- Physical Review Journals
All JournalsPhysics Magazine

PRX Quantum

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Efficient Magic State Distillation by Zero-Level Distillation

Tomohiro Itogawa1,*, Yugo Takada1, Yutaka Hirano1, and Keisuke Fujii1,2,3,4

  • 1Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan
  • 2Center for Quantum Information and Quantum Biology, Osaka University, 1-2 Machikaneyama, Toyonaka 560-0043, Japan
  • 3RIKEN Center for Quantum Computing (RQC), Hirosawa 2-1, Wako, Saitama 351-0198, Japan
  • 4Fujitsu Quantum Computing Joint Research Division, Center for Quantum Information and Quantum Biology, Osaka University, 1-2 Machikaneyama, Toyonaka, Osaka 565-8531, Japan

  • *Contact author: u348506d@ecs.osaka-u.ac.jp
PDF

PRX Quantum 6, 020356 – Published 20 June, 2025

DOI: https://doi.org/10.1103/thxx-njr6

Abstract

Magic state distillation (MSD) is an essential element for universal fault-tolerant quantum computing, which distills a high-fidelity magic state from noisy magic states using ideal (error-corrected) Clifford operations. For ideal Clifford operations, it needs to be performed on the logical qubits and hence incurs a large spatiotemporal overhead, which is one of the major bottlenecks for the realization of fault-tolerant quantum computers (FTQCs). Here we propose zero-level distillation, which prepares a high-fidelity logical magic state at the physical level, namely zero level, using physical qubits and nearest-neighbor two-qubit gates on a square lattice. We develop a zero-level distillation circuit and show that distillation can be made even more efficient than the conventional sophisticated approaches with logical level distillations. The key idea involves the Knill et al.-type distillation using the Steane code and its careful mapping to the square-lattice architecture with error detection. The distilled magic state on the Steane-code state is then teleported or converted to surface codes. We numerically find that the error rate of the logical magic state scales as approximately 100×p2 in terms of the physical error rate p. For example, with a physical error rate of p=10−4 (10−3), the logical error rate is reduced to pL=10−6 (10−4), resulting in an improvement of 2 (1) orders of magnitude. This contributes to reducing both space and time overhead for early FTQC as well as full-fledged FTQC combined with conventional multilevel distillation protocols.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Magic state distillation is an essential step toward realizing fault-tolerant quantum computers (FTQCs), which promise revolutionary advances by tackling complex problems beyond the capabilities of classical computers. Despite its importance, current distillation methods typically require significant resources, in particular, large numbers of logical qubits. This requirement poses a significant practical challenge to realizing large-scale FTQCs.

In this paper, we propose a novel approach called zero-level distillation, aimed at significantly improving the efficiency of magic-state distillation. Unlike traditional approaches reliant on logical qubits, our method operates directly at the physical qubit level. By employing physical qubits and carefully designed circuits, our approach substantially reduces the number of qubits needed. Specifically, we utilize the Steane code with nearest-neighbor interactions on a two-dimensional square lattice, effectively managing stringent hardware constraints. We present detailed circuits for this zero-level distillation process, demonstrating through numerical simulations that our method significantly decreases the logical error rate, achieving 2 orders of magnitude improvement under realistic physical error rates.

The potential implications of zero-level distillation are profound. It provides a practical approach suitable for both early FTQCs and more advanced quantum computing applications. Combining our method with established distillation techniques, such as (0+1)-distillation or recently proposed magic-state cultivation, significantly enhances the capabilities of fault-tolerant quantum computing systems. This integrated approach represents a promising pathway toward achieving practical quantum computing with substantially reduced overhead, encouraging further research and technological advancements.

Article Text

References (35)

  1. Peter W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Comput. 26, 1484 (1997).
  2. Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd, Quantum algorithm for linear systems of equations, Phys. Rev. Lett. 103, 150502 (2009).
  3. Alán Aspuru-Guzik, Anthony D. Dutoi, Peter J. Love, and Martin Head-Gordon, Simulated quantum computation of molecular energies, Science 309, 1704 (2005).
  4. A. Morvan et al., Phase transition in random circuit sampling, (2023).
  5. Youngseok Kim, Andrew Eddins, Sajant Anand, Ken Xuan Wei, Ewout van den Berg, Sami Rosenblatt, Hasan Nayfeh, Yantao Wu, Michael Zaletel, Kristan Temme, and Abhinav Kandala, Evidence for the utility of quantum computing before fault tolerance, Nature 618, 500 (2023).
  6. John Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  7. M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles, Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
  8. Peter W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493 (1995).
  9. Keisuke Fujii, Quantum computation with topological codes: From qubit to topological fault-tolerance, (2015).
  10. A. Yu. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. (N. Y) 303, 2 (2003).
  11. Sergey Bravyi and Robert Raussendorf, Measurement-based quantum computation with the toric code states, Phys. Rev. A 76, 022304 (2007).
  12. Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland, Surface codes: Towards practical large-scale quantum computation, Phys. Rev. A 86, 032324 (2012).
  13. Robert Raussendorf and Hans J. Briegel, A one-way quantum computer, Phys. Rev. Lett. 86, 5188 (2001).
  14. Emanuel Knill and Bryan Eastin, Restrictions on transversal encoded quantum gate sets, Phys. Rev. Lett. 102, 110502 (2009).
  15. Sergey Bravyi and Alexei Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, Phys. Rev. A 71, 022316 (2005).
  16. Daniel Gottesman and Isaac L. Chuang, Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations, Nature 402, 390 (1999).
  17. Craig Gidney and Martin Ekerå, How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits, Quantum 5, 433 (2021).
  18. Hayato Goto, Minimizing resource overheads for fault-tolerant preparation of encoded states of the steane code, Sci. Rep. 6, 19578 (2016).
  19. Christopher Chamberland and Andrew W. Cross, Fault-tolerant magic state preparation with flag qubits, Quantum 3, 143 (2019).
  20. Christopher Chamberland and Kyungjoo Noh, Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits, Npj Quantum Inf. 6, 91 (2020).
  21. Lukas Postler, Sascha Heuβen, Ivan Pogorelov, Manuel Rispler, Thomas Feldker, Michael Meth, Christian D Marciniak, Roman Stricker, Martin Ringbauer, and Rainer Blatt et al., Demonstration of fault-tolerant universal quantum gate operations, Nature 605, 675 (2022).
  22. Emanuel Knill, Raymond Laflamme, and Wojciech H. Zurek, Resilient quantum computation, Science 279, 342 (1998).
  23. Yutaro Akahoshi, Kazunori Maruyama, Hirotaka Oshima, Shintaro Sato, and Keisuke Fujii, Partially fault-tolerant quantum computing architecture with error-corrected clifford gates and space-time efficient analog rotations, (2023).
  24. Daniel Litinski, Magic state distillation: Not as costly as you think, Quantum 3, 205 (2019).
  25. Emanuel Knill, Raymond Laflamme, and Wojciech H. Zurek, Resilient quantum computation: Error models and thresholds, Proc. R. Soc. London. Ser. A: Math. Phys. Eng. Sci. 454, 365 (1998).
  26. Hendrik Poulsen Nautrup, Nicolai Friis, and Hans J. Briegel, Fault-tolerant interface between quantum memories and quantum processors, Nat. Commun. 8, 1321 (2017).
  27. Lingling Lao and Carmen G. Almudever, Fault-tolerant quantum error correction on near-term quantum processors using flag and bridge qubits, Phys. Rev. A 101, 032333 (2020).
  28. Yasunari Suzuki, Yoshiaki Kawase, Yuya Masumura, Yuria Hiraga, Masahiro Nakadai, Jiabao Chen, Ken M. Nakanishi, Kosuke Mitarai, Ryosuke Imai, Shiro Tamiya, Takahiro Yamamoto, Tennin Yan, Toru Kawakubo, Yuya O. Nakagawa, Yohei Ibe, Youyuan Zhang, Hirotsugu Yamashita, Hikaru Yoshimura, Akihiro Hayashi, and Keisuke Fujii, Qulacs: A fast and versatile quantum circuit simulator for research purpose, Quantum 5, 559 (2021).
  29. Mitsuki Katsuda, Kosuke Mitarai, and Keisuke Fujii, Simulation and performance analysis of quantum error correction with a rotated surface code under a realistic noise model, Phys. Rev. Res. 6, 013024 (2024).
  30. Craig Gidney, Michael Newman, Austin Fowler, and Michael Broughton, A fault-tolerant honeycomb memory, Quantum 5, 605 (2021).
  31. Craig Gidney, Stim: a fast stabilizer circuit simulator, Quantum 5, 497 (2021).
  32. John Preskill, Beyond NISQ: The megaquop machine, ACM Trans. Quantum Comput. 6, 18 (2025).
  33. Neil J. Ross and Peter Selinger, Optimal ancilla-free Clifford+T approximation of z-rotations, Quantum Info. Comput. 16, 901 (2016).
  34. Yutaka Hirano, Tomohiro Itogawa, and Keisuke Fujii, in 2024 IEEE International Conference on Quantum Computing and Engineering (QCE), (IEEE, Montréal, 2024), Vol. 1, p. 843.
  35. Craig Gidney, Noah Shutty, and Cody Jones, Magic state cultivation: Growing T states as cheap as CNOT gates, arXiv:2409.17595.

Outline

Information

PRX Quantum 6, 020356– Published 20 June, 2025
  • Received 21 November 2024
  • Revised 28 February 2025
  • Accepted 22 May 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation