

[40] C. Hadfield, S. Bravyi, R. Raymond, and A. Mezzacapo, Commun. Math. Phys. 391, 951 (2022).
https://doi.org/10.1007/s00220-022-04343-8

[41] A. A. Akhtar, H.-Y. Hu, and Y.-Z. You, Quantum 7, 1026 (2023).
https://doi.org/10.22331/q-2023-06-01-1026

[42] H.-Y. Hu, S. Choi, and Y.-Z. You, Phys. Rev. Res. 5, 023027 (2023).
https://doi.org/10.1103/PhysRevResearch.5.023027

[43] J. Kunjummen, M. C. Tran, D. Carney, and J. M. Taylor, Phys. Rev. A 107, 042403 (2023).
https://doi.org/10.1103/PhysRevA.107.042403

[44] S. Gandhari, V. V. Albert, T. Gerrits, J. M. Taylor, and M. J. Gullans, PRX Quantum 5, 010346 (2024).
https://doi.org/10.1103/PRXQuantum.5.010346

[45] R. Levy, D. Luo, and B. K. Clark, Phys. Rev. Res. 6, 013029 (2024).
https://doi.org/10.1103/PhysRevResearch.6.013029

[46] Z. Zhu, J. M. Lukens, and B. T. Kirby, Quantum 8, 1455 (2024).
https://doi.org/10.22331/q-2024-08-29-1455

[47] Z. Cai, A. Chapman, H. Jnane, and B. Koczor, arXiv preprint arXiv:2402.09511 (2024).
https://doi.org/10.48550/arXiv.2402.09511
arXiv:2402.09511

[48] H.-Y. Hu, A. Gu, S. Majumder, H. Ren, Y. Zhang, D. S. Wang, Y.-Z. You, Z. Minev, S. F. Yelin, and A. Seif, Nat Commun. 16, 2943 (2025).
https://doi.org/10.1038/s41467-025-57349-w

[49] M. K. Joshi, A. Elben, B. Vermersch, T. Brydges, C. Maier, P. Zoller, R. Blatt, and C. F. Roos, Phys. Rev. Lett. 124, 240505 (2020).
https://doi.org/10.1103/PhysRevLett.124.240505

[50] L. K. Joshi, J. Franke, A. Rath, F. Ares, S. Murciano, F. Kranzl, R. Blatt, P. Zoller, B. Vermersch, P. Calabrese, C. F. Roos, and M. K. Joshi, Phys. Rev. Lett. 133, 010402 (2024), publisher: American Physical Society.
https://doi.org/10.1103/PhysRevLett.133.010402

[51] T. Zhang, J. Sun, X.-X. Fang, X.-M. Zhang, X. Yuan, and H. Lu, Phys. Rev. Lett. 127, 200501 (2021).
https://doi.org/10.1103/PhysRevLett.127.200501

[52] G. Struchalin, Y. A. Zagorovskii, E. Kovlakov, S. Straupe, and S. Kulik, PRX Quantum 2, 010307 (2021).
https://doi.org/10.1103/PRXQuantum.2.010307

[53] V. Vitale, A. Rath, P. Jurcevic, A. Elben, C. Branciard, and B. Vermersch, PRX Quantum 5, 030338 (2024).
https://doi.org/10.1103/PRXQuantum.5.030338

[54] H. Dong, P. Zhang, C. B. Dağ, Y. Gao, N. Wang, J. Deng, X. Zhang, J. Chen, S. Xu, K. Wang, Y. Wu, C. Zhang, F. Jin, X. Zhu, A. Zhang, Y. Zou, Z. Tan, Z. Cui, Z. Zhu, F. Shen, T. Li, J. Zhong, Z. Bao, H. Li, Z. Wang, Q. Guo, C. Song, F. Liu, A. Chan, L. Ying, and H. Wang, Phys. Rev. Lett. 134, 010402 (2025).
https://doi.org/10.1103/PhysRevLett.134.010402

[55] T. I. Andersen, N. Astrakhantsev, A. H. Karamlou, J. Berndtsson, J. Motruk, A. Szasz, J. A. Gross, A. Schuckert, T. Westerhout, Y. Zhang, et al., Nature 638, 79 (2025).
https://doi.org/10.1038/s41586-024-08460-3

