Operational interpretation of the Stabilizer Entropy

Lennart Bittel; Lorenzo Leone

doi:10.22331/q-2026-04-15-2069

Operational interpretation of the Stabilizer Entropy

Lennart Bittel¹ and Lorenzo Leone^2,1

¹Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
²Dipartimento di Ingegneria Industriale, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (SA), Italy

Published:	2026-04-15, volume 10, page 2069
Editor:	Ulysse Chabaud
Eprint:	arXiv:2507.22883v3
Doi:	https://doi.org/10.22331/q-2026-04-15-2069
Citation:	Quantum 10, 2069 (2026).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Magic-state resource theory is a fundamental framework with far-reaching applications in quantum error correction and the classical simulation of quantum systems. Recent advances have significantly deepened our understanding of magic as a resource across diverse domains, including many-body physics, nuclear and particle physics, and quantum chemistry. Central to this progress is the stabilizer Rényi entropy, a computable and experimentally accessible magic monotone. Despite its widespread adoption, a rigorous operational interpretation of the stabilizer entropy has remained an open problem. In this work, we provide such an interpretation in the context of quantum property testing. By showing that the stabilizer entropy is the most robust measurable magic monotone, we demonstrate that the Clifford orbit of a quantum state becomes exponentially indistinguishable from Haar-random states, at a rate governed by the stabilizer entropy $M_{\alpha}(\psi)$ and the number of available copies. This implies that the Clifford orbit forms an approximate state $k$-design, with an approximation error $\exp(-\Theta (M_{\alpha}(\psi)))$ for $\alpha\ge2$. Conversely, we establish that the optimal probability of distinguishing a given quantum state from the set of stabilizer states is also governed by its stabilizer entropy. These results reveal that the stabilizer entropy quantitatively characterizes the transition from stabilizer states to universal quantum states, thereby offering a comprehensive operational perspective of the stabilizer entropy as a quantum resource.

► BibTeX data

@article{Bittel2026operational,
  doi = {10.22331/q-2026-04-15-2069},
  url = {https://doi.org/10.22331/q-2026-04-15-2069},
  title = {Operational interpretation of the {S}tabilizer {E}ntropy},
  author = {Bittel, Lennart and Leone, Lorenzo},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2069},
  month = apr,
  year = {2026}
}

► References

[1] Scott Aaronson. Shadow tomography of quantum states, 2018. URL https://arxiv.org/abs/1711.01053.
arXiv:1711.01053

[2] Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Physical Review A, 70: 052328–052328, November 2004. 10.1103/PhysRevA.70.052328.
https://doi.org/10.1103/PhysRevA.70.052328

[3] Noga Alon. Combinatorial nullstellensatz. Combinatorics, Probability and Computing, 8 (1-2): 7–29, 1999.

[4] Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. Entanglement in many-body systems. Rev. Mod. Phys., 80: 517–576, May 2008. 10.1103/RevModPhys.80.517. URL.
https://doi.org/10.1103/RevModPhys.80.517

[5] Rafael Aoude, Hannah Banks, Chris D. White, and Martin J. White. Probing new physics in the top sector using quantum information, 2025. URL https://arxiv.org/abs/2505.12522.
arXiv:2505.12522

[6] Srinivasan Arunachalam and Arkopal Dutt. Polynomial-time tolerant testing stabilizer states, 2024. URL https://arxiv.org/abs/2408.06289.
arXiv:2408.06289

[7] Srinivasan Arunachalam, Sergey Bravyi, and Arkopal Dutt. A note on polynomial-time tolerant testing stabilizer states, 2024. URL https://arxiv.org/abs/2410.22220.
arXiv:2410.22220

[8] Zongbo Bao, Philippe van Dordrecht, and Jonas Helsen. Tolerant testing of stabilizer states with a polynomial gap via a generalized uncertainty relation, 2024. URL https://arxiv.org/abs/2410.21811.
arXiv:2410.21811

[9] Michael Beverland, Earl Campbell, Mark Howard, and Vadym Kliuchnikov. Lower bounds on the non-clifford resources for quantum computations. Quantum Science and Technology, 5 (3): 035009, June 2020. ISSN 2058-9565. 10.1088/2058-9565/ab8963. URL http://dx.doi.org/10.1088/2058-9565/ab8963.
https://doi.org/10.1088/2058-9565/ab8963

[10] Lennart Bittel and al. The theory of the real clifford commutant. In preparation, 2025.

