Computing quantum magic of state vectors

Piotr Sierant; Jofre Vallès-Muns; Artur Garcia-Saez

doi:10.22331/q-2026-04-10-2059

Computing quantum magic of state vectors

Piotr Sierant¹, Jofre Vallès-Muns¹, and Artur Garcia-Saez^1,2

¹Barcelona Supercomputing Center, Barcelona 08034, Spain
²Qilimanjaro Quantum Tech, 08019 Barcelona, Spain

Published:	2026-04-10, volume 10, page 2059
Editor:	Dax Enshan Koh
Eprint:	arXiv:2601.07824v3
Doi:	https://doi.org/10.22331/q-2026-04-10-2059
Citation:	Quantum 10, 2059 (2026).

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

Abstract

Non-stabilizerness, also known as “magic,'' quantifies how far a quantum state departs from the stabilizer set. It is a central resource behind quantum advantage and a useful probe of the complexity of quantum many-body states. Yet standard magic quantifiers, such as the stabilizer Rényi entropy (SRE) for qubits and the mana for qutrits, are costly to evaluate numerically, with the computational complexity growing rapidly with the number $N$ of qudits. Here we introduce efficient, numerically exact algorithms that exploit the fast Hadamard transform to compute the SRE for qubits ($d=2$) and the mana for qutrits ($d=3$) for pure states given as state vectors. Our methods compute SRE and mana at cost $O(N d^{2N})$, providing an exponential improvement over the naive $O(d^{3N})$ scaling, with substantial parallelism and straightforward GPU acceleration. We further show how to combine the fast Hadamard transform with Monte Carlo sampling to estimate the SRE of state vectors, and we extend the approach to compute the mana of mixed states. All algorithms are implemented in the open-source Julia package HadaMAG, which provides a high-performance toolbox for computing SRE and mana with built-in support for multithreading, MPI-based distributed parallelism, and GPU acceleration. The package, together with the methods developed in this work, offers a practical route to large-scale numerical studies of magic in quantum many-body systems.

Featured image: HadaMAG workflow: a quantum state vector $|\psi\rangle$ with $d^N$ amplitudes is fed through $d^N$ fast Hadamard transforms, i.e., butterfly networks of additions and subtractions, to efficiently extract all $d^{2N}$ Pauli expectation values $\langle P \rangle$, from which measures of quantum magic, the stabilizer Rényi entropy $M_2(|\psi\rangle)$ for qubits ($d=2$) and the mana $\mathcal{M}(|\psi\rangle)$ for qutrits ($d=3$), are obtained.

Popular summary

Stabilizer states form a special class of quantum states that align with a discrete set of privileged directions in Hilbert space and can therefore be simulated efficiently on a classical computer. Magic, or non-stabilizerness, measures how far a state departs from this classically tractable set, and is a key resource behind the enhanced computational power of quantum systems. Characterizing this feature in concrete many-body states requires computing suitable measures of magic. Yet this quickly becomes difficult in practice, because standard magic measures require summing exponentially many expectation values. Here we show that this computation can be reorganized using fast Hadamard and Fourier transforms, yielding exact algorithms that are exponentially faster than straightforward approaches. This enables the computation of measures of magic: "stabilizer Rényi entropy" for qubits and "mana" for qutrits in significantly larger systems than previously accessible from state-vector data. We also develop approximate sampling methods and extend the same framework to mixed states. All methods are implemented in the open-source Julia package HadaMAG.jl, providing a practical toolbox for large-scale studies of quantum magic.

► BibTeX data

@article{Sierant2026computingquantum,
  doi = {10.22331/q-2026-04-10-2059},
  url = {https://doi.org/10.22331/q-2026-04-10-2059},
  title = {Computing quantum magic of state vectors},
  author = {Sierant, Piotr and Vall{\`{e}}s-Muns, Jofre and Garcia-Saez, Artur},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2059},
  month = apr,
  year = {2026}
}

► References

[1] J. Preskill, ``Quantum computing and the entanglement frontier,'' (2012), arXiv:1203.5813.
https://doi.org/10.48550/arXiv.1203.5813
arXiv:1203.5813

[2] 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

[3] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003).
https://doi.org/10.1103/PhysRevLett.91.147902

