Low Overhead Qutrit Magic State Distillation

Shiroman Prakash; Tanay Saha

doi:10.22331/q-2025-06-12-1768

Low Overhead Qutrit Magic State Distillation

Shiroman Prakash¹ and Tanay Saha^1,2

¹Department of Physics and Computer Science, Dayalbagh Educational Institute, Agra-282005, India
²Present Address: Department of Mathematics, Simon Fraser University, Burnaby, B.C., Canada

Published:	2025-06-12, volume 9, page 1768
Editor:	Narayanan Rengaswamy
Eprint:	arXiv:2403.06228v3
Doi:	https://doi.org/10.22331/q-2025-06-12-1768
Citation:	Quantum 9, 1768 (2025).

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Abstract

We show that using qutrits rather than qubits leads to a substantial reduction in the overhead cost associated with an approach to fault-tolerant quantum computing known as magic state distillation. We construct a family of $[[9m-k, k, 2]]_3$ triorthogonal qutrit error-correcting codes for any positive integers $m$ and $k$ with $k \leq 3m-2$ that are suitable for magic state distillation. In magic state distillation, the number of ancillae required to produce a magic state with target error rate $\epsilon$ is $O(\log^\gamma \epsilon^{-1})$, where the yield parameter $\gamma$ characterizes the overhead cost. For $k=3m-2$, our codes have $\gamma = \log_2 (2+\frac{6}{3 m-2})$, which tends to $1$ as $m \to \infty$. Moreover, the $[[20,7,2]]_3$ qutrit code that arises from our construction when $m=3$ already has a yield parameter of $1.51$ which outperforms all known qubit triorthogonal codes of size less than a few hundred qubits.

► BibTeX data

@article{Prakash2025lowoverheadqutrit,
  doi = {10.22331/q-2025-06-12-1768},
  url = {https://doi.org/10.22331/q-2025-06-12-1768},
  title = {Low {O}verhead {Q}utrit {M}agic {S}tate {D}istillation},
  author = {Prakash, Shiroman and Saha, Tanay},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1768},
  month = jun,
  year = {2025}
}

► References

[1] Sergey Bravyi and Alexei Kitaev. ``Universal quantum computation with ideal Clifford gates and noisy ancillas''. Phys. Rev. A 71, 022316 (2005).
https://doi.org/10.1103/PhysRevA.71.022316

[2] E. Knill. ``Fault-tolerant postselected quantum computation: Schemes'' (2004). arXiv:quant-ph/0402171.
https://doi.org/10.48550/arXiv.quant-ph/0402171
arXiv:quant-ph/0402171

[3] Alexandre M. Souza et al. ``Experimental magic state distillation for fault-tolerant quantum computing''. Nature Commun. 2, 1–5 (2011).
https://doi.org/10.1038/ncomms1166

[4] Lukas Postler et al. ``Demonstration of fault-tolerant universal quantum gate operations''. Nature 605, 675–680 (2022).
https://doi.org/10.1038/s41586-022-04721-1

[5] Yangsen Ye et al. ``Logical magic state preparation with fidelity beyond the distillation threshold on a superconducting quantum processor''. Phys. Rev. Lett. 131, 210603 (2023).
https://doi.org/10.1103/PhysRevLett.131.210603

[6] Riddhi S. Gupta et al. ``Encoding a magic state with beyond break-even fidelity''. Nature 625, 259–263 (2024).
https://doi.org/10.1038/s41586-023-06846-3

[7] Pedro Sales Rodriguez et al. ``Experimental demonstration of logical magic state distillation'' (2024). arXiv:2412.15165.
https://doi.org/10.48550/arXiv.2412.15165
arXiv:2412.15165

[8] Joe O'Gorman and Earl T. Campbell. ``Quantum computation with realistic magic-state factories''. Phys. Rev. A 95, 032338 (2017).
https://doi.org/10.1103/PhysRevA.95.032338

[9] Earl T. Campbell and Mark Howard. ``Unifying gate synthesis and magic state distillation''. Phys. Rev. Lett. 118, 060501 (2017).
https://doi.org/10.1103/PhysRevLett.118.060501

[10] Daniel Litinski. ``Magic state distillation: Not as costly as you think''. Quantum 3, 205 (2019).
https://doi.org/10.22331/q-2019-12-02-205

[11] Christopher Chamberland and Kyungjoo Noh. ``Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits''. npj Quantum Information 6, 91 (2020).
https://doi.org/10.1038/s41534-020-00319-5

[12] Héctor Bombín, Mihir Pant, Sam Roberts, and Karthik I Seetharam. ``Fault-tolerant postselection for low-overhead magic state preparation''. PRX Quantum 5, 010302 (2024).
https://doi.org/10.1103/PRXQuantum.5.010302

[13] Sergey Bravyi and Jeongwan Haah. ``Magic-state distillation with low overhead''. Phys. Rev. A 86, 052329 (2012).
https://doi.org/10.1103/PhysRevA.86.052329

[14] Matthew B. Hastings and Jeongwan Haah. ``Distillation with sublogarithmic overhead''. Phys. Rev. Lett. 120, 050504 (2018).
https://doi.org/10.1103/PhysRevLett.120.050504

[15] Cody Jones. ``Multilevel distillation of magic states for quantum computing''. Phys. Rev. A 87, 042305 (2013).
https://doi.org/10.1103/PhysRevA.87.042305

[16] Jeongwan Haah, Matthew B. Hastings, D. Poulin, and D. Wecker. ``Magic state distillation with low space overhead and optimal asymptotic input count''. Quantum 1, 31 (2017).
https://doi.org/10.22331/q-2017-10-03-31

[17] Jeongwan Haah and Matthew B. Hastings. ``Codes and Protocols for Distilling $T$, controlled-$S$, and Toffoli Gates''. Quantum 2, 71 (2018).
https://doi.org/10.22331/q-2018-06-07-71

[18] Adam Wills, Min-Hsiu Hsieh, and Hayata Yamasaki. ``Constant-overhead magic state distillation'' (2024). arXiv:2408.07764.
https://doi.org/10.48550/arXiv.2408.07764
arXiv:2408.07764

[19] Louis Golowich and Venkatesan Guruswami. ``Asymptotically good quantum codes with transversal non-Clifford gates'' (2024). arXiv:2408.09254.
https://doi.org/10.48550/arXiv.2408.09254
arXiv:2408.09254

[20] Quynh T. Nguyen. ``Good binary quantum codes with transversal CCZ gate'' (2024). arXiv:2408.10140.
https://doi.org/10.48550/arXiv.2408.10140
arXiv:2408.10140

[21] Sepehr Nezami and Jeongwan Haah. ``Classification of small triorthogonal codes''. Phys. Rev. A 106, 012437 (2022).
https://doi.org/10.1103/PhysRevA.106.012437

[22] Mark Howard, Joel Wallman, Victor Veitch, and Joseph Emerson. ``Contextuality supplies the ‘magic’ for quantum computation''. Nature 510, 351–355 (2014).
https://doi.org/10.1038/nature13460

[23] Dayal Pyari Srivastava, Vishal Sahni, and Prem Saran Satsangi. ``Modelling microtubules in the brain as n-qudit quantum Hopfield network and beyond''. Int. J. Gen. Syst. 45, 41–54 (2016).
https://doi.org/10.1080/03081079.2015.1076405

[24] Dayal Pyari Srivastava, Vishal Sahni, and Prem Saran Satsangi. ``From n-qubit multi-particle quantum teleportation modelling to n-qudit contextuality based quantum teleportation and beyond''. Int. J. Gen. Syst. 46, 414–435 (2017).
https://doi.org/10.1080/03081079.2017.1308361

[25] Shiroman Prakash and Aashi Gupta. ``Contextual bound states for qudit magic state distillation''. Phys. Rev. A 101, 010303 (2020).
https://doi.org/10.1103/PhysRevA.101.010303

