{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T01:48:43Z","timestamp":1784166523389,"version":"3.55.0"},"reference-count":128,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T00:00:00Z","timestamp":1775779200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Red.es, Ministerio para la Transformaci\u00f3n Digital y de la Funci\u00f3n P\u00fablica","award":["C005\/24-ED CV1"],"award-info":[{"award-number":["C005\/24-ED CV1"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>\n                    Non-stabilizerness, also known as \u201cmagic,'' 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 R\u00e9nyi entropy (SRE) for qubits and the mana for qutrits, are costly to evaluate numerically, with the computational complexity growing rapidly with the number\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mi>N<\/mml:mi>\n                    <\/mml:math>\n                    of qudits. Here we introduce efficient, numerically exact algorithms that exploit the fast Hadamard transform to compute the SRE for qubits (\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mi>d<\/mml:mi>\n                      <mml:mo>=<\/mml:mo>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:math>\n                    ) and the mana for qutrits (\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mi>d<\/mml:mi>\n                      <mml:mo>=<\/mml:mo>\n                      <mml:mn>3<\/mml:mn>\n                    <\/mml:math>\n                    ) for pure states given as state vectors. Our methods compute SRE and mana at cost\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mi>O<\/mml:mi>\n                      <mml:mo stretchy=\"false\">(<\/mml:mo>\n                      <mml:mi>N<\/mml:mi>\n                      <mml:msup>\n                        <mml:mi>d<\/mml:mi>\n                        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                          <mml:mn>2<\/mml:mn>\n                          <mml:mi>N<\/mml:mi>\n                        <\/mml:mrow>\n                      <\/mml:msup>\n                      <mml:mo stretchy=\"false\">)<\/mml:mo>\n                    <\/mml:math>\n                    , providing an exponential improvement over the naive\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mi>O<\/mml:mi>\n                      <mml:mo stretchy=\"false\">(<\/mml:mo>\n                      <mml:msup>\n                        <mml:mi>d<\/mml:mi>\n                        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                          <mml:mn>3<\/mml:mn>\n                          <mml:mi>N<\/mml:mi>\n                        <\/mml:mrow>\n                      <\/mml:msup>\n                      <mml:mo stretchy=\"false\">)<\/mml:mo>\n                    <\/mml:math>\n                    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.\n                  <\/jats:p>","DOI":"10.22331\/q-2026-04-10-2059","type":"journal-article","created":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T08:24:05Z","timestamp":1775809445000},"page":"2059","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":4,"title":["Computing quantum magic of state vectors"],"prefix":"10.22331","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9219-7274","authenticated-orcid":false,"given":"Piotr","family":"Sierant","sequence":"first","affiliation":[{"name":"Barcelona Supercomputing Center, Barcelona 08034, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-0139-9744","authenticated-orcid":false,"given":"Jofre","family":"Vall\u00e8s-Muns","sequence":"additional","affiliation":[{"name":"Barcelona Supercomputing Center, Barcelona 08034, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3561-0223","authenticated-orcid":false,"given":"Artur","family":"Garcia-Saez","sequence":"additional","affiliation":[{"name":"Barcelona Supercomputing Center, Barcelona 08034, Spain"},{"name":"Qilimanjaro Quantum Tech, 08019 Barcelona, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2026,4,10]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"J. Preskill, ``Quantum computing and the entanglement frontier,&apos;&apos; (2012), arXiv:1203.5813.","DOI":"10.48550\/arXiv.1203.5813"},{"key":"1","doi-asserted-by":"publisher","unstructured":"A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Nature 607, 667 (2022).","DOI":"10.1038\/s41586-022-04940-6"},{"key":"2","doi-asserted-by":"publisher","unstructured":"G. Vidal, Phys. Rev. Lett. 91, 147902 (2003).","DOI":"10.1103\/PhysRevLett.91.147902"},{"key":"3","doi-asserted-by":"publisher","unstructured":"R. Or\u00fas, Ann. Phys. 349, 117 (2014).","DOI":"10.1016\/j.aop.2014.06.013"},{"key":"4","doi-asserted-by":"publisher","unstructured":"R. Or\u00fas, Nat. Rev. Phys. 1, 538\u2013550 (2019).","DOI":"10.1038\/s42254-019-0086-7"},{"key":"5","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1007\/978-3-030-34489-4"},{"key":"6","doi-asserted-by":"publisher","unstructured":"M. C. Ba\u00f1uls, Annu. Rev. Condens. Matter Phys. 14, 173\u2013191 (2023).","DOI":"10.1146\/annurev-conmatphys-040721-022705"},{"key":"7","doi-asserted-by":"publisher","unstructured":"F. Verstraete, V. Murg, and J. Cirac, Adv. Phys. 57, 143 (2008).","DOI":"10.1080\/14789940801912366"},{"key":"8","doi-asserted-by":"publisher","unstructured":"N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 100, 030504 (2008).","DOI":"10.1103\/PhysRevLett.100.030504"},{"key":"9","doi-asserted-by":"crossref","unstructured":"U. Schollw\u00f6ck, , 96 (2011).","DOI":"10.1016\/j.aop.2010.09.012"},{"key":"10","doi-asserted-by":"publisher","unstructured":"L. G. Valiant, in Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, STOC &apos;01 (Association for Computing Machinery, New York, NY, USA, 2001) p. 114\u2013123.","DOI":"10.1145\/380752.380785"},{"key":"11","doi-asserted-by":"publisher","unstructured":"B. M. Terhal and D. P. DiVincenzo, Phys. Rev. A 65, 032325 (2002).","DOI":"10.1103\/PhysRevA.65.032325"},{"key":"12","doi-asserted-by":"publisher","unstructured":"S. Bravyi, Quantum Info. Comput. 5, 216\u2013238 (2005).","DOI":"10.48550\/arXiv.quant-ph\/0404180"},{"key":"13","doi-asserted-by":"publisher","unstructured":"D. Gottesman, ``The heisenberg representation of quantum computers,&apos;&apos; (1998), arXiv:quant-ph\/9807006.","DOI":"10.48550\/arXiv.quant-ph\/9807006"},{"key":"14","doi-asserted-by":"publisher","unstructured":"S. Aaronson and D. Gottesman, Phys. Rev. A 70, 052328 (2004).","DOI":"10.1103\/PhysRevA.70.052328"},{"key":"15","doi-asserted-by":"publisher","unstructured":"P. Sierant, P. Stornati, and X. Turkeshi, PRX Quantum 7, 010302 (2026).","DOI":"10.1103\/3yx4-1j27"},{"key":"16","doi-asserted-by":"publisher","unstructured":"Z.-W. Liu and A. Winter, PRX Quantum 3, 020333 (2022).","DOI":"10.1103\/PRXQuantum.3.020333"},{"key":"17","doi-asserted-by":"publisher","unstructured":"V. Veitch, S. A. H. Mousavian, D. Gottesman, and J. Emerson, New J. Phys. 16, 013009 (2014).","DOI":"10.1088\/1367-2630\/16\/1\/013009"},{"key":"18","doi-asserted-by":"publisher","unstructured":"A. Heimendahl, M. Heinrich, and D. Gross, Journal of Mathematical Physics 63, 112201 (2022).","DOI":"10.1063\/5.0085774"},{"key":"19","doi-asserted-by":"publisher","unstructured":"T. Haug and L. Piroli, Quantum 7, 1092 (2023a).","DOI":"10.22331\/q-2023-08-28-1092"},{"key":"20","doi-asserted-by":"publisher","unstructured":"M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (2000).","DOI":"10.1017\/CBO9780511976667"},{"key":"21","doi-asserted-by":"publisher","unstructured":"S. Bravyi and A. Kitaev, Phys. Rev. A 71, 022316 (2005).","DOI":"10.1103\/PhysRevA.71.022316"},{"key":"22","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1103\/PhysRevA.52.3457"},{"key":"23","doi-asserted-by":"publisher","unstructured":"S. Bravyi, G. Smith, and J. A. Smolin, Phys. Rev. X 6, 021043 (2016).","DOI":"10.1103\/PhysRevX.6.021043"},{"key":"24","doi-asserted-by":"publisher","unstructured":"S. Bravyi and D. Gosset, Phys. Rev. Lett. 116, 250501 (2016).","DOI":"10.1103\/PhysRevLett.116.250501"},{"key":"25","doi-asserted-by":"publisher","unstructured":"S. Bravyi, D. Browne, P. Calpin, E. Campbell, D. Gosset, and M. Howard, Quantum 3, 181 (2019).","DOI":"10.22331\/q-2019-09-02-181"},{"key":"26","doi-asserted-by":"publisher","unstructured":"M. Howard and E. Campbell, Phys. Rev. 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D 110, 116016 (2024).","DOI":"10.1103\/PhysRevD.110.116016"},{"key":"79","doi-asserted-by":"publisher","unstructured":"I. Chernyshev, C. E. P. Robin, and M. J. Savage, Phys. Rev. Res. 7, 023228 (2025).","DOI":"10.1103\/PhysRevResearch.7.023228"},{"key":"80","doi-asserted-by":"crossref","unstructured":"Q. Liu, I. Low, and Z. Yin, ``Quantum magic in quantum electrodynamics,&apos;&apos; (2025a), arXiv:2503.03098 [quant-ph].","DOI":"10.1103\/l8vw-kwhz"},{"key":"81","doi-asserted-by":"publisher","unstructured":"A. Gu, H.-Y. Hu, D. Luo, T. L. Patti, N. C. Rubin, and S. F. Yelin, Quantum 8, 1422 (2024).","DOI":"10.22331\/q-2024-07-23-1422"},{"key":"82","doi-asserted-by":"publisher","unstructured":"C. Cao, G. Cheng, A. Hamma, L. Leone, W. Munizzi, and S. F. Oliviero, PRX Quantum 6, 040375 (2025).","DOI":"10.1103\/z3vr-w5c5"},{"key":"83","unstructured":"F. B. 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