[4] R. Orús, Ann. Phys. 349, 117 (2014).
https://doi.org/10.1016/j.aop.2014.06.013

[5] R. Orús, Nat. Rev. Phys. 1, 538–550 (2019).
https://doi.org/10.1038/s42254-019-0086-7

[6] S.-J. Ran, E. Tirrito, C. Peng, X. Chen, L. Tagliacozzo, G. Su, and M. Lewenstein, Tensor network contractions: methods and applications to quantum many-body systems (Springer Nature, 2020).
https://doi.org/10.1007/978-3-030-34489-4

[7] M. C. Bañuls, Annu. Rev. Condens. Matter Phys. 14, 173–191 (2023).
https://doi.org/10.1146/annurev-conmatphys-040721-022705

[8] F. Verstraete, V. Murg, and J. Cirac, Adv. Phys. 57, 143 (2008).
https://doi.org/10.1080/14789940801912366

[9] N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 100, 030504 (2008).
https://doi.org/10.1103/PhysRevLett.100.030504

[10] U. Schollwöck, , 96 (2011).

[11] L. G. Valiant, in Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, STOC '01 (Association for Computing Machinery, New York, NY, USA, 2001) p. 114–123.
https://doi.org/10.1145/380752.380785

[12] B. M. Terhal and D. P. DiVincenzo, Phys. Rev. A 65, 032325 (2002).
https://doi.org/10.1103/PhysRevA.65.032325

[13] S. Bravyi, Quantum Info. Comput. 5, 216–238 (2005).
https://doi.org/10.48550/arXiv.quant-ph/0404180
arXiv:quant-ph/0404180

[14] D. Gottesman, ``The heisenberg representation of quantum computers,'' (1998), arXiv:quant-ph/9807006.
https://doi.org/10.48550/arXiv.quant-ph/9807006
arXiv:quant-ph/9807006

[15] S. Aaronson and D. Gottesman, Phys. Rev. A 70, 052328 (2004).
https://doi.org/10.1103/PhysRevA.70.052328

[16] P. Sierant, P. Stornati, and X. Turkeshi, PRX Quantum 7, 010302 (2026).
https://doi.org/10.1103/3yx4-1j27

[17] Z.-W. Liu and A. Winter, PRX Quantum 3, 020333 (2022).
https://doi.org/10.1103/PRXQuantum.3.020333

[18] V. Veitch, S. A. H. Mousavian, D. Gottesman, and J. Emerson, New J. Phys. 16, 013009 (2014).
https://doi.org/10.1088/1367-2630/16/1/013009

[19] A. Heimendahl, M. Heinrich, and D. Gross, Journal of Mathematical Physics 63, 112201 (2022).
https://doi.org/10.1063/5.0085774

[20] T. Haug and L. Piroli, Quantum 7, 1092 (2023a).
https://doi.org/10.22331/q-2023-08-28-1092

[21] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (2000).
https://doi.org/10.1017/CBO9780511976667

[22] S. Bravyi and A. Kitaev, Phys. Rev. A 71, 022316 (2005).
https://doi.org/10.1103/PhysRevA.71.022316

[23] A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Phys. Rev. A 52, 3457 (1995).
https://doi.org/10.1103/PhysRevA.52.3457

[24] S. Bravyi, G. Smith, and J. A. Smolin, Phys. Rev. X 6, 021043 (2016).
https://doi.org/10.1103/PhysRevX.6.021043

[25] S. Bravyi and D. Gosset, Phys. Rev. Lett. 116, 250501 (2016).
https://doi.org/10.1103/PhysRevLett.116.250501

[26] S. Bravyi, D. Browne, P. Calpin, E. Campbell, D. Gosset, and M. Howard, Quantum 3, 181 (2019).
https://doi.org/10.22331/q-2019-09-02-181

[27] M. Howard and E. Campbell, Phys. Rev. Lett. 118, 090501 (2017).
https://doi.org/10.1103/PhysRevLett.118.090501

[28] M. Heinrich and D. Gross, Quantum 3, 132 (2019).
https://doi.org/10.22331/q-2019-04-08-132