[11] Lennart Bittel, Jens Eisert, Lorenzo Leone, Antonio A. Mele, and Salvatore F. E. Oliviero. A complete theory of the clifford commutant, 2025. URL https://arxiv.org/abs/2504.12263.
arXiv:2504.12263

[12] Zvika Brakerski, Nir Magrafta, and Tomer Solomon. State-based classical shadows, 2025. URL https://arxiv.org/abs/2507.10362.
arXiv:2507.10362

[13] Sergey Bravyi and Jeongwan Haah. Magic-state distillation with low overhead. Physical Review A, 86: 052329–052329, November 2012. 10.1103/PhysRevA.86.052329.
https://doi.org/10.1103/PhysRevA.86.052329

[14] Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal Clifford gates and noisy ancillas. Physical Review A, 71: 022316–022316, February 2005. 10.1103/PhysRevA.71.022316.
https://doi.org/10.1103/PhysRevA.71.022316

[15] Raphael Brieger, Markus Heinrich, Ingo Roth, and Martin Kliesch. Stability of classical shadows under gate-dependent noise. Physical Review Letters, 134 (9), March 2025. ISSN 1079-7114. 10.1103/physrevlett.134.090801. URL http://dx.doi.org/10.1103/PhysRevLett.134.090801.
https://doi.org/10.1103/physrevlett.134.090801

[16] Florian Brökemeier, S. Momme Hengstenberg, James W. T. Keeble, Caroline E. P. Robin, Federico Rocco, and Martin J. Savage. Quantum magic and multipartite entanglement in the structure of nuclei. Phys. Rev. C, 111: 034317, Mar 2025. 10.1103/PhysRevC.111.034317.
https://doi.org/10.1103/PhysRevC.111.034317

[17] Kaifeng Bu, Weichen Gu, and Arthur Jaffe. Stabilizer testing and magic entropy, 2023. URL https://arxiv.org/abs/2306.09292. 10.1007/s00220-025-05421-3.
https://doi.org/10.1007/s00220-025-05421-3
arXiv:2306.09292

[18] Giorgio Busoni, John Gargalionis, Ewan N. V. Wallace, and Martin J. White. Emergent symmetry in a two-higgs-doublet model from quantum information and magic, 2025. URL https://arxiv.org/abs/2506.01314. 10.1103/r5ps-pmh3.
https://doi.org/10.1103/r5ps-pmh3
arXiv:2506.01314

[19] Earl T. Campbell and Dan E. Browne. Bound States for Magic State Distillation in Fault-Tolerant Quantum Computation. Physical Review Letters, 104: 030503–030503, January 2010. 10.1103/PhysRevLett.104.030503.
https://doi.org/10.1103/PhysRevLett.104.030503

[20] Ivan Chernyshev, Caroline E. P. Robin, and Martin J. Savage. Quantum magic and computational complexity in the neutrino sector. Phys. Rev. Res., 7: 023228, Jun 2025. 10.1103/PhysRevResearch.7.023228.
https://doi.org/10.1103/PhysRevResearch.7.023228

[21] Eric Chitambar and Gilad Gour. Quantum resource theories. Review of Modern Physics, 91: 025001–025001, April 2019. 10.1103/RevModPhys.91.025001.
https://doi.org/10.1103/RevModPhys.91.025001

[22] Mario Collura, Jacopo De Nardis, Vincenzo Alba, and Guglielmo Lami. The quantum magic of fermionic gaussian states, 2025. URL https://arxiv.org/abs/2412.05367. 10.22331/q-2026-03-23-2036.
https://doi.org/10.22331/q-2026-03-23-2036
arXiv:2412.05367

[23] Janek Denzler, Santiago Varona, Tommaso Guaita, and Jose Carrasco. Highly-entangled, highly-doped states that are efficiently cross-device verifiable. Phys. Rev. Res., 7: 023320, Jun 2025. 10.1103/b6rb-yr6c.
https://doi.org/10.1103/b6rb-yr6c

[24] Bryan Eastin and Emanuel Knill. Restrictions on transversal encoded quantum gate sets. Phys. Rev. Lett., 102: 110502, Mar 2009. 10.1103/PhysRevLett.102.110502.
https://doi.org/10.1103/PhysRevLett.102.110502

[25] Pedro R. Nicácio Falcão, Poetri Sonya Tarabunga, Martina Frau, Emanuele Tirrito, Jakub Zakrzewski, and Marcello Dalmonte. Nonstabilizerness in u(1) lattice gauge theory. Phys. Rev. B, 111: L081102, Feb 2025. 10.1103/PhysRevB.111.L081102.
https://doi.org/10.1103/PhysRevB.111.L081102

[26] Kun Fang and Zi-Wen Liu. Surpassing the fundamental limits of distillation with catalysts, 2024. URL https://arxiv.org/abs/2410.14547.
arXiv:2410.14547

[27] Martina Frau, Poetri Sonya Tarabunga, Mario Collura, Emanuele Tirrito, and Marcello Dalmonte. Stabilizer disentangling of conformal field theories. SciPost Phys., 18: 165, 2025. 10.21468/SciPostPhys.18.5.165. URL https://scipost.org/10.21468/SciPostPhys.18.5.165.
https://doi.org/10.21468/SciPostPhys.18.5.165

[28] Daniel Gottesman. The Heisenberg Representation of Quantum Computers, July 1998.

[29] Sabee Grewal, Vishnu Iyer, William Kretschmer, and Daniel Liang. Improved stabilizer estimation via bell difference sampling, 2024. URL https://arxiv.org/abs/2304.13915. 10.1145/3618260.3649738.
https://doi.org/10.1145/3618260.3649738
arXiv:2304.13915

[30] D. Gross, K. Audenaert, and J. Eisert. Evenly distributed unitaries: On the structure of unitary designs. Journal of Mathematical Physics, 48 (5), May 2007. ISSN 1089-7658. 10.1063/1.2716992. URL http://dx.doi.org/10.1063/1.2716992.
https://doi.org/10.1063/1.2716992

[31] David Gross, Sepehr Nezami, and Michael Walter. Schur–weyl duality for the clifford group with applications: Property testing, a robust hudson theorem, and de finetti representations. Communications in Mathematical Physics, 385 (3): 1325–1393, June 2021. ISSN 1432-0916. 10.1007/s00220-021-04118-7. URL http://dx.doi.org/10.1007/s00220-021-04118-7.
https://doi.org/10.1007/s00220-021-04118-7

[32] Andi Gu, Hong-Ye Hu, Di Luo, Taylor L. Patti, Nicholas C. Rubin, and Susanne F. Yelin. Zero and finite temperature quantum simulations powered by quantum magic. Quantum, 8: 1422, July 2024. ISSN 2521-327X. 10.22331/q-2024-07-23-1422. URL http://dx.doi.org/10.22331/q-2024-07-23-1422.
https://doi.org/10.22331/q-2024-07-23-1422

[33] Alioscia Hamma, Radu Ionicioiu, and Paolo Zanardi. Bipartite entanglement and entropic boundary law in lattice spin systems. Phys. Rev. A, 71: 022315, Feb 2005. 10.1103/PhysRevA.71.022315.
https://doi.org/10.1103/PhysRevA.71.022315

[34] Tobias Haug and Lorenzo Piroli. Quantifying nonstabilizerness of matrix product states. Phys. Rev. B, 107 (3): 035148, January 2023a. 10.1103/PhysRevB.107.035148.
https://doi.org/10.1103/PhysRevB.107.035148

[35] Tobias Haug and Lorenzo Piroli. Stabilizer entropies and nonstabilizerness monotones. Quantum, 7: 1092, August 2023b. ISSN 2521-327X. 10.22331/q-2023-08-28-1092. URL http://dx.doi.org/10.22331/q-2023-08-28-1092.
https://doi.org/10.22331/q-2023-08-28-1092

[36] Tobias Haug and Poetri Sonya Tarabunga. Efficient witnessing and testing of magic in mixed quantum states, 2025. URL https://arxiv.org/abs/2504.18098.
arXiv:2504.18098

[37] Tobias Haug, Soovin Lee, and M. S. Kim. Efficient quantum algorithms for stabilizer entropies. Phys. Rev. Lett., 132: 240602, Jun 2024. 10.1103/PhysRevLett.132.240602.
https://doi.org/10.1103/PhysRevLett.132.240602

[38] Markus Heinrich and David Gross. Robustness of Magic and Symmetries of the Stabiliser Polytope. Quantum, 3: 132, April 2019. ISSN 2521-327X. 10.22331/q-2019-04-08-132. URL https://doi.org/10.22331/q-2019-04-08-132.
https://doi.org/10.22331/q-2019-04-08-132

[39] Marcel Hinsche and Jonas Helsen. Single-copy stabilizer testing. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing, page 439–450. ACM, June 2025. 10.1145/3717823.3718169. URL http://dx.doi.org/10.1145/3717823.3718169.
https://doi.org/10.1145/3717823.3718169

[40] Masahiro Hoshino and Yuto Ashida. Stabilizer rényi entropy encodes fusion rules of topological defects and boundaries, 2025. URL https://arxiv.org/abs/2507.10656. 10.1103/1tyr-rlbb.
https://doi.org/10.1103/1tyr-rlbb
arXiv:2507.10656

[41] Masahiro Hoshino, Masaki Oshikawa, and Yuto Ashida. Stabilizer rényi entropy and conformal field theory, 2025. URL https://arxiv.org/abs/2503.13599. 10.1103/ylsz-dm3y.
https://doi.org/10.1103/ylsz-dm3y
arXiv:2503.13599

