High-performance repetition cat code using fast noisy operations

Francois-Marie Le Régent; Camille Berdou; Zaki Leghtas; Jérémie Guillaud; Mazyar Mirrahimi

doi:10.22331/q-2023-12-06-1198

High-performance repetition cat code using fast noisy operations

Francois-Marie Le Régent^1,2, Camille Berdou², Zaki Leghtas², Jérémie Guillaud¹, and Mazyar Mirrahimi²

¹Alice&Bob, 53 boulevard du Général Martial Valin, 75015 Paris
²Laboratoire de Physique de l'Ecole Normale Supérieure, Ecole normale supérieure, MINES Paris, Université PSL, Sorbonne Université, CNRS, Inria, 75005 Paris

Published:	2023-12-06, volume 7, page 1198
Eprint:	arXiv:2212.11927v5
Doi:	https://doi.org/10.22331/q-2023-12-06-1198
Citation:	Quantum 7, 1198 (2023).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Updated after initial publication: This publication was updated to version v5 after the initial publication. The authors left the following comment on the arXiv:

19 pages, 12 figures

Abstract

Bosonic cat qubits stabilized by two-photon driven dissipation benefit from exponential suppression of bit-flip errors and an extensive set of gates preserving this protection. These properties make them promising building blocks of a hardware-efficient and fault-tolerant quantum processor. In this paper, we propose a performance optimization of the repetition cat code architecture using fast but noisy CNOT gates for stabilizer measurements. This optimization leads to high thresholds for the physical figure of merit, given as the ratio between intrinsic single-photon loss rate of the bosonic mode and the engineered two-photon loss rate, as well as an improved scaling below threshold of the required overhead, to reach an expected level of logical error rate. Relying on the specific error models for cat qubit operations, this optimization exploits fast parity measurements, using accelerated low-fidelity CNOT gates, combined with fast ancilla parity-check qubits. The significant enhancement in the performance is explained by: 1- the highly asymmetric error model of cat qubit CNOT gates with a major component on control (ancilla) qubits, and 2- the robustness of the repetition cat code error correction performance in presence of the leakage induced by fast operations. In order to demonstrate these performances, we develop a method to sample the repetition code under circuit-level noise that also takes into account cat qubit state leakage.

Featured image: QEC circuit for the distance-$d$ repetition cat code.
(a) a repetition cat code consists of a 1D array of stabilized cat qubits, where the phase-flip error correction is performed by repetitive $XX$ parity measurements between neighboring cat qubits.
(b) The QEC on a distance-$d$ phase-flip repetition code is performed by repeated $XX$ parity measurements.
(c) Each QEC cycle is composed of $d-1$ $XX$ parity checks between neighboring cat qubits. Each parity check is performed by applying bias-preserving CNOT gates between the two associated data cat qubits (as targets) and one ancilla cat qubit (as control). The ancilla cat qubit is finally measured in its $X$ basis.
(d) The measurement of an ancilla cat qubit in the $X$ basis is performed through a photon number parity measurement. This is done by turning off the two-photon dissipation mechanism on the ancilla cat qubit and following a Ramsey type experiment. In this scheme the ancilla cat mode is coupled dispersively to a qubit, e.g. a transmon, where the interaction Hamiltonian is given by $\hat{H}_{\mathrm{int}}=\hbar \frac{\chi}{2} \hat{\sigma}_z \hat{a}^{\dagger} \hat{a}$. The Ramsey sequence consists then of applying two $\pi/2$-pulses on the qubit separated by a waiting time of $\pi/\chi$. During the waiting time, the qubit accumulates a phase of $0$ or $\pi$ depending on the photon number parity of the cat mode. The final readout of the qubit thus measures this observable.

Popular summary

The cat qubits are bosonic codes benefitting from exponentially suppressed bit-flip rates with increasing number of encoding photons. In conjunction with bias-preserving operations, they can be used to drastically reduce the hardware overhead required for error correction towards fault-tolerant quantum computation. In this paper we investigate another useful property of such qubits, the asymmetry in the phase-flip errors of a bias-preserving CNOT gate. While performing a fast CNOT gate, the probability of such an error increases on the control cat qubit while it decreases on the target one. By exploiting this property, we demonstrate an improved performance of error correction in a concatenated repetition cat code architecture. A central contribution of our paper is to propose a way to handle the further leakage out of code space induced by such fast operations. The numerical tools developed in the paper demonstrate the viability of this approach towards further reduction of hardware overhead.