[29] S. Sarkar, C. Mukhopadhyay, and A. Bayat, New Journal of Physics 22, 083077 (2020).
https://doi.org/10.1088/1367-2630/aba919

[30] H. Hamaguchi, K. Hamada, and N. Yoshioka, Quantum 8, 1461 (2024).
https://doi.org/10.22331/q-2024-09-05-1461

[31] L. Leone, S. F. E. Oliviero, and A. Hamma, Phys. Rev. Lett. 128, 050402 (2022).
https://doi.org/10.1103/PhysRevLett.128.050402

[32] L. Leone and L. Bittel, Phys. Rev. A 110, L040403 (2024).
https://doi.org/10.1103/PhysRevA.110.L040403

[33] T. Haug and L. Piroli, Phys. Rev. B 107, 035148 (2023b).
https://doi.org/10.1103/PhysRevB.107.035148

[34] P. S. Tarabunga, E. Tirrito, T. Chanda, and M. Dalmonte, PRX Quantum 4, 040317 (2023).
https://doi.org/10.1103/PRXQuantum.4.040317

[35] J. Odavić, T. Haug, G. Torre, A. Hamma, F. Franchini, and S. M. Giampaolo, SciPost Phys. 15, 131 (2023).
https://doi.org/10.21468/SciPostPhys.15.4.131

[36] G. Passarelli, R. Fazio, and P. Lucignano, Phys. Rev. A 110, 022436 (2024).
https://doi.org/10.1103/PhysRevA.110.022436

[37] P. S. Tarabunga and C. Castelnovo, Quantum 8, 1347 (2024).
https://doi.org/10.22331/q-2024-05-14-1347

[38] P. R. N. Falcão, P. S. Tarabunga, M. Frau, E. Tirrito, J. Zakrzewski, and M. Dalmonte, Phys. Rev. B 111, L081102 (2025a).
https://doi.org/10.1103/PhysRevB.111.L081102

[39] B. Jasser, J. Odavić, and A. Hamma, Phys. Rev. B 112, 174204 (2025).
https://doi.org/10.1103/rz86-47h3

[40] S. Bera and M. Schirò, SciPost Phys. 19, 159 (2025).
https://doi.org/10.21468/SciPostPhys.19.6.159

[41] Y.-M. Ding, Z. Wang, and Z. Yan, PRX Quantum 6, 030328 (2025).
https://doi.org/10.1103/pyzr-jmvw

[42] M. Hoshino, M. Oshikawa, and Y. Ashida, Phys. Rev. X 16, 011037 (2026).
https://doi.org/10.1103/ylsz-dm3y

[43] M. Hoshino and Y. Ashida, Phys. Rev. Lett. 136, 080402 (2026).
https://doi.org/10.1103/1tyr-rlbb

[44] D. Rattacaso, L. Leone, S. F. E. Oliviero, and A. Hamma, Phys. Rev. A 108, 042407 (2023).
https://doi.org/10.1103/PhysRevA.108.042407

[45] X. Turkeshi, A. Dymarsky, and P. Sierant, Phys. Rev. B 111, 054301 (2025a).
https://doi.org/10.1103/PhysRevB.111.054301

[46] X. Turkeshi, E. Tirrito, and P. Sierant, Nature Communications 16, 2575 (2025b).
https://doi.org/10.1038/s41467-025-57704-x

[47] E. Tirrito, X. Turkeshi, and P. Sierant, Phys. Rev. Lett. 135, 220401 (2025a).
https://doi.org/10.1103/1jzy-sk9r

[48] J. Odavić, M. Viscardi, and A. Hamma, Phys. Rev. B 112, 104301 (2025).
https://doi.org/10.1103/y9r6-dx7p

[49] T. Haug, L. Aolita, and M. Kim, Quantum 9, 1801 (2025).
https://doi.org/10.22331/q-2025-07-21-1801

[50] J. A. Montañà López and P. Kos, J. Phys. A: Math. and Theor. 57, 475301 (2024).
https://doi.org/10.1088/1751-8121/ad85b0

[51] P. S. Tarabunga and E. Tirrito, npj Quantum Information 11, 166 (2025).
https://doi.org/10.1038/s41534-025-01104-y

[52] G. C. Santra, A. Windey, S. Bandyopadhyay, A. Legramandi, and P. Hauke, ``Complexity transitions in chaotic quantum systems,'' (2025), arXiv:2505.09707 [quant-ph].
arXiv:2505.09707

[53] G. Passarelli, A. Russomanno, and P. Lucignano, Phys. Rev. A 111, 062417 (2025).
https://doi.org/10.1103/d7tm-9hkp

[54] Z.-Y. Hou, C. Cao, and Z.-C. Yang, ``Stabilizer entanglement as a magic highway,'' (2025), arXiv:2503.20873 [quant-ph].
arXiv:2503.20873

[55] Y. Zhang and Y. Gu, ``Quantum magic dynamics in random circuits,'' (2024), arXiv:2410.21128 [quant-ph].
arXiv:2410.21128

[56] B. Magni, A. Christopoulos, A. De Luca, and X. Turkeshi, Phys. Rev. X 15, 031071 (2025).
https://doi.org/10.1103/p8dn-glcw

[57] B. Magni and X. Turkeshi, Quantum 9, 1956 (2025).
https://doi.org/10.22331/q-2025-12-24-1956

[58] Y. Zhang, S. Vijay, Y. Gu, and Y. Bao, PRX Quantum 7, 010344 (2026).
https://doi.org/10.1103/myrb-nyhf

[59] D. Szombathy, A. Valli, C. P. Moca, L. Farkas, and G. Zaránd, Phys. Rev. Res. 7, 043072 (2025).
https://doi.org/10.1103/jplh-zl35

[60] S. F. E. Oliviero, L. Leone, A. Hamma, and S. Lloyd, npj Quantum Info. 8, 148 (2022).
https://doi.org/10.1038/s41534-022-00666-5

[61] T. Haug and M. Kim, PRX Quantum 4, 010301 (2023).
https://doi.org/10.1103/PRXQuantum.4.010301

[62] T. Haug, S. Lee, and M. S. Kim, Phys. Rev. Lett. 132, 240602 (2024).
https://doi.org/10.1103/PhysRevLett.132.240602

[63] R. Sarkar and T. J. Yoder, Quantum 8, 1307 (2024).
https://doi.org/10.22331/q-2024-04-04-1307

[64] D. Gross, J. Math. Phys. 47, 122107 (2006).
https://doi.org/10.1063/1.2393152

[65] D. Gross, Applied Physics B 86, 367 (2007).
https://doi.org/10.1007/s00340-006-2510-9

[66] V. Veitch, C. Ferrie, D. Gross, and J. Emerson, New Journal of Physics 14, 113011 (2012).
https://doi.org/10.1088/1367-2630/14/11/113011

[67] H. Pashayan, J. J. Wallman, and S. D. Bartlett, Phys. Rev. Lett. 115, 070501 (2015).
https://doi.org/10.1103/PhysRevLett.115.070501

[68] C. D. White, C. Cao, and B. Swingle, Phys. Rev. B 103, 075145 (2021).
https://doi.org/10.1103/PhysRevB.103.075145

[69] P. S. Tarabunga, Quantum 8, 1413 (2024).
https://doi.org/10.22331/q-2024-07-17-1413

[70] C. D. White and J. H. Wilson, ``Mana in haar-random states,'' (2020), arXiv:2011.13937 [quant-ph].
arXiv:2011.13937

[71] K. Goto, T. Nosaka, and M. Nozaki, Phys. Rev. D 106, 126009 (2022).
https://doi.org/10.1103/PhysRevD.106.126009

[72] T. J. Sewell and C. D. White, Phys. Rev. B 106, 125130 (2022).
https://doi.org/10.1103/PhysRevB.106.125130

[73] N. Koukoulekidis and D. Jennings, npj Quantum Information 8, 42 (2022).
https://doi.org/10.1038/s41534-022-00551-1

[74] R. Basu, A. Ganguly, S. Nath, and O. Parrikar, Journal of High Energy Physics 2024, 264 (2024).
https://doi.org/10.1007/JHEP05(2024)264

[75] A. Ahmadi and E. Greplova, SciPost Phys. 16, 043 (2024).
https://doi.org/10.21468/SciPostPhys.16.2.043