[42] Hsin-Yuan Huang, Richard Kueng, and John Preskill. Predicting many properties of a quantum system from very few measurements. Nature Physics, 16 (10): 1050–1057, June 2020. ISSN 1745-2481. 10.1038/s41567-020-0932-7. URL http://dx.doi.org/10.1038/s41567-020-0932-7.
https://doi.org/10.1038/s41567-020-0932-7

[43] Hsin-Yuan Huang, Richard Kueng, and John Preskill. Information-theoretic bounds on quantum advantage in machine learning. Physical Review Letters, 126 (19), May 2021. ISSN 1079-7114. 10.1103/physrevlett.126.190505. URL http://dx.doi.org/10.1103/PhysRevLett.126.190505.
https://doi.org/10.1103/physrevlett.126.190505

[44] Daniele Iannotti, Gianluca Esposito, Lorenzo Campos Venuti, and Alioscia Hamma. Entanglement and stabilizer entropies of random bipartite pure quantum states. Quantum, 9: 1797, July 2025. ISSN 2521-327X. 10.22331/q-2025-07-21-1797. URL http://dx.doi.org/10.22331/q-2025-07-21-1797.
https://doi.org/10.22331/q-2025-07-21-1797

[45] Marc Illa, Martin J. Savage, and Xiaojun Yao. Dynamical local tadpole-improvement in quantum simulations of gauge theories, 2025. URL https://arxiv.org/abs/2504.21575.
arXiv:2504.21575

[46] Barbara Jasser, Jovan Odavić, and Alioscia Hamma. Stabilizer entropy and entanglement complexity in the sachdev-ye-kitaev model. Physical Review B, 112 (17), November 2025. ISSN 2469-9969. 10.1103/rz86-47h3. URL http://dx.doi.org/10.1103/rz86-47h3.
https://doi.org/10.1103/rz86-47h3

[47] Robbie King, Kianna Wan, and Jarrod R. McClean. Exponential learning advantages with conjugate states and minimal quantum memory. PRX Quantum, 5 (4), October 2024. ISSN 2691-3399. 10.1103/prxquantum.5.040301. URL http://dx.doi.org/10.1103/PRXQuantum.5.040301.
https://doi.org/10.1103/prxquantum.5.040301

[48] Robbie King, David Gosset, Robin Kothari, and Ryan Babbush. Triply efficient shadow tomography. PRX Quantum, 6 (1), February 2025. ISSN 2691-3399. 10.1103/prxquantum.6.010336. URL http://dx.doi.org/10.1103/PRXQuantum.6.010336.
https://doi.org/10.1103/prxquantum.6.010336

[49] Alexei Kitaev and John Preskill. Topological entanglement entropy. Physical Review Letters, 96 (11), March 2006. ISSN 1079-7114. 10.1103/physrevlett.96.110404. URL http://dx.doi.org/10.1103/PhysRevLett.96.110404.
https://doi.org/10.1103/physrevlett.96.110404

[50] Guglielmo Lami and Mario Collura. Nonstabilizerness via perfect pauli sampling of matrix product states. Phys. Rev. Lett., 131: 180401, Oct 2023. 10.1103/PhysRevLett.131.180401.
https://doi.org/10.1103/PhysRevLett.131.180401

[51] Guglielmo Lami, Tobias Haug, and Jacopo De Nardis. Quantum state designs with clifford enhanced matrix product states, 2024. URL https://arxiv.org/abs/2404.18751.
arXiv:2404.18751

[52] Wonjun Lee, Minki Hhan, Gil Young Cho, and Hyukjoon Kwon. Shallow quantum circuit for generating o(1)-entangled approximate state designs, 2025. URL https://arxiv.org/abs/2507.17871.
arXiv:2507.17871

[53] Lorenzo Leone and Lennart Bittel. Stabilizer entropies are monotones for magic-state resource theory. Phys. Rev. A, 110: L040403, Oct 2024a. 10.1103/PhysRevA.110.L040403.
https://doi.org/10.1103/PhysRevA.110.L040403

[54] Lorenzo Leone and Lennart Bittel. Stabilizer entropies are monotones for magic-state resource theory. Phys. Rev. A, 110: L040403, Oct 2024b. 10.1103/PhysRevA.110.L040403.
https://doi.org/10.1103/PhysRevA.110.L040403

[55] Lorenzo Leone, Salvatore F. E. Oliviero, You Zhou, and Alioscia Hamma. Quantum Chaos is Quantum. Quantum, 5: 453–453, May 2021. 10.22331/q-2021-05-04-453.
https://doi.org/10.22331/q-2021-05-04-453

[56] Lorenzo Leone, Salvatore F. E. Oliviero, and Alioscia Hamma. Stabilizer Rényi Entropy. Physical Review Letters, 128: 050402–050402, February 2022. 10.1103/PhysRevLett.128.050402.
https://doi.org/10.1103/PhysRevLett.128.050402

[57] Lorenzo Leone, Salvatore F. E. Oliviero, Alioscia Hamma, Jens Eisert, and Lennart Bittel. The non-clifford cost of random unitaries, 2025. URL https://arxiv.org/abs/2505.10110.
arXiv:2505.10110

[58] Zi-Wen Liu and Andreas Winter. Many-Body Quantum Magic. PRX Quantum, 3: 020333–020333, May 2022. 10.1103/PRXQuantum.3.020333.
https://doi.org/10.1103/PRXQuantum.3.020333

[59] Beatrice Magni and Xhek Turkeshi. Quantum complexity and chaos in many-qudit doped clifford circuits, 2025. URL https://arxiv.org/abs/2506.02127. 10.22331/q-2025-12-24-1956.
https://doi.org/10.22331/q-2025-12-24-1956
arXiv:2506.02127

[60] Beatrice Magni, Alexios Christopoulos, Andrea De Luca, and Xhek Turkeshi. Anticoncentration in clifford circuits and beyond: From random tensor networks to pseudo-magic states, 2025a. URL https://arxiv.org/abs/2502.20455. 10.1103/p8dn-glcw.
https://doi.org/10.1103/p8dn-glcw
arXiv:2502.20455

[61] Beatrice Magni, Markus Heinrich, Lorenzo Leone, and Xhek Turkeshi. Anticoncentration and state design of doped real clifford circuits and tensor networks, 2025b. URL https://arxiv.org/abs/2512.15880.
arXiv:2512.15880

[62] Vikash Mittal and Yi-Ping Huang. Quantum magic in discrete-time quantum walk, 2025. URL https://arxiv.org/abs/2506.17783. 10.1103/7rwg-lhpv.
https://doi.org/10.1103/7rwg-lhpv
arXiv:2506.17783

[63] Vikash Mittal and Yi-Ping Huang. Quantum magic in discrete-time quantum walk. Physical Review Research, 8 (1), February 2026. ISSN 2643-1564. 10.1103/7rwg-lhpv. URL http://dx.doi.org/10.1103/7rwg-lhpv.
https://doi.org/10.1103/7rwg-lhpv

[64] J. Odavić, M. Viscardi, and A. Hamma. Stabilizer entropy in nonintegrable quantum evolutions, September 2025. ISSN 2469-9969. URL http://dx.doi.org/10.1103/y9r6-dx7p.
https://doi.org/10.1103/y9r6-dx7p

[65] Salvatore F. E. Oliviero, Lorenzo Leone, and Alioscia Hamma. Magic-state resource theory for the ground state of the transverse-field Ising model. Phys. Rev. A, 106 (4): 042426, October 2022a. 10.1103/PhysRevA.106.042426.
https://doi.org/10.1103/PhysRevA.106.042426

[66] Salvatore F. E. Oliviero, Lorenzo Leone, Alioscia Hamma, and Seth Lloyd. Measuring magic on a quantum processor. npj Quantum Inf, 8 (1): 1–8, December 2022b. ISSN 2056-6387. 10.1038/s41534-022-00666-5.
https://doi.org/10.1038/s41534-022-00666-5

[67] G. Passarelli, R. Fazio, and P. Lucignano. Nonstabilizerness of permutationally invariant systems. Phys. Rev. A, 110: 022436, Aug 2024. 10.1103/PhysRevA.110.022436.
https://doi.org/10.1103/PhysRevA.110.022436

[68] Gianluca Passarelli, Procolo Lucignano, Davide Rossini, and Angelo Russomanno. Chaos and magic in the dissipative quantum kicked top. Quantum, 9: 1653, March 2025a. ISSN 2521-327X. 10.22331/q-2025-03-05-1653. URL https://doi.org/10.22331/q-2025-03-05-1653.
https://doi.org/10.22331/q-2025-03-05-1653

[69] Gianluca Passarelli, Angelo Russomanno, and Procolo Lucignano. Nonstabilizerness of a boundary time crystal. Phys. Rev. A, 111: 062417, Jun 2025b. 10.1103/d7tm-9hkp.
https://doi.org/10.1103/d7tm-9hkp

[70] Patrick Rall, Daniel Liang, Jeremy Cook, and William Kretschmer. Simulation of qubit quantum circuits via pauli propagation. Physical Review A, 99 (6), June 2019. ISSN 2469-9934. 10.1103/physreva.99.062337. URL http://dx.doi.org/10.1103/PhysRevA.99.062337.
https://doi.org/10.1103/physreva.99.062337