[26] Hussain Anwar, Earl T Campbell, and Dan E Browne. ``Qutrit magic state distillation''. New J. Phys. 14, 063006 (2012).
https://doi.org/10.1088/1367-2630/14/6/063006

[27] Earl T. Campbell, Hussain Anwar, and Dan E. Browne. ``Magic-state distillation in all prime dimensions using quantum Reed-Muller codes''. Phys. Rev. X 2, 041021 (2012).
https://doi.org/10.1103/PhysRevX.2.041021

[28] Earl T. Campbell. ``Enhanced fault-tolerant quantum computing in $d$-level systems''. Phys. Rev. Lett. 113, 230501 (2014).
https://doi.org/10.1103/PhysRevLett.113.230501

[29] Anirudh Krishna and Jean-Pierre Tillich. ``Towards low overhead magic state distillation''. Phys. Rev. Lett. 123, 070507 (2019).
https://doi.org/10.1103/PhysRevLett.123.070507

[30] Hillary Dawkins and Mark Howard. ``Qutrit magic state distillation tight in some directions''. Phys. Rev. Lett. 115, 030501 (2015).
https://doi.org/10.1103/PhysRevLett.115.030501

[31] Mark Howard and Hillary Dawkins. ``Small codes for magic state distillation''. Eur. Phys. J. D 70, 55 (2016).
https://doi.org/10.1140/epjd/e2016-60682-y

[32] Shiroman Prakash. ``Magic state distillation with the ternary Golay code''. Proceedings of the Royal Society A 476, 20200187 (2020).
https://doi.org/10.1098/rspa.2020.0187

[33] Mark Howard and Jiri Vala. ``Qudit versions of the qubit ${\pi}/8$ gate''. Phys. Rev. A 86, 022316 (2012).
https://doi.org/10.1103/PhysRevA.86.022316

[34] Wim van Dam and Mark Howard. ``Noise thresholds for higher-dimensional systems using the discrete Wigner function''. Phys. Rev. A 83, 032310 (2011).
https://doi.org/10.1103/PhysRevA.83.032310

[35] Shiroman Prakash, Akalank Jain, Bhakti Kapur, and Shubangi Seth. ``Normal form for single-qutrit Clifford+${T}$ operators and synthesis of single-qutrit gates''. Phys. Rev. A 98, 032304 (2018).
https://doi.org/10.1103/PhysRevA.98.032304

[36] Andrew N. Glaudell, Neil J. Ross, and Jacob M. Taylor. ``Canonical forms for single-qutrit Clifford+${T}$ operators''. Annals of Physics 406, 54–70 (2019).
https://doi.org/10.1016/j.aop.2019.04.001

[37] Shawn X. Cui, Daniel Gottesman, and Anirudh Krishna. ``Diagonal gates in the Clifford hierarchy''. Phys. Rev. A 95, 012329 (2017).
https://doi.org/10.1103/PhysRevA.95.012329

[38] Amolak Ratan Kalra, Dinesh Valluri, and Michele Mosca. ``Synthesis and arithmetic of single qutrit circuits'' (2024). arXiv:2311.08696.
https://doi.org/10.48550/arXiv.2311.08696
arXiv:2311.08696

[39] Akalank Jain and Shiroman Prakash. ``Qutrit and ququint magic states''. Phys. Rev. A 102, 042409 (2020).
https://doi.org/10.1103/PhysRevA.102.042409

[40] Victor Veitch, Christopher Ferrie, David Gross, and Joseph Emerson. ``Negative quasi-probability as a resource for quantum computation''. New J. Phys. 14, 113011 (2012).
https://doi.org/10.1088/1367-2630/14/11/113011

[41] Victor Veitch, S A Hamed Mousavian, Daniel Gottesman, and Joseph Emerson. ``The resource theory of stabilizer quantum computation''. New J. Phys. 16, 013009 (2014).
https://doi.org/10.1088/1367-2630/16/1/013009