[76] R. Nyström, N. Pranzini, and E. Keski-Vakkuri, Phys. Rev. Res. 7, 033085 (2025).
https://doi.org/10.1103/9ph7-cyzh

[77] C. E. P. Robin and M. J. Savage, Phys. Rev. C 112, 044004 (2025).
https://doi.org/10.1103/r8rq-y9tb

[78] F. Brökemeier, S. M. Hengstenberg, J. W. T. Keeble, C. E. P. Robin, F. Rocco, and M. J. Savage, Phys. Rev. C 111, 034317 (2025).
https://doi.org/10.1103/PhysRevC.111.034317

[79] C. D. White and M. J. White, Phys. Rev. D 110, 116016 (2024).
https://doi.org/10.1103/PhysRevD.110.116016

[80] I. Chernyshev, C. E. P. Robin, and M. J. Savage, Phys. Rev. Res. 7, 023228 (2025).
https://doi.org/10.1103/PhysRevResearch.7.023228

[81] Q. Liu, I. Low, and Z. Yin, ``Quantum magic in quantum electrodynamics,'' (2025a), arXiv:2503.03098 [quant-ph].
arXiv:2503.03098

[82] A. Gu, H.-Y. Hu, D. Luo, T. L. Patti, N. C. Rubin, and S. F. Yelin, Quantum 8, 1422 (2024).
https://doi.org/10.22331/q-2024-07-23-1422

[83] C. Cao, G. Cheng, A. Hamma, L. Leone, W. Munizzi, and S. F. Oliviero, PRX Quantum 6, 040375 (2025).
https://doi.org/10.1103/z3vr-w5c5

[84] F. B. Trigueros, ``Stabilizer renyi entropies,'' https://github.com/FabianBallar7/StabilizerRenyiEntropy (2025).
https://github.com/FabianBallar7/StabilizerRenyiEntropy

[85] F. B. Trigueros and J. A. M. Guzmán, ``Nonstabilizerness and error resilience in noisy quantum circuits,'' (2025), arXiv:2506.18976 [quant-ph].
arXiv:2506.18976

[86] G. Lami and M. Collura, Phys. Rev. Lett. 131, 180401 (2023).
https://doi.org/10.1103/PhysRevLett.131.180401

[87] P. S. Tarabunga, E. Tirrito, M. C. Bañuls, and M. Dalmonte, Phys. Rev. Lett. 133, 010601 (2024).
https://doi.org/10.1103/PhysRevLett.133.010601

[88] L.-Y.-N. Liu, S. Yi, and J. Cui, ``Stabilizer rényi entropy for translation-invariant matrix product states,'' (2025b), arXiv:2508.03534 [quant-ph].
arXiv:2508.03534

[89] D. J. Luitz and Y. Bar Lev, Phys. Rev. B 96, 020406 (2017).
https://doi.org/10.1103/PhysRevB.96.020406

[90] P. Sierant, M. Lewenstein, A. Scardicchio, L. Vidmar, and J. Zakrzewski, Reports on Progress in Physics 88, 026502 (2025).
https://doi.org/10.1088/1361-6633/ad9756

[91] P. Sierant, M. Lewenstein, and J. Zakrzewski, Phys. Rev. Lett. 125, 156601 (2020).
https://doi.org/10.1103/PhysRevLett.125.156601

[92] A. Morningstar, L. Colmenarez, V. Khemani, D. J. Luitz, and D. A. Huse, Phys. Rev. B 105, 174205 (2022).
https://doi.org/10.1103/PhysRevB.105.174205

[93] D. J. Luitz and Y. Bar Lev, Phys. Rev. A 99, 010105 (2019).
https://doi.org/10.1103/PhysRevA.99.010105

[94] N. Defenu, A. Lerose, and S. Pappalardi, Physics Reports 1074, 1 (2024), out-of-equilibrium dynamics of quantum many-body systems with long-range interactions.
https://doi.org/10.1016/j.physrep.2024.04.005

[95] M. Brenes, V. K. Varma, A. Scardicchio, and I. Girotto, Computer Physics Communications 235, 477 (2019).
https://doi.org/10.1016/j.cpc.2018.08.010

[96] Fino and Algazi, IEEE Transactions on Computers C-25, 1142 (1976).
https://doi.org/10.1109/TC.1976.1674569

[97] L. Bittel and L. Leone, ``Operational interpretation of the stabilizer entropy,'' (2025), arXiv:2507.22883 [quant-ph].
arXiv:2507.22883

[98] D. E. Knuth, The art of computer programming, Vol. 3 (Pearson Education, 1997).