[56] M. Votto, M. Ljubotina, C. Lancien, J. I. Cirac, P. Zoller, M. Serbyn, L. Piroli, and B. Vermersch, arXiv:preprint arXiv:2507.12550 (2025).
https://doi.org/10.1103/rbg2-f61m
arXiv:2507.12550

[57] A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller, Nat. Rev. Phys. 5, 9 (2023).
https://doi.org/10.1038/s42254-022-00535-2

[58] A. Rath, R. van Bijnen, A. Elben, P. Zoller, and B. Vermersch, Phys. Rev. Lett. 127, 200503 (2021).
https://doi.org/10.1103/PhysRevLett.127.200503

[59] H.-Y. Huang, R. Kueng, and J. Preskill, Phys. Rev. Lett. 127, 030503 (2021b).
https://doi.org/10.1103/PhysRevLett.127.030503

[60] G. Hao Low, arXiv preprint arXiv:2208.08964 (2022).
arXiv:2208.08964

[61] J. Bringewatt, J. Kunjummen, and N. Mueller, Quantum 8, 1300 (2024).
https://doi.org/10.22331/q-2024-03-27-1300

[62] S. N. Hearth, M. O. Flynn, A. Chandran, and C. R. Laumann, Phys. Rev. Lett. 133, 060802 (2024).
https://doi.org/10.1103/PhysRevLett.133.060802

[63] F. Sauvage and M. Larocca, arXiv preprint arXiv:2408.05279 (2024).
https://doi.org/10.48550/arXiv.2408.05279
arXiv:2408.05279

[64] R. Arvind, K. Bharti, J. Y. Khoo, D. E. Koh, and J. F. Kong, Phys. Rev. Res. 7, 013105 (2025).
https://doi.org/10.1103/PhysRevResearch.7.013105

[65] J. C. Halimeh, N. Mueller, J. Knolle, Z. Papić, and Z. Davoudi, arXiv preprint arXiv:2509.03586 (2025).
https://doi.org/10.48550/arXiv.2509.03586
arXiv:2509.03586

[66] M. C. Banuls, R. Blatt, J. Catani, A. Celi, J. I. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, S. Montangero, et al., Eur. Phys. J. D 74, 1 (2020).
https://doi.org/10.1140/epjd/e2020-100571-8

[67] N. Klco, A. Roggero, and M. J. Savage, Rept. Prog. Phys. 85, 064301 (2022).
https://doi.org/10.1088/1361-6633/ac58a4

[68] C. W. Bauer, Z. Davoudi, A. B. Balantekin, T. Bhattacharya, M. Carena, W. A. de Jong, P. Draper, A. El-Khadra, N. Gemelke, M. Hanada, D. Kharzeev, H. Lamm, Y.-Y. Li, J. Liu, M. Lukin, Y. Meurice, C. Monroe, B. Nachman, G. Pagano, J. Preskill, E. Rinaldi, A. Roggero, D. I. Santiago, M. J. Savage, I. Siddiqi, G. Siopsis, D. Van Zanten, N. Wiebe, Y. Yamauchi, K. Yeter-Aydeniz, and S. Zorzetti, PRX Quantum 4, 027001 (2023a).
https://doi.org/10.1103/PRXQuantum.4.027001

[69] C. W. Bauer, Z. Davoudi, N. Klco, and M. J. Savage, Nature Rev. Phys. 5, 420 (2023b).
https://doi.org/10.1038/s42254-023-00599-8

[70] A. Di Meglio, K. Jansen, I. Tavernelli, C. Alexandrou, S. Arunachalam, C. W. Bauer, K. Borras, S. Carrazza, A. Crippa, V. Croft, R. de Putter, A. Delgado, V. Dunjko, D. J. Egger, E. Fernández-Combarro, E. Fuchs, L. Funcke, D. González-Cuadra, M. Grossi, J. C. Halimeh, Z. Holmes, S. Kühn, D. Lacroix, R. Lewis, D. Lucchesi, M. L. Martinez, F. Meloni, A. Mezzacapo, S. Montangero, L. Nagano, V. R. Pascuzzi, V. Radescu, E. R. Ortega, A. Roggero, J. Schuhmacher, J. Seixas, P. Silvi, P. Spentzouris, F. Tacchino, K. Temme, K. Terashi, J. Tura, C. Tüysüz, S. Vallecorsa, U.-J. Wiese, S. Yoo, and J. Zhang, PRX Quantum 5, 037001 (2024).
https://doi.org/10.1103/PRXQuantum.5.037001

[71] E. Fradkin, Field theories of condensed matter physics (Cambridge University Press, 2013).