[71] Davide Rattacaso, Lorenzo Leone, Salvatore F. E. Oliviero, and Alioscia Hamma. Stabilizer entropy dynamics after a quantum quench, October 2023. ISSN 2469-9934. URL http://dx.doi.org/10.1103/PhysRevA.108.042407.
https://doi.org/10.1103/PhysRevA.108.042407

[72] Caroline E. P. Robin and Martin J. Savage. Quantum complexity fluctuations from nuclear and hypernuclear forces. Physical Review C, 112 (4), October 2025. ISSN 2469-9993. 10.1103/r8rq-y9tb. URL http://dx.doi.org/10.1103/r8rq-y9tb.
https://doi.org/10.1103/r8rq-y9tb

[73] Angelo Russomanno, Gianluca Passarelli, Davide Rossini, and Procolo Lucignano. Nonstabilizerness in the unitary and monitored quantum dynamics of xxz-staggered and sachdev-ye-kitaev models. Physical Review B, 112 (6), August 2025. ISSN 2469-9969. 10.1103/njgn-fksh. URL http://dx.doi.org/10.1103/njgn-fksh.
https://doi.org/10.1103/njgn-fksh

[74] Matthieu Sarkis and Alexandre Tkatchenko. Are molecules magical? non-stabilizerness in molecular bonding, 2025. URL https://arxiv.org/abs/2504.06673.
arXiv:2504.06673

[75] Annarita Scocco, Wai-Keong Mok, Leandro Aolita, Mario Collura, and Tobias Haug. Rise and fall of nonstabilizerness via random measurements, 2025. URL https://arxiv.org/abs/2507.11619. 10.1103/31sq-k4m3.
https://doi.org/10.1103/31sq-k4m3
arXiv:2507.11619

[76] Doru Sticlet, Balázs Dóra, Dominik Szombathy, Gergely Zaránd, and Cătălin Paşcu Moca. Nonstabilizerness in open xxz spin chains: Universal scaling and dynamics. Physical Review Research, 7 (4), November 2025. ISSN 2643-1564. 10.1103/96bk-xf8p. URL http://dx.doi.org/10.1103/96bk-xf8p.
https://doi.org/10.1103/96bk-xf8p

[77] Poetri Sonya Tarabunga. Critical behaviors of non-stabilizerness in quantum spin chains. Quantum, 8: 1413, July 2024. ISSN 2521-327X. 10.22331/q-2024-07-17-1413. URL https://doi.org/10.22331/q-2024-07-17-1413.
https://doi.org/10.22331/q-2024-07-17-1413

[78] Poetri Sonya Tarabunga, Emanuele Tirrito, Mari Carmen Bañuls, and Marcello Dalmonte. Nonstabilizerness via matrix product states in the pauli basis. Phys. Rev. Lett., 133: 010601, Jul 2024. 10.1103/PhysRevLett.133.010601.
https://doi.org/10.1103/PhysRevLett.133.010601

[79] Emanuele Tirrito, Xhek Turkeshi, and Piotr Sierant. Anticoncentration and magic spreading under ergodic quantum dynamics, 2024. URL https://arxiv.org/abs/2412.10229.
arXiv:2412.10229

[80] Emanuele Tirrito, Luca Lumia, Alessio Paviglianiti, Guglielmo Lami, Alessandro Silva, Xhek Turkeshi, and Mario Collura. Magic phase transitions in monitored gaussian fermions, 2025a. URL https://arxiv.org/abs/2507.07179.
arXiv:2507.07179

[81] Emanuele Tirrito, Poetri Sonya Tarabunga, Devendra Singh Bhakuni, Marcello Dalmonte, Piotr Sierant, and Xhek Turkeshi. Universal spreading of nonstabilizerness and quantum transport, 2025b. URL https://arxiv.org/abs/2506.12133.
arXiv:2506.12133

[82] Xhek Turkeshi, Anatoly Dymarsky, and Piotr Sierant. Pauli spectrum and nonstabilizerness of typical quantum many-body states. Phys. Rev. B, 111: 054301, Feb 2025a. 10.1103/PhysRevB.111.054301.
https://doi.org/10.1103/PhysRevB.111.054301

[83] Xhek Turkeshi, Emanuele Tirrito, and Piotr Sierant. Magic spreading in random quantum circuits. Nature Communications, 16 (1), March 2025b. ISSN 2041-1723. 10.1038/s41467-025-57704-x. URL http://dx.doi.org/10.1038/s41467-025-57704-x.
https://doi.org/10.1038/s41467-025-57704-x

[84] Naga Dileep Varikuti, Soumik Bandyopadhyay, and Philipp Hauke. Impact of clifford operations on non-stabilizing power and quantum chaos. Quantum, 10: 2017, March 2026. ISSN 2521-327X. 10.22331/q-2026-03-10-2017. URL http://dx.doi.org/10.22331/q-2026-03-10-2017.
https://doi.org/10.22331/q-2026-03-10-2017

[85] Michele Viscardi, Marcello Dalmonte, Alioscia Hamma, and Emanuele Tirrito. Interplay of entanglement structures and stabilizer entropy in spin models. SciPost Physics Core, 9 (1), February 2026. ISSN 2666-9366. 10.21468/scipostphyscore.9.1.012. URL http://dx.doi.org/10.21468/SciPostPhysCore.9.1.012.
https://doi.org/10.21468/scipostphyscore.9.1.012

[86] Xin Wang, Mark M. Wilde, and Yuan Su. Efficiently computable bounds for magic state distillation. Physical Review Letters, 124 (9), March 2020. ISSN 1079-7114. 10.1103/physrevlett.124.090505. URL http://dx.doi.org/10.1103/PhysRevLett.124.090505.
https://doi.org/10.1103/physrevlett.124.090505

[87] Chris D. White and Martin J. White. Magic states of top quarks. Phys. Rev. D, 110: 116016, Dec 2024. 10.1103/PhysRevD.110.116016.
https://doi.org/10.1103/PhysRevD.110.116016

[88] Huangjun Zhu, Richard Kueng, Markus Grassl, and David Gross. The clifford group fails gracefully to be a unitary 4-design, 2016. URL https://arxiv.org/abs/1609.08172.
arXiv:1609.08172

Cited by

[1] Daniel Miller, Kyano Levi, Lukas Postler, Alex Steiner, Lennart Bittel, Gregory A. L. White, Yifan Tang, Eric J. Kuehnke, Antonio A. Mele, Sumeet Khatri, Lorenzo Leone, Jose Carrasco, Christian D. Marciniak, Ivan Pogorelov, Milena Guevara-Bertsch, Robert Freund, Rainer Blatt, Philipp Schindler, Thomas Monz, Martin Ringbauer, and Jens Eisert, "Experimental measurement and a physical interpretation of quantum shadow enumerators", Physical Review Research 8 2, 023318 (2026).

[2] Lennart Bittel and Lorenzo Leone, "Adaptively Secure Unitary Designs with Constant Non-Clifford Cost", Physical Review Letters 136 21, 210802 (2026).

[3] Marcel Hinsche, Zongbo Bao, Philippe van Dordrecht, Jens Eisert, Jop Briët, and Jonas Helsen, Proceedings of the 58th Annual ACM Symposium on Theory of Computing 869 (2026) ISBN:9798400725364.

[4] Caroline E. P. Robin and Martin J. Savage, "Antiflatness and nonlocal nonstabilizerness in two-particle scattering processes", Physical Review D 114 1, 014007 (2026).

[5] Zong-Yue Hou, ChunJun Cao, and Zhi-Cheng Yang, "Stabilizer entanglement enhances magic injection", npj Quantum Information 12 1, 113 (2026).

[6] Wenjing Hou, Zhanyuan Fu, Lei Li, Yiran Wang, and Yongming Li, "The convolution-based quantum Jensen-Shannon divergence and its application in quantifying magic resource", Physics Letters A 589, 131793 (2026).

[7] Beatrice Magni, Markus Heinrich, Lorenzo Leone, and Xhek Turkeshi, "Anticoncentration and state design of doped real Clifford circuits and tensor networks", Physical Review A 113 6, 062446 (2026).

[8] Masahiro Hoshino, Masaki Oshikawa, and Yuto Ashida, "Stabilizer Rényi Entropy and Conformal Field Theory", Physical Review X 16 1, 011037 (2026).

[9] Stefano Cusumano, Lorenzo Campos Venuti, Simone Cepollaro, Immacolata De Simone, Gianluca Esposito, Daniele Iannotti, Barbara Jasser, Jovan Odavi\' c, Michele Viscardi, and Alioscia Hamma, "Non-stabilizerness and violations of CHSH inequalities", arXiv:2504.03351, (2025).

[10] Hugo Lóio, Guglielmo Lami, Lorenzo Leone, Max McGinley, Xhek Turkeshi, and Jacopo De Nardis, "Quantum State Designs via Magic Teleportation", arXiv:2510.13950, (2025).

[11] Piotr Sierant, Jofre Vallès-Muns, and Artur Garcia-Saez, "Computing quantum magic of state vectors", Quantum 10, 2059 (2026).