► BibTeX data

@article{Regent2023highperformance,
  doi = {10.22331/q-2023-12-06-1198},
  url = {https://doi.org/10.22331/q-2023-12-06-1198},
  title = {High-performance repetition cat code using fast noisy operations},
  author = {R{\'{e}}gent, Francois-Marie Le and Berdou, Camille and Leghtas, Zaki and Guillaud, J{\'{e}}r{\'{e}}mie and Mirrahimi, Mazyar},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1198},
  month = dec,
  year = {2023}
}

► References

[1] Atharv Joshi, Kyungjoo Noh, and Yvonne Y Gao. ``Quantum information processing with bosonic qubits in circuit QED''. Quantum Science and Technology 6, 033001 (2021).
https://doi.org/10.1088/2058-9565/abe989

[2] Weizhou Cai, Yuwei Ma, Weiting Wang, Chang-Ling Zou, and Luyan Sun. ``Bosonic quantum error correction codes in superconducting quantum circuits''. Fundamental Research 1, 50–67 (2021).
https://doi.org/10.1016/j.fmre.2020.12.006

[3] Mazyar Mirrahimi, Zaki Leghtas, Victor V Albert, Steven Touzard, Robert J Schoelkopf, Liang Jiang, and Michel H Devoret. ``Dynamically protected cat-qubits: a new paradigm for universal quantum computation''. New Journal of Physics 16, 045014 (2014).
https://doi.org/10.1088/1367-2630/16/4/045014

[4] Jérémie Guillaud and Mazyar Mirrahimi. ``Repetition cat qubits for fault-tolerant quantum computation''. Physical Review X 9 (2019).
https://doi.org/10.1103/physrevx.9.041053

[5] Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, Earl T. Campbell, Connor T. Hann, Joseph Iverson, Harald Putterman, Thomas C. Bohdanowicz, Steven T. Flammia, Andrew Keller, Gil Refael, John Preskill, Liang Jiang, Amir H. Safavi-Naeini, Oskar Painter, and Fernando G.S.L. Brandão. ``Building a fault-tolerant quantum computer using concatenated cat codes''. PRX Quantum 3 (2022).
https://doi.org/10.1103/prxquantum.3.010329

[6] Raphaël Lescanne, Marius Villiers, Théau Peronnin, Alain Sarlette, Matthieu Delbecq, Benjamin Huard, Takis Kontos, Mazyar Mirrahimi, and Zaki Leghtas. ``Exponential suppression of bit-flips in a qubit encoded in an oscillator''. Nature Physics 16, 509–513 (2020).
https://doi.org/10.1038/s41567-020-0824-x

[7] C. Berdou, A. Murani, U. Réglade, W.C. Smith, M. Villiers, J. Palomo, M. Rosticher, A. Denis, P. Morfin, M. Delbecq, T. Kontos, N. Pankratova, F. Rautschke, T. Peronnin, L.-A. Sellem, P. Rouchon, A. Sarlette, M. Mirrahimi, P. Campagne-Ibarcq, S. Jezouin, R. Lescanne, and Z. Leghtas. ``One hundred second bit-flip time in a two-photon dissipative oscillator''. PRX Quantum 4 (2023).
https://doi.org/10.1103/prxquantum.4.020350

[8] L. G. Lutterbach and L. Davidovich. ``Method for direct measurement of the wigner function in cavity qed and ion traps''. Phys. Rev. Lett. 78, 2547–2550 (1997).
https://doi.org/10.1103/PhysRevLett.78.2547

[9] P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche. ``Direct measurement of the wigner function of a one-photon fock state in a cavity''. Phys. Rev. Lett. 89, 200402 (2002).
https://doi.org/10.1103/PhysRevLett.89.200402

[10] L. Sun, A. Petrenko, Z. Leghtas, B. Vlastakis, G. Kirchmair, K. M. Sliwa, A. Narla, M. Hatridge, S. Shankar, J. Blumoff, L. Frunzio, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf. ``Tracking photon jumps with repeated quantum non-demolition parity measurements''. Nature 511, 444–+ (2014).
https://doi.org/10.1038/nature13436

[11] 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).
https://doi.org/10.1103/PhysRevA.86.032324

[12] Shruti Puri, Lucas St-Jean, Jonathan A. Gross, Alexander Grimm, Nicholas E. Frattini, Pavithran S. Iyer, Anirudh Krishna, Steven Touzard, Liang Jiang, Alexandre Blais, Steven T. Flammia, and S. M. Girvin. ``Bias-preserving gates with stabilized cat qubits''. Science Advances 6 (2020).
https://doi.org/10.1126/sciadv.aay5901

[13] Qian Xu, Joseph K. Iverson, Fernando G. S. L. Brandão, and Liang Jiang. ``Engineering fast bias-preserving gates on stabilized cat qubits''. Phys. Rev. Res. 4, 013082 (2022).
https://doi.org/10.1103/PhysRevResearch.4.013082

[14] Harald Putterman, Joseph Iverson, Qian Xu, Liang Jiang, Oskar Painter, Fernando G. S. L. Brandão, and Kyungjoo Noh. ``Stabilizing a bosonic qubit using colored dissipation''. Physical Review Letters 128 (2022).
https://doi.org/10.1103/physrevlett.128.110502

[15] Ronan Gautier, Alain Sarlette, and Mazyar Mirrahimi. ``Combined dissipative and hamiltonian confinement of cat qubits''. PRX Quantum 3, 020339 (2022).
https://doi.org/10.1103/PRXQuantum.3.020339

[16] Diego Ruiz, Ronan Gautier, Jérémie Guillaud, and Mazyar Mirrahimi. ``Two-photon driven kerr quantum oscillator with multiple spectral degeneracies''. Physical Review A 107 (2023).
https://doi.org/10.1103/physreva.107.042407

[17] Qian Xu, Guo Zheng, Yu-Xin Wang, Peter Zoller, Aashish A. Clerk, and Liang Jiang. ``Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits''. npj Quantum Information 9, 78 (2023).
https://doi.org/10.1038/s41534-023-00746-0

[18] Jérémie Guillaud and Mazyar Mirrahimi. ``Error rates and resource overheads of repetition cat qubits''. Physical Review A 103 (2021).
https://doi.org/10.1103/physreva.103.042413

[19] Austin G. Fowler, Adam C. Whiteside, and Lloyd C. L. Hollenberg. ``Towards practical classical processing for the surface code''. Phys. Rev. Lett. 108, 180501 (2012).
https://doi.org/10.1103/PhysRevLett.108.180501

[20] Oscar Higgott. ``Pymatching: A python package for decoding quantum codes with minimum-weight perfect matching''. ACM Transactions on Quantum Computing (2021).
https://doi.org/10.1145/3505637

[21] P. Aliferis and B.M. Terhal. ``Fault-tolerant quantum computation for local leakage faults''. Quantum Information and Computation 7, 139–156 (2007).
https://doi.org/10.26421/qic7.1-2-9

[22] F. Battistel, B.M. Varbanov, and B.M. Terhal. ``Hardware-efficient leakage-reduction scheme for quantum error correction with superconducting transmon qubits''. PRX Quantum 2 (2021).
https://doi.org/10.1103/prxquantum.2.030314

[23] M. McEwen et al. ``Removing leakage-induced correlated errors in superconducting quantum error correction''. Nature Communications 12 (2021).
https://doi.org/10.1038/s41467-021-21982-y

[24] Zijun Chen et al. ``Exponential suppression of bit or phase errors with cyclic error correction''. Nature 595, 383–387 (2021).
https://doi.org/10.1038/s41586-021-03588-y

[25] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. ``Topological quantum memory''. Journal of Mathematical Physics 43, 4452–4505 (2002).
https://doi.org/10.1063/1.1499754

[26] Jérémie Guillaud, Joachim Cohen, and Mazyar Mirrahimi. ``Quantum computation with cat qubits''. SciPost Physics Lecture Notes (2023).
https://doi.org/10.21468/scipostphyslectnotes.72

[27] Giacomo Pantaleoni, Ben Q. Baragiola, and Nicolas C. Menicucci. ``Modular bosonic subsystem codes''. Physical Review Letters 125 (2020).
https://doi.org/10.1103/physrevlett.125.040501

[28] Andrew S. Darmawan, Benjamin J. Brown, Arne L. Grimsmo, David K. Tuckett, and Shruti Puri. ``Practical quantum error correction with the XZZX code and kerr-cat qubits''. PRX Quantum 2 (2021).
https://doi.org/10.1103/prxquantum.2.030345

[29] Scott Aaronson and Daniel Gottesman. ``Improved simulation of stabilizer circuits''. Phys. Rev. A 70, 052328 (2004).
https://doi.org/10.1103/PhysRevA.70.052328