[72] H. Kleinert, Gauge Fields in Condensed Matter: Vol. 1: Superflow and Vortex Lines (Disorder Fields, Phase Transitions) Vol. 2: Stresses and Defects (Differential Geometry, Crystal Melting) (World Scientific, 1989).

[73] X.-G. Wen, Int. J. Mod. Phys. A 4, 239 (1990).
https://doi.org/10.1142/S0217979290000139

[74] M. A. Levin and X.-G. Wen, Phys. Rev. B 71, 045110 (2005).
https://doi.org/10.1103/PhysRevB.71.045110

[75] Y.-A. Chen, A. Kapustin, and DJ. Radičević, Annals of Physics 393, 234 (2018).
https://doi.org/10.1016/j.aop.2018.03.024

[76] Y.-A. Chen, Phys. Rev. Research 2, 033527 (2020).
https://doi.org/10.1103/PhysRevResearch.2.033527

[77] Y.-A. Chen and Y. Xu, PRX Quantum 4, 010326 (2023).
https://doi.org/10.1103/PRXQuantum.4.010326

[78] S. Das Sarma, M. Freedman, and C. Nayak, Physics Today 59, 32 (2006).
https://doi.org/10.1063/1.2337825

[79] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).
https://doi.org/10.1103/RevModPhys.80.1083

[80] V. Lahtinen and J. K. Pachos, SciPost Phys. 3, 021 (2017).
https://doi.org/10.21468/SciPostPhys.3.3.021

[81] J. Kogut and L. Susskind, Phys. Rev. D 11, 395 (1975).
https://doi.org/10.1103/PhysRevD.11.395

[82] J. B. Kogut, Rev. Mod. Phys. 51, 659 (1979).
https://doi.org/10.1103/RevModPhys.51.659

[83] Z. Davoudi, arXiv preprint arXiv:2507.15840 (2025).
https://doi.org/10.48550/arXiv.2507.15840
arXiv:2507.15840

[84] F. J. Wegner, J. Math. Phys. 12, 2259 (1971).
https://doi.org/10.1063/1.1665530

[85] D. Horn, M. Weinstein, and S. Yankielowicz, Phys. Rev. D 19, 3715 (1979).
https://doi.org/10.1103/PhysRevD.19.3715

[86] N. Mueller, T. V. Zache, and R. Ott, Phys. Rev. Lett. 129, 011601 (2022).
https://doi.org/10.1103/PhysRevLett.129.011601

[87] J. Hartse, L. Fidkowski, and N. Mueller, Phys. Rev. Lett. 135, 060402 (2025).
https://doi.org/10.1103/n5hb-l5p5

[88] P. Black, Dictionary of algorithms and data structures (2012).
https://xlinux.nist.gov/dads/HTML/greedyalgo.html

[89] A. A. Mele, Quantum 8, 1340 (2024).
https://doi.org/10.22331/q-2024-05-08-1340

[90] A. Smith, J. Knolle, D. L. Kovrizhin, and R. Moessner, Phys. Rev. Lett. 118, 266601 (2017).
https://doi.org/10.1103/PhysRevLett.118.266601

[91] J. C. Halimeh, U. E. Khodaeva, D. L. Kovrizhin, R. Moessner, and J. Knolle, arXiv preprint arXiv:2410.08268 (2024).
https://doi.org/10.48550/arXiv.2410.08268
arXiv:2410.08268

[92] https://github.com/frolandh/lgtshallowshadows.
https://github.com/frolandh/LGTShallowShadows

[93] P. Handler, S. Jasperson, S. Koeppen, et al., Zh. Eksp. Teor. Fiz 75, 1083 (1978).