[12] Qiaofeng Liu, Ian Low, and Zhewei Yin, "Maximal magic for two-qubit states", Quantum Science and Technology 11 1, 015035 (2026).

[13] Marcel Hinsche, Zongbo Bao, Philippe van Dordrecht, Jens Eisert, Jop Briët, and Jonas Helsen, "Clifford testing: algorithms and lower bounds", arXiv:2510.07164, (2025).

[14] C. E. P. Robin and M. J. Savage, "Anti-Flatness and Non-Local Magic in Two-Particle Scattering Processes", arXiv:2510.23426, (2025).

[15] Daniele Iannotti, Lorenzo Campos Venuti, and Alioscia Hamma, "Van Hove singularities in stabilizer entropy densities", Journal of Physics A Mathematical General 59 7, 075301 (2026).

[16] Zhiyao Li, Marc Illa, and Martin J. Savage, "A Framework for Quantum Simulations of Energy-Loss and Hadronization in Non-Abelian Gauge Theories: SU(2) Lattice Gauge Theory in 1+1D", arXiv:2512.05210, (2025).

[17] Wonjun Lee, Minki Hhan, Gil Young Cho, and Hyukjoon Kwon, "Shallow Quantum Circuit for Generating Extremely Low-Entangled Approximate State Designs", Physical Review Letters 136 15, 150603 (2026).

[18] Piotr Sierant and Xhek Turkeshi, "Theory of Magic Phase Transitions in Encoding-Decoding Circuits", arXiv:2603.00235, (2026).

[19] Naga Dileep Varikuti, Soumik Bandyopadhyay, and Philipp Hauke, "Deep Thermalization and Measurements of Quantum Resources", arXiv:2512.09999, (2025).

[20] Lorenzo Leone, Jens Eisert, and Salvatore F. E. Oliviero, "The unbearable hardness of deciding about magic", arXiv:2602.22330, (2026).

[21] Benjamin Stratton, "Algorithm for estimating <inline-formula><mml:math><mml:mi>α</mml:mi></mml:math></inline-formula>-stabilizer Rényi entropies via purity", Physical Review A 113 2, 022434 (2026).

[22] Han-Ze Li, Yi-Rui Zhang, Yu-Jun Zhao, Xuyang Huang, and Jian-Xin Zhong, "Slow growth of quantum nonstabilizerness in disorder-free Stark many-body localization", Physical Review B 113 10, 104305 (2026).

[23] Ryota Matsuda, Masahiro Hoshino, and Yuto Ashida, "Quantum Computational Resources and Conformal Field Theory: Unifying Spins, Bosons, and Fermions", arXiv:2607.05343, (2026).

[24] Wonjun Lee, Hyukjoon Kwon, and Gil Young Cho, "Gluing Randomness via Entanglement: Tight Bound from Second Rényi Entropy", arXiv:2601.16454, (2026).

[25] Daniele Iannotti, Angelo Russotto, Barbara Jasser, Jovan Odavić, and Alioscia Hamma, "Non-stabilizerness and U(1) symmetry in chaotic many-body quantum systems", arXiv:2603.28870, (2026).

[26] Hsueh-Hao Lu, Yasunari Suzuki, Yasunobu Nakamura, and En-Jui Kuo, "Quantum Magic in early FTQC: From Diagonal Clifford Hierarchy No-Go Theorems to Architecture Design Blueprints", arXiv:2605.04758, (2026).

[27] Poetri Sonya Tarabunga, Bernhard Jobst, Raúl Morral-Yepes, Marc Langer, Barbara Kraus, Frank Pollmann, and Sheng-Hsuan Lin, "Computable fermionic non-Gaussianity from the covariance matrix", arXiv:2607.02242, (2026).

[28] Yi-Rui Zhang, Han-Ze Li, Xu-Yang Huang, Yu-Jun Zhao, and Jian-Xin Zhong, "Stark Many-Body Localization-Induced Quantum Mpemba Effect", Entropy 28 3, 348 (2026).

[29] Alejandro Borda Kuhlmann and Julian Rincon, "Magic-protected entanglement and Clifford-irreducible structure in magic state space", arXiv:2607.18400, (2026).

[30] Gianluca Esposito, Michele Viscardi, and Alioscia Hamma, "Stabilizer entropy is trustworthy for mixed states", arXiv:2606.29443, (2026).

[31] Jinwei Chu and Savan Kharel, "Magic without a phase: phase-independent stabilizer Rényi entropy in gluon scattering", arXiv:2607.13134, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-17 23:54:53) and SAO/NASA ADS (last updated successfully 2026-08-17 23:54:54). The list may be incomplete as not all publishers provide suitable and complete citation data.

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.