A Mathematical Structure for Amplitude-Mixing Error-Transparent Gates for Binomial Codes

Owen C. Wetherbee; Saswata Roy; Baptiste Royer; Valla Fatemi

doi:10.22331/q-2025-10-21-1890

A Mathematical Structure for Amplitude-Mixing Error-Transparent Gates for Binomial Codes

Owen C. Wetherbee¹, Saswata Roy¹, Baptiste Royer², and Valla Fatemi³

¹Department of Physics, Cornell University, Ithaca, NY, 14853, USA
²Département de Physique and Institut Quantique, Université de Sherbrooke, Sherbrooke J1K 2R1, QC, Canada
³School of Applied and Engineering Physics, Cornell University, Ithaca, NY, 14853, USA

Published:	2025-10-21, volume 9, page 1890
Editor:	Pei Zeng
Eprint:	arXiv:2412.08870v2
Doi:	https://doi.org/10.22331/q-2025-10-21-1890
Citation:	Quantum 9, 1890 (2025).

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

Abstract

Bosonic encodings of quantum information offer hardware-efficient, noise-biased approaches to quantum error correction relative to qubit register encodings. Implementations have focused in particular on error correction of stored, idle quantum information, whereas quantum algorithms are likely to desire high duty cycles of active control. Error-transparent operations are one way to preserve error rates during operations, but, to the best of our knowledge, only phase gates have so far been given an explicitly error-transparent formulation for binomial encodings. Here, we introduce the concept of 'parity nested' operations, and show how these operations can be designed to achieve continuous amplitude-mixing logical gates for binomial encodings that are fully error-transparent to the photon loss channel. For a binomial encoding that protects against $l$ photon losses, the construction requires $\textit{$\lfloor$l/2$\rfloor$ + 1}$ orders of generalized squeezing in the parity nested operation to fully preserve this protection. We further show that error-transparency to all the correctable photon jumps, but not the no-jump errors, can be achieved with just a single order of squeezing. Finally, we comment on possible approaches to experimental realization of this concept.

Featured image: General procedure for constructing a 'parity-nested' error-transparent $\bar{X}(\theta)$ gate for a binomial code, depicted here with order $N = 3$ and cutoff $K = 3$. After breaking the Fock states into generalized mod $N$ parity manifolds, the couplings in each manifold are tuned to achieve identical $\bar{X}(\theta)$ evolution in the codespace and all error subspaces. The solid colored arrows depict the couplings required for error-transparency to the errors $\hat{a}$ and $\hat{a}^{2}$, and the dotted arrow is the extra coupling required to achieve a full ET distance of $2$ (i.e. also ET to the no-jump error $\hat{n}$). We note that couplings between Fock states separated by an odd number of states within the generalized parity manifolds do not need to be used. This is a consequence of the Fock-space structure of rotation-symmetric codes and in general results in fewer required orders of squeezing to achieve a given ET distance.

Popular summary

Quantum computers must protect their fragile quantum information from noise. One hardware-efficient approach to providing this protection involves encoding the information in a single high-dimensional system — a bosonic mode — rather than a collection of two-dimensional qubits. While such encodings can reliably protect idle quantum information, operations often transform correctable errors into destructive ones, greatly reducing the effectiveness of these encodings during actual computation.

In this theory work, we design universal operations for bosonic encodings that are 'error-transparent,' meaning correctable errors remain correctable throughout the operation. We achieve this by conceptually dividing the high-dimensional system into distinct blocks and mirroring the operation in each block so that regardless of when an error occurs, the system always ends up in the same final state. This shows that universal single-qubit operations can be made error-transparent, constituting an important step towards fault-tolerant quantum computing with bosonic encodings.