[42] Hussain Anwar, Benjamin J. Brown, Earl T. Campbell, and Dan E. Browne. ``Fast decoders for qudit topological codes''. New J. Phys. 16, 063038 (2014).
https://doi.org/10.1088/1367-2630/16/6/063038

[43] Fern H. E. Watson, Hussain Anwar, and Dan E. Browne. ``Fast fault-tolerant decoder for qubit and qudit surface codes''. Phys. Rev. A 92, 032309 (2015).
https://doi.org/10.1103/PhysRevA.92.032309

[44] Fern H. E. Watson, Earl T. Campbell, Hussain Anwar, and Dan E. Browne. ``Qudit color codes and gauge color codes in all spatial dimensions''. Phys. Rev. A 92, 022312 (2015).
https://doi.org/10.1103/PhysRevA.92.022312

[45] Adam M. Meier, Bryan Eastin, and Emanuel Knill. ``Magic-state distillation with the four-qubit code'' (2012). arXiv:1204.4221.
https://doi.org/10.48550/arXiv.1204.4221
arXiv:1204.4221

[46] A. Mari and J. Eisert. ``Positive Wigner functions render classical simulation of quantum computation efficient''. Phys. Rev. Lett. 109, 230503 (2012).
https://doi.org/10.1103/PhysRevLett.109.230503

[47] Mark Howard and Earl Campbell. ``Application of a resource theory for magic states to fault-tolerant quantum computing''. Phys. Rev. Lett. 118, 090501 (2017).
https://doi.org/10.1103/PhysRevLett.118.090501

[48] Xin Wang, Mark M. Wilde, and Yuan Su. ``Efficiently computable bounds for magic state distillation''. Phys. Rev. Lett. 124, 090505 (2020).
https://doi.org/10.1103/PhysRevLett.124.090505

[49] Nikolaos Koukoulekidis and David Jennings. ``Constraints on magic state protocols from the statistical mechanics of Wigner negativity''. npj Quantum Information 8, 42 (2022).
https://doi.org/10.1038/s41534-022-00551-1

[50] Austin G. Fowler, Simon J. Devitt, and Cody Jones. ``Surface code implementation of block code state distillation''. Scientific Reports 3, 1939 (2013).
https://doi.org/10.1038/srep01939

[51] B. P. Lanyon, T. J. Weinhold, N. K. Langford, J. L. O'Brien, K. J. Resch, A. Gilchrist, and A. G. White. ``Manipulating biphotonic qutrits''. Phys. Rev. Lett. 100, 060504 (2008).
https://doi.org/10.1103/PhysRevLett.100.060504

[52] R. Bianchetti, S. Filipp, M. Baur, J. M. Fink, C. Lang, L. Steffen, M. Boissonneault, A. Blais, and A. Wallraff. ``Control and tomography of a three level superconducting artificial atom''. Phys. Rev. Lett. 105, 223601 (2010).
https://doi.org/10.1103/PhysRevLett.105.223601

[53] M.A. Yurtalan, J. Shi, G.J.K. Flatt, and A. Lupascu. ``Characterization of multilevel dynamics and decoherence in a high-anharmonicity capacitively shunted flux circuit''. Phys. Rev. Appl. 16, 054051 (2021).
https://doi.org/10.1103/PhysRevApplied.16.054051

[54] M. Kononenko, M. A. Yurtalan, S. Ren, J. Shi, S. Ashhab, and A. Lupascu. ``Characterization of control in a superconducting qutrit using randomized benchmarking''. Phys. Rev. Res. 3, L042007 (2021).
https://doi.org/10.1103/PhysRevResearch.3.L042007

[55] Martin Ringbauer, Michael Meth, Lukas Postler, Roman Stricker, Rainer Blatt, Philipp Schindler, and Thomas Monz. ``A universal qudit quantum processor with trapped ions''. Nature Phys. 18, 1053–1057 (2022). arXiv:2109.06903.
https://doi.org/10.1038/s41567-022-01658-0
arXiv:2109.06903