[99] P. S. Tarabunga and T. Haug, SciPost Phys. 19, 085 (2025a).
https://doi.org/10.21468/SciPostPhys.19.4.085

[100] J. de Boer, J. Järvelä, and E. Keski-Vakkuri, Phys. Rev. D 99, 066012 (2019).
https://doi.org/10.1103/PhysRevD.99.066012

[101] H. Yao and X.-L. Qi, Phys. Rev. Lett. 105, 080501 (2010).
https://doi.org/10.1103/PhysRevLett.105.080501

[102] J. Schliemann, Phys. Rev. B 83, 115322 (2011).
https://doi.org/10.1103/PhysRevB.83.115322

[103] M. P. Zaletel, J. H. Bardarson, and J. E. Moore, Phys. Rev. Lett. 107, 020402 (2011).
https://doi.org/10.1103/PhysRevLett.107.020402

[104] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953).
https://doi.org/10.1063/1.1699114

[105] W. K. Hastings, Biometrika 57, 97 (1970).
https://doi.org/10.2307/2334940

[106] T. Haug and P. S. Tarabunga, npj Quantum Information 12, 40 (2026).
https://doi.org/10.1038/s41534-026-01189-z

[107] L. Hantzko, L. Binkowski, and S. Gupta, Physica Scripta 99, 085128 (2024).
https://doi.org/10.1088/1402-4896/ad6499

[108] T. Jones, ``Decomposing dense matrices into dense Pauli tensors,'' arXiv preprint (2024), arXiv:2401.16378 [quant-ph].
arXiv:2401.16378

[109] S. Vidal Romero and J. Santos-Suárez, Quantum Information Processing 22, 449 (2023).
https://doi.org/10.1007/s11128-023-04204-w

[110] F. Mezzadri, ``How to generate random matrices from the classical compact groups,'' (2007), arXiv:math-ph/0609050 [math-ph].
https://doi.org/10.48550/arXiv.math-ph/0609050
arXiv:math-ph/0609050

[111] X. Turkeshi, M. Schirò, and P. Sierant, Phys. Rev. A 108, 042408 (2023).
https://doi.org/10.1103/PhysRevA.108.042408

[112] S. Aditya, X. Turkeshi, and P. Sierant, ``Growth and spreading of quantum resources under random circuit dynamics,'' (2025a), arXiv:2512.14827 [quant-ph].
arXiv:2512.14827

[113] P. S. Tarabunga and T. Haug, SciPost Phys. 19, 085 (2025b).
https://doi.org/10.21468/SciPostPhys.19.4.085

[114] E. Tirrito, P. S. Tarabunga, D. S. Bhakuni, M. Dalmonte, P. Sierant, and X. Turkeshi, ``Universal spreading of nonstabilizerness and quantum transport,'' (2025b), arXiv:2506.12133 [quant-ph].
arXiv:2506.12133

[115] S. Aditya, A. Summer, P. Sierant, and X. Turkeshi, ``Mpemba effects in quantum complexity,'' (2025b), arXiv:2509.22176 [quant-ph].
arXiv:2509.22176

[116] H. Tal‐Ezer and R. Kosloff, J. Chem. Phys. 81, 3967 (1984).
https://doi.org/10.1063/1.448136

[117] P. R. N. Falcão, P. Sierant, J. Zakrzewski, and E. Tirrito, Phys. Rev. Lett. 135, 240404 (2025b).
https://doi.org/10.1103/xfp5-hhs4

[118] M. Serbyn, D. A. Abanin, and Z. Papić, Nature Physics 17, 675 (2021).
https://doi.org/10.1038/s41567-021-01230-2

[119] R. Smith, Z. Papić, and A. Hallam, Phys. Rev. B 111, 245148 (2025).
https://doi.org/10.1103/jz4d-vdhj