► BibTeX data

@article{Wetherbee2025mathematical,
  doi = {10.22331/q-2025-10-21-1890},
  url = {https://doi.org/10.22331/q-2025-10-21-1890},
  title = {A {M}athematical {S}tructure for {A}mplitude-{M}ixing {E}rror-{T}ransparent {G}ates for {B}inomial {C}odes},
  author = {Wetherbee, Owen C. and Roy, Saswata and Royer, Baptiste and Fatemi, Valla},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1890},
  month = oct,
  year = {2025}
}

► References

[1] 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

[2] B. M. Terhal, J. Conrad, and C. Vuillot. ``Towards scalable bosonic quantum error correction''. Quantum Science and Technology 5, 043001 (2020).
https://doi.org/10.1088/2058-9565/ab98a5

[3] Yuan Liu, Shraddha Singh, Kevin C. Smith, Eleanor Crane, John M. Martyn, Alec Eickbusch, Alexander Schuckert, Richard D. Li, Jasmine Sinanan-Singh, Micheline B. Soley, Takahiro Tsunoda, Isaac L. Chuang, Nathan Wiebe, and Steven M. Girvin. ``Hybrid oscillator-qubit quantum processors: Instruction set architectures, abstract machine models, and applications''. PRX QuantumPages – (2025).
https://doi.org/10.1103/4rf7-9tfx

[4] Matthew Reagor, Wolfgang Pfaff, Christopher Axline, Reinier W. Heeres, Nissim Ofek, Katrina Sliwa, Eric Holland, Chen Wang, Jacob Blumoff, Kevin Chou, Michael J. Hatridge, Luigi Frunzio, Michel H. Devoret, Liang Jiang, and Robert J. Schoelkopf. ``Quantum memory with millisecond coherence in circuit QED''. Physical Review B 94, 014506 (2016).
https://doi.org/10.1103/PhysRevB.94.014506

[5] Chan U Lei, Lev Krayzman, Suhas Ganjam, Luigi Frunzio, and Robert J. Schoelkopf. ``High coherence superconducting microwave cavities with indium bump bonding''. Applied Physics Letters 116, 154002 (2020).
https://doi.org/10.1063/5.0003907

[6] 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, 030336 (2023).
https://doi.org/10.1103/PRXQuantum.4.030336

[7] Suhas Ganjam, Yanhao Wang, Yao Lu, Archan Banerjee, Chan U. Lei, Lev Krayzman, Kim Kisslinger, Chenyu Zhou, Ruoshui Li, Yichen Jia, Mingzhao Liu, Luigi Frunzio, and Robert J. Schoelkopf. ``Surpassing millisecond coherence in on chip superconducting quantum memories by optimizing materials and circuit design''. Nature Communications 15, 3687 (2024).
https://doi.org/10.1038/s41467-024-47857-6

[8] L. Hu, Y. Ma, W. Cai, X. Mu, Y. Xu, W. Wang, Y. Wu, H. Wang, Y. P. Song, C.-L. Zou, S. M. Girvin, L.-M. Duan, and L. Sun. ``Quantum error correction and universal gate set operation on a binomial bosonic logical qubit''. Nature Physics 15, 503–508 (2019).
https://doi.org/10.1038/s41567-018-0414-3

[9] Zhongchu Ni, Sai Li, Xiaowei Deng, Yanyan Cai, Libo Zhang, Weiting Wang, Zhen-Biao Yang, Haifeng Yu, Fei Yan, Song Liu, Chang-Ling Zou, Luyan Sun, Shi-Biao Zheng, Yuan Xu, and Dapeng Yu. ``Beating the break-even point with a discrete-variable-encoded logical qubit''. Nature 616, 56–60 (2023).
https://doi.org/10.1038/s41586-023-05784-4

[10] V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsioutsios, S. Ganjam, A. Miano, B. L. Brock, A. Z. Ding, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret. ``Real-time quantum error correction beyond break-even''. Nature 616, 50–55 (2023).
https://doi.org/10.1038/s41586-023-05782-6

[11] Nissim Ofek, Andrei Petrenko, Reinier Heeres, Philip Reinhold, Zaki Leghtas, Brian Vlastakis, Yehan Liu, Luigi Frunzio, S. M. Girvin, L. Jiang, Mazyar Mirrahimi, M. H. Devoret, and R. J. Schoelkopf. ``Extending the lifetime of a quantum bit with error correction in superconducting circuits''. Nature 536, 441–445 (2016).
https://doi.org/10.1038/nature18949