[56] Noah Goss et al. ``High-fidelity qutrit entangling gates for superconducting circuits''. Nature Commun. 13, 7481 (2022). arXiv:2206.07216.
https://doi.org/10.1038/s41467-022-34851-z
arXiv:2206.07216

[57] Jan Schütz, Alexander Martin, Sanah Laschinger, and Gerhard Birkl. ``Coherent dynamics in a five-level atomic system''. J. Phys. B 55, 234004 (2022). arXiv:2210.11893.
https://doi.org/10.1088/1361-6455/ac9c3a
arXiv:2210.11893

[58] Joseph Lindon, Arina Tashchilina, Logan W. Cooke, and Lindsay J. LeBlanc. ``Complete unitary qutrit control in ultracold atoms''. Phys. Rev. Appl. 19, 034089 (2023).
https://doi.org/10.1103/PhysRevApplied.19.034089

[59] Noah Goss, Samuele Ferracin, Akel Hashim, Arnaud Carignan-Dugas, John Mark Kreikebaum, Ravi K. Naik, David I. Santiago, and Irfan Siddiqi. ``Extending the computational reach of a superconducting qutrit processor'' (2023). arXiv:2305.16507.
https://doi.org/10.48550/arXiv.2305.16507
arXiv:2305.16507

[60] Mahadevan Subramanian and Adrian Lupascu. ``Efficient two-qutrit gates in superconducting circuits using parametric coupling'' (2023). arXiv:2309.05766.
https://doi.org/10.48550/arXiv.2309.05766
arXiv:2309.05766

[61] Wieb Bosma, John Cannon, and Catherine Playoust. ``The Magma algebra system I: The user language''. J. Symb. Comput. 24, 235–265 (1997).
https://doi.org/10.1006/jsco.1996.0125

[62] F.J. MacWilliams and N.J.A. Sloane. ``Chapter 5: Dual codes and their weight distribution''. In The Theory of Error-Correcting Codes. Volume 16 of North-Holland Mathematical Library, pages 125–154. Elsevier (1977).
https://doi.org/10.1016/S0924-6509(08)70530-0

Cited by

[1] Tanay Saha and Shiroman Prakash, "Sublogarithmic distillation in all prime dimensions using punctured Reed-Muller codes", Physical Review A 113 4, 042405 (2026).

[2] Sarah Meng Li, Michele Mosca, Neil J. Ross, John van de Wetering, and Yuming Zhao, "A Complete and Natural Rule Set for Multi-Qutrit Clifford Circuits", Electronic Proceedings in Theoretical Computer Science 426, 23 (2025).

[3] Shiroman Prakash and Rishabh Singhal, "Search for high-threshold qutrit magic-state distillation routines", Physical Review A 113 4, 042404 (2026).

[4] Amolak Ratan Kalra, Michele Mosca, and Dinesh Valluri, "Synthesis and Arithmetic of Single Qutrit Circuits", Quantum 9, 1647 (2025).

[5] Amolak Ratan Kalra and Shiroman Prakash, "Invariant Theory, Magic State Distillation, and Bounds on Classical Codes", arXiv:2501.10163, (2025).

[6] Sarah Meng Li, Michele Mosca, Neil J. Ross, John van de Wetering, and Yuming Zhao, "A Complete and Natural Rule Set for Multi-Qutrit Clifford Circuits", arXiv:2508.14670, (2025).

[7] Victor V. Albert and Philippe Faist, "Handbook of Error-Correcting Codes", arXiv:2606.11484, (2026).

[8] Michael J. Cervia, Henry Lamm, Diyi Liu, Edison M. Murairi, and Shuchen Zhu, "Magic State Distillation using Asymptotically Good Codes on Qudits", arXiv:2512.21874, (2025).

[9] Michael Zurel, Santanil Jana, and Nadish de Silva, "High-threshold magic state distillation with quantum quadratic residue codes", arXiv:2603.18560, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-07 19:53:16) and SAO/NASA ADS (last updated successfully 2026-08-07 19:53:17). The list may be incomplete as not all publishers provide suitable and complete citation data.

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