{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T03:23:11Z","timestamp":1778815391268,"version":"3.51.4"},"reference-count":43,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2025,9,15]],"date-time":"2025-09-15T00:00:00Z","timestamp":1757894400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"Magic State Distillation (MSD) has been a research focus for fault-tolerant quantum computing due to the need for non-Clifford resource in gaining quantum advantage. Although many of the MSD protocols so far are based on stabilizer codes with transversal T<\/mml:mi><\/mml:math> gates, there exists quite several protocols that don't fall into this class. Here we propose a method to map MSD protocols to iterative dynamical systems under the framework of stabilizer reduction. With the proposed mapping, we are able to analyze the performance of MSD protocols using techniques from dynamical systems theory, easily simulate the distillation process of input states under arbitrary noise and visualize it using flow diagram. We apply our mapping to common MSD protocols for |<\/mml:mo><\/mml:mrow>T<\/mml:mi>&#x27E9;<\/mml:mo><\/mml:math> state and find some interesting properties: The [<\/mml:mo>[<\/mml:mo>15<\/mml:mn>,<\/mml:mo>1<\/mml:mn>,<\/mml:mo>3<\/mml:mn>]<\/mml:mo>]<\/mml:mo><\/mml:math> code may distill states corresponding to T<\/mml:mi><\/mml:msqrt><\/mml:math> gate and the [<\/mml:mo>[<\/mml:mo>5<\/mml:mn>,<\/mml:mo>1<\/mml:mn>,<\/mml:mo>3<\/mml:mn>]<\/mml:mo>]<\/mml:mo><\/mml:math> code can distill the magic state corresponding to the T<\/mml:mi><\/mml:math> gate. Besides, we examine the exotic MSD protocols that may distill into other magic states proposed in [Eur. Phys. J. D 70, 55 (2016)] and identify the condition for distillable magic states. We also study new MSD protocols generated by concatenating different codes and numerically demonstrate that concatenation can generate MSD protocols with various magic states. By concatenating efficient codes with exotic codes, we can reduce the overhead of the exotic MSD protocols. We believe our proposed method will be a useful tool for simulating and visualization MSD protocols for canonical MSD protocols on |<\/mml:mo><\/mml:mrow>T<\/mml:mi>&#x27E9;<\/mml:mo><\/mml:math> as well as other unexplored MSD protocols for other states.<\/jats:p>","DOI":"10.22331\/q-2025-09-15-1858","type":"journal-article","created":{"date-parts":[[2025,9,15]],"date-time":"2025-09-15T09:54:14Z","timestamp":1757930054000},"page":"1858","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["From Magic State Distillation to Dynamical Systems"],"prefix":"10.22331","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6419-986X","authenticated-orcid":false,"given":"Yunzhe","family":"Zheng","sequence":"first","affiliation":[{"name":"Department of Physics, Tsinghua University, Beijing, 100084, China"},{"name":"Department of Applied Physics, Yale University, New Haven, Connecticut, 06511, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dong E.","family":"Liu","sequence":"additional","affiliation":[{"name":"Department of Physics, Tsinghua University, Beijing, 100084, China"},{"name":"Frontier Science Center for Quantum Information, Beijing 100084, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2025,9,15]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Google Quantum AI. ``Quantum error correction below the surface code threshold'' (2024).","DOI":"10.1038\/s41586-024-08449-y"},{"key":"1","unstructured":"Hengyun Zhou, Chen Zhao, Madelyn Cain, Dolev Bluvstein, Casey Duckering, Hong-Ye Hu, Sheng-Tao Wang, Aleksander Kubica, and Mikhail D. 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