[12] Marios H. Michael, Matti Silveri, R. T. Brierley, Victor V. Albert, Juha Salmilehto, Liang Jiang, and S. M. Girvin. ``New Class of Quantum Error-Correcting Codes for a Bosonic Mode''. Physical Review X 6, 031006 (2016).
https://doi.org/10.1103/PhysRevX.6.031006

[13] Arne L. Grimsmo, Joshua Combes, and Ben Q. Baragiola. ``Quantum Computing with Rotation-Symmetric Bosonic Codes''. Physical Review X 10, 011058 (2020).
https://doi.org/10.1103/PhysRevX.10.011058

[14] Eliot Kapit. ``Error-Transparent Quantum Gates for Small Logical Qubit Architectures''. Physical Review Letters 120, 050503 (2018).
https://doi.org/10.1103/PhysRevLett.120.050503

[15] Wen-Long Ma, Mengzhen Zhang, Yat Wong, Kyungjoo Noh, Serge Rosenblum, Philip Reinhold, Robert J. Schoelkopf, and Liang Jiang. ``Path-Independent Quantum Gates with Noisy Ancilla''. Physical Review Letters 125, 110503 (2020).
https://doi.org/10.1103/PhysRevLett.125.110503

[16] Y. Ma, Y. Xu, X. Mu, W. Cai, L. Hu, W. Wang, X. Pan, H. Wang, Y. P. Song, C.-L. Zou, and L. Sun. ``Error-transparent operations on a logical qubit protected by quantum error correction''. Nature Physics 16, 827–831 (2020).
https://doi.org/10.1038/s41567-020-0893-x

[17] Philip Reinhold, Serge Rosenblum, Wen-Long Ma, Luigi Frunzio, Liang Jiang, and Robert J. Schoelkopf. ``Error-corrected gates on an encoded qubit''. Nature Physics 16, 822–826 (2020).
https://doi.org/10.1038/s41567-020-0931-8

[18] Isaac L. Chuang, Debbie W. Leung, and Yoshihisa Yamamoto. ``Bosonic quantum codes for amplitude damping''. Physical Review A 56, 1114–1125 (1997).
https://doi.org/10.1103/PhysRevA.56.1114

[19] M. Ueda. ``Probability-density-functional description of quantum photodetection processes''. Quantum Optics: Journal of the European Optical Society Part B 1, 131 (1989).
https://doi.org/10.1088/0954-8998/1/2/005

[20] Ching Tsung Lee. ``Superoperators and their implications in the hybrid model for photodetection''. Physical Review A 49, 4888–4894 (1994).
https://doi.org/10.1103/PhysRevA.49.4888

[21] Markus Grassl, Linghang Kong, Zhaohui Wei, Zhang-Qi Yin, and Bei Zeng. ``Quantum Error-Correcting Codes for Qudit Amplitude Damping''. IEEE Transactions on Information Theory 64, 4674–4685 (2018).
https://doi.org/10.1109/TIT.2018.2790423

[22] Os Vy, Xiaoting Wang, and Kurt Jacobs. ``Error-transparent evolution: the ability of multi-body interactions to bypass decoherence''. New Journal of Physics 15, 053002 (2013).
https://doi.org/10.1088/1367-2630/15/5/053002

[23] Takahiro Tsunoda, James D. Teoh, William D. Kalfus, Stijn J. de Graaf, Benjamin J. Chapman, Jacob C. Curtis, Neel Thakur, Steven M. Girvin, and Robert J. Schoelkopf. ``Error-Detectable Bosonic Entangling Gates with a Noisy Ancilla''. PRX Quantum 4, 020354 (2023).
https://doi.org/10.1103/PRXQuantum.4.020354

[24] Ivan Rojkov, Matteo Simoni, Elias Zapusek, Florentin Reiter, and Jonathan Home. ``Stabilization of cat-