From Magic State Distillation to Dynamical Systems

Yunzhe Zheng; Dong E. Liu

doi:10.22331/q-2025-09-15-1858

From Magic State Distillation to Dynamical Systems

Yunzhe Zheng^1,2 and Dong E. Liu^1,3

¹Department of Physics, Tsinghua University, Beijing, 100084, China
²Department of Applied Physics, Yale University, New Haven, Connecticut, 06511, USA
³Frontier Science Center for Quantum Information, Beijing 100084, China

Published:	2025-09-15, volume 9, page 1858
Editor:	Alessio Benavoli
Eprint:	arXiv:2412.04402v2
Doi:	https://doi.org/10.22331/q-2025-09-15-1858
Citation:	Quantum 9, 1858 (2025).

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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 $T$ 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 $|T\rangle$ state and find some interesting properties: The $[[15, 1, 3]]$ code may distill states corresponding to $\sqrt{T}$ gate and the $[[5, 1, 3]]$ code can distill the magic state corresponding to the $T$ 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 $|T\rangle$ as well as other unexplored MSD protocols for other states.

Popular summary

Magic state distillation is a critical routine for fault-tolerant quantum computing that distill high-quality "magic" resource states at cost of many lower-quality states. Even though various magic state distillation protocols have been analyzed individually so far, there still doesn't exist a unified method for analyzing any generic protcols.

In this work, we propose a new method to connect distillation protocols with iterative dynamical systems. Our method is totally generic for any distillation protocols and allows to analyze biased noise presented in the input states. Plotting the dynamical systems using flow diagram also provides intuitive visualization of distillation behavior of various protocols. We demonstrate our method using canonical well-known protocols as well as small protocols that distill exotic "magic" states. Furthermore, we showcase the potential for distilling more general magic states.

As magic state distillation becomes one of the central topics for fault-tolerant quantum computing, our method would provide a convenient framework that assists to find and analyze new distillation protocols.

► BibTeX data

@article{Zheng2025frommagicstate,
  doi = {10.22331/q-2025-09-15-1858},
  url = {https://doi.org/10.22331/q-2025-09-15-1858},
  title = {From {M}agic {S}tate {D}istillation to {D}ynamical {S}ystems},
  author = {Zheng, Yunzhe and Liu, Dong E.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1858},
  month = sep,
  year = {2025}
}

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Cited by

[1] Cai-Hong Wang, Jiang-Tao Yuan, Zhi-Hao Ma, Shao-Ming Fei, and Shang-Quan Bu, "Generalized symplectic representation of local Clifford operators", Physical Review A 113 2, 022454 (2026).

[2] Samyak Surti, Lucas Daguerre, and Isaac H. Kim, "Efficient Simulation of Logical Magic State Preparation Protocols", PRX Quantum 7 2, 020329 (2026).

[3] Yunzhe Zheng, Yuanchen Zhao, and Dong E. Liu, "Fragility of Magic State Distillation under Imperfect Measurements", arXiv:2503.01165, (2025).

[4] Simon Burton and Hussain Anwar, "Meromorphic Quantum Computing", arXiv:2605.06251, (2026).

[5] Yunzhe Zheng, Allen Zang, and Aleksander Kubica, "Magic Gate Teleportation: Structure, Useful Resource States, and Simpler Feedforward", arXiv:2607.08508, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-09 01:20:51) and SAO/NASA ADS (last updated successfully 2026-08-09 01:20:53). The list may be incomplete as not all publishers provide suitable and complete citation data.

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