[30] Jack Edmonds. ``Paths, trees, and flowers''. Canadian Journal of Mathematics 17, 449–467 (1965).
https://doi.org/10.4153/CJM-1965-045-4

[31] Vladimir Kolmogorov. ``Blossom v: a new implementation of a minimum cost perfect matching algorithm''. Mathematical Programming Computation 1, 43–67 (2009).
https://doi.org/10.1007/s12532-009-0002-8

Cited by

[1] A. Marquet, A. Essig, J. Cohen, N. Cottet, A. Murani, E. Albertinale, S. Dupouy, A. Bienfait, T. Peronnin, S. Jezouin, R. Lescanne, and B. Huard, "Autoparametric Resonance Extending the Bit-Flip Time of a Cat Qubit up to 0.3 s", Physical Review X 14 2, 021019 (2024).

[2] Florian Hopfmueller, Maxime Tremblay, Philippe St-Jean, Baptiste Royer, and Marc-Antoine Lemonde, "Bosonic Pauli+: Efficient Simulation of Concatenated Gottesman-Kitaev-Preskill Codes", Quantum 8, 1539 (2024).

[3] Jaehak Lee, Nuri Kang, Seok-Hyung Lee, Hyunseok Jeong, Liang Jiang, and Seung-Woo Lee, "Fault-Tolerant Quantum Computation by Hybrid Qubits with Bosonic Cat Code and Single Photons", PRX Quantum 5 3, 030322 (2024).

[4] Adrian Copetudo, Clara Yun Fontaine, Fernando Valadares, and Yvonne Y. Gao, "Shaping photons: Quantum information processing with bosonic cQED", Applied Physics Letters 124 8, 080502 (2024).

[5] Diego Ruiz, Jérémie Guillaud, Anthony Leverrier, Mazyar Mirrahimi, and Christophe Vuillot, "LDPC-cat codes for low-overhead quantum computing in 2D", Nature Communications 16 1, 1040 (2025).

[6] Shubham P. Jain, Joseph T. Iosue, Alexander Barg, and Victor V. Albert, "Quantum spherical codes", Nature Physics 20 8, 1300 (2024).

[7] Lucas Berent, Timo Hillmann, Jens Eisert, Robert Wille, and Joschka Roffe, "Analog Information Decoding of Bosonic Quantum Low-Density Parity-Check Codes", PRX Quantum 5 2, 020349 (2024).

[8] Harald Putterman, Kyungjoo Noh, Connor T. Hann, Gregory S. MacCabe, Shahriar Aghaeimeibodi, Rishi N. Patel, Menyoung Lee, William M. Jones, Hesam Moradinejad, Roberto Rodriguez, Neha Mahuli, Jefferson Rose, John Clai Owens, Harry Levine, Emma Rosenfeld, Philip Reinhold, Lorenzo Moncelsi, Joshua Ari Alcid, Nasser Alidoust, Patricio Arrangoiz-Arriola, James Barnett, Przemyslaw Bienias, Hugh A. Carson, Cliff Chen, Li Chen, Harutiun Chinkezian, Eric M. Chisholm, Ming-Han Chou, Aashish Clerk, Andrew Clifford, R. Cosmic, Ana Valdes Curiel, Erik Davis, Laura DeLorenzo, J. Mitchell D’Ewart, Art Diky, Nathan D’Souza, Philipp T. Dumitrescu, Shmuel Eisenmann, Essam Elkhouly, Glen Evenbly, Michael T. Fang, Yawen Fang, Matthew J. Fling, Warren Fon, Gabriel Garcia, Alexey V. Gorshkov, Julia A. Grant, Mason J. Gray, Sebastian Grimberg, Arne L. Grimsmo, Arbel Haim, Justin Hand, Yuan He, Mike Hernandez, David Hover, Jimmy S. C. Hung, Matthew Hunt, Joe Iverson, Ignace Jarrige, Jean-Christophe Jaskula, Liang Jiang, Mahmoud Kalaee, Rassul Karabalin, Peter J. Karalekas, Andrew J. Keller, Amirhossein Khalajhedayati, Aleksander Kubica, Hanho Lee, Catherine Leroux, Simon Lieu, Victor Ly, Keven Villegas Madrigal, Guillaume Marcaud, Gavin McCabe, Cody Miles, Ashley Milsted, Joaquin Minguzzi, Anurag Mishra, Biswaroop Mukherjee, Mahdi Naghiloo, Eric Oblepias, Gerson Ortuno, Jason Pagdilao, Nicola Pancotti, Ashley Panduro, JP Paquette, Minje Park, Gregory A. Peairs, David Perello, Eric C. Peterson, Sophia Ponte, John Preskill, Johnson Qiao, Gil Refael, Rachel Resnick, Alex Retzker, Omar A. Reyna, Marc Runyan, Colm A. Ryan, Abdulrahman Sahmoud, Ernesto Sanchez, Rohan Sanil, Krishanu Sankar, Yuki Sato, Thomas Scaffidi, Salome Siavoshi, Prasahnt Sivarajah, Trenton Skogland, Chun-Ju Su, Loren J. Swenson, Stephanie M. Teo, Astrid Tomada, Giacomo Torlai, E. Alex Wollack, Yufeng Ye, Jessica A. Zerrudo, Kailing Zhang, Fernando G. S. L. Brandão, Matthew H. Matheny, and Oskar Painter, "Hardware-efficient quantum error correction via concatenated bosonic qubits", Nature 638 8052, 927 (2025).