[94] J. Drouffe, C. Itzykson, and J. Zuber, Nuc. Phys. B 147, 132 (1979).
https://doi.org/10.1016/0550-3213(79)90418-8

[95] E. Alvarez, L. Alvarez-Gaume, and Y. Lozano, Nuc. Phys. B 424, 155 (1994).
https://doi.org/10.1016/0550-3213(94)90093-0

[96] V. M. Buchstaber and M. I. Monastyrsky, J. Phys. A: Math. Gen. 36, 7679 (2003).
https://doi.org/10.1088/0305-4470/36/28/301

[97] E. Cobanera, G. Ortiz, and E. Knill, Nuc. Phys. B 877, 574 (2013).
https://doi.org/10.1016/j.nuclphysb.2013.10.018

[98] W. Ji and X.-G. Wen, Phys. Rev. Research 2, 033417 (2020).
https://doi.org/10.1103/PhysRevResearch.2.033417

[99] D. Aasen, P. Fendley, and R. S. Mong, arXiv preprint arXiv:2008.08598 (2020).
https://doi.org/10.48550/arXiv.2008.08598
arXiv:2008.08598

[100] A. Chakraborty, R. Lewis, and C. A. Muschik, arXiv preprint arXiv:2510.08718 (2025).
https://doi.org/10.48550/arXiv.2510.08718
arXiv:2510.08718

[101] A. T. Than, Y. Y. Atas, A. Chakraborty, J. Zhang, M. T. Diaz, K. Wen, X. Liu, R. Lewis, A. M. Green, C. A. Muschik, et al., arXiv preprint arXiv:2501.00579 (2024).
https://doi.org/10.48550/arXiv.2501.00579
arXiv:2501.00579

[102] Y. Zhan, A. Elben, H.-Y. Huang, and Y. Tong, PRX Quantum 5, 010350 (2024).
https://doi.org/10.1103/PRXQuantum.5.010350

[103] Z. Ji, Y.-K. Liu, and F. Song, in Annual International Cryptology Conference (Springer, 2018) pp. 126–152.
https://doi.org/10.1007/978-3-319-96878-0_5

[104] F. Ma and H.-Y. Huang, in Proceedings of the 57th Annual ACM Symposium on Theory of Computing (2025) pp. 806–809.
https://doi.org/10.1145/3717823.3718254

[105] T. Schuster, J. Haferkamp, and H.-Y. Huang, Science 389, 92 (2025).
https://doi.org/10.1126/science.adv8590

[106] L. Grevink, J. Haferkamp, M. Heinrich, J. Helsen, M. Hinsche, T. Schuster, and Z. Zimborás, arXiv preprint arXiv:2506.23925 (2025).
arXiv:2506.23925

[107] L. Cui, T. Schuster, L. Mao, H.-Y. Huang, and F. Brandao, arXiv preprint arXiv:2510.08434 (2025).
arXiv:2510.08434

[108] M. Ippoliti, Quantum 8, 1293 (2024).
https://doi.org/10.22331/q-2024-03-21-1293

[109] S. Chen, W. Yu, P. Zeng, and S. T. Flammia, PRX Quantum 2, 030348 (2021).
https://doi.org/10.1103/PRXQuantum.2.030348

[110] D. E. Koh and S. Grewal, Quantum 6, 776 (2022).
https://doi.org/10.22331/q-2022-08-16-776

[111] P.-G. Rozon, N. Bao, and K. Agarwal, Phys. Rev. Lett. 133, 130803 (2024).
https://doi.org/10.1103/PhysRevLett.133.130803

[112] R. Brieger, M. Heinrich, I. Roth, and M. Kliesch, Phys. Rev. Lett. 134, 090801 (2025).
https://doi.org/10.1103/PhysRevLett.134.090801

[113] H.-Y. Hu, R. LaRose, Y.-Z. You, E. Rieffel, and Z. Wang, arXiv prepreint arXiv:2203.07263 (2022).
arXiv:2203.07263

Cited by

[1] Maxwell West, Frederic Sauvage, Aniruddha Sen, Roy Forestano, David Wierichs, Nathan Killoran, Dmitry Grinko, M. Cerezo, and Martin Larocca, "Classical shadows with arbitrary group representations", arXiv:2604.01429, (2026).

[2] Maxwell West, M. Cerezo, and Martin Larocca, "Particle-preserving fermionic shadows with mode-independent sample complexity", arXiv:2606.27254, (2026).

[3] Matteo Turco, Luca Spagnoli, and Alessandro Roggero, "Binary Gauss Stabilizers for Abelian Lattice Gauge Theories", arXiv:2607.14861, (2026).

The above citations are from SAO/NASA ADS (last updated successfully 2026-08-09 22:03:05). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2026-08-09 22:03:03).

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