[120] G. E. Fux, E. Tirrito, M. Dalmonte, and R. Fazio, Phys. Rev. Res. 6, L042030 (2024).
https://doi.org/10.1103/PhysRevResearch.6.L042030

[121] M. Bejan, C. McLauchlan, and B. Béri, PRX Quantum 5, 030332 (2024).
https://doi.org/10.1103/PRXQuantum.5.030332

[122] A. Paviglianiti, G. Lami, M. Collura, and A. Silva, PRX Quantum 6, 030320 (2025).
https://doi.org/10.1103/msm2-vmg7

[123] A. Scocco, W.-K. Mok, L. Aolita, M. Collura, and T. Haug, ``Rise and fall of nonstabilizerness via random measurements,'' (2025), arXiv:2507.11619 [quant-ph].
arXiv:2507.11619

[124] H. Lóio, G. Lami, L. Leone, M. McGinley, X. Turkeshi, and J. D. Nardis, ``Quantum state designs via magic teleportation,'' (2025), arXiv:2510.13950 [quant-ph].
arXiv:2510.13950

[125] S. Cepollaro, G. Cuffaro, M. B. Weiss, S. Cusumano, A. Hamma, and S. Lloyd, ``Stabilizer entropy of subspaces,'' (2025), arXiv:2512.23013 [quant-ph].
arXiv:2512.23013

[126] Z. Xiao and S. Ryu, ``Exponentially accelerated sampling of pauli strings for nonstabilizerness,'' (2026), arXiv:2601.00761 [quant-ph].
arXiv:2601.00761

[127] X. Huang, H.-Z. Li, and J.-X. Zhong, ``A fast and exact approach for stabilizer rényi entropy via the xor-fwht algorithm,'' (2026), arXiv:2512.24685 [quant-ph].
arXiv:2512.24685

[128] P. S. Tarabunga, E. Tirrito, M. C. Bañuls, and M. Dalmonte, Quantum Sci. Technol. 10, 045026 (2025).
https://doi.org/10.1088/2058-9565/adfd0d

Cited by

[1] Sergi Masot-Llima, Piotr Sierant, Paolo Stornati, and Artur Garcia-Saez, "Limits of Clifford disentangling in tensor network states", Physical Review B 114 2, 024311 (2026).

[2] Sabhyata Gupta, Piotr Sierant, Luis Santos, and Paolo Stornati, "Exact stabilizer scars in two-dimensional U(1) lattice gauge theory", Physical Review D 113 9, 094509 (2026).

[3] Ning Sun and Pengfei Zhang, "Connecting Magic Dynamics in Thermofield Double States to Spectral Form Factors", Physical Review Letters 137 3, 030401 (2026).

[4] Sreemayee Aditya, Xhek Turkeshi, and Piotr Sierant, "Growth and spreading of quantum resources under random circuit dynamics", Physical Review Research 8 3, 033062 (2026).

[5] Zhenyu Xiao and Shinsei Ryu, "Exponentially Accelerated Sampling of Pauli Strings for Nonstabilizerness", arXiv:2601.00761, (2026).

[6] Xuyang Huang, Han-Ze Li, Ching Hua Lee, and Jian-Xin Zhong, "A fast and exact approach for stabilizer Rényi entropy via the XOR-FWHT algorithm", arXiv:2512.24685, (2025).

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

[8] Caroline E. P. Robin and Martin J. Savage, "Quantum Complexity and New Directions in Nuclear Physics and High-Energy Physics Phenomenology", arXiv:2604.26376, (2026).

[9] Daniele Iannotti, Beatrice Magni, Riccardo Cioli, Alioscia Hamma, and Xhek Turkeshi, "Non-Local Magic Resources for Fermionic Gaussian States", arXiv:2604.27049, (2026).

[10] Daichi Kagamihara and Shunji Tsuchiya, "Stabilizer Rényi entropy of 3-uniform hypergraph states", arXiv:2602.23687, (2026).

[11] Henry Froland and Dorota M. Grabowska, "Measuring Non-Stabilizerness in an SU(2) Lattice Gauge Theory", arXiv:2606.14842, (2026).

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

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-08 01:09:59) and SAO/NASA ADS (last updated successfully 2026-08-08 01:10:01). 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.