[9] Harald Putterman, Kyungjoo Noh, Rishi N. Patel, Gregory A. Peairs, Gregory S. MacCabe, Menyoung Lee, Shahriar Aghaeimeibodi, Connor T. Hann, Ignace Jarrige, Guillaume Marcaud, Yuan He, Hesam Moradinejad, John Clai Owens, Thomas Scaffidi, Patricio Arrangoiz-Arriola, Joe Iverson, Harry Levine, Fernando G. S. L. Brandão, Matthew H. Matheny, and Oskar Painter, "Preserving Phase Coherence and Linearity in Cat Qubits with Exponential Bit-Flip Suppression", Physical Review X 15 1, 011070 (2025).

[10] Vojtěch Kuchař and Petr Marek, "Nonlinear squeezing of superpositions of quadrature eigenstates", Physical Review A 112 5, 053706 (2025).

[11] Connor T. Hann, Kyungjoo Noh, Harald Putterman, Matthew H. Matheny, Joseph K. Iverson, Michael T. Fang, Christopher Chamberland, Oskar Painter, and Fernando G.S.L. Brandão, "Hybrid Cat-Transmon Architecture for Scalable, Hardware-Efficient Quantum Error Correction", PRX Quantum 6 3, 030305 (2025).

[12] Ziang Wang, Feng Wu, Hui-Hai Zhao, Xin Wan, and Xiaotong Ni, "SuperGrad: a differentiable simulator for superconducting processors", Quantum 9, 1722 (2025).

[13] Iivari Pietikäinen, Ondřej Černotík, Alec Eickbusch, Aniket Maiti, John W.O. Garmon, Radim Filip, and Steven M. Girvin, "Strategies and Trade-Offs for Controllability and Memory Time of Ultra-High-Quality Microwave Cavities in Circuit Quantum Electrodynamics", PRX Quantum 5 4, 040307 (2024).

[14] Yongnan Chen, Zaixu Fan, Haopeng Wang, Cewen Tian, and Hongyang Ma, "Probing Threshold Behavior of Adaptive Cascaded Quantum Codes Under Variable Biased Noise for Practical Fault-Tolerant Quantum Computing", Electronics 15 2, 436 (2026).

[15] Ofir Milul, Barkay Guttel, Uri Goldblatt, Sergey Hazanov, Lalit M. Joshi, Daniel Chausovsky, Nitzan Kahn, Engin çiftyürek, Fabien Lafont, and Serge Rosenblum, "Superconducting Cavity Qubit with Tens of Milliseconds Single-Photon Coherence Time", PRX Quantum 4 3, 030336 (2023).

[16] Élie Gouzien, Diego Ruiz, Francois-Marie Le Régent, Jérémie Guillaud, and Nicolas Sangouard, "Performance Analysis of a Repetition Cat Code Architecture: Computing 256-bit Elliptic Curve Logarithm in 9 Hours with 126 133 Cat Qubits", Physical Review Letters 131 4, 040602 (2023).

[17] Ronan Gautier, Mazyar Mirrahimi, and Alain Sarlette, "Designing High-Fidelity Zeno Gates for Dissipative Cat Qubits", PRX Quantum 4 4, 040316 (2023).

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

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-09 18:12:31) and SAO/NASA ADS (last updated successfully 2026-08-09 18:12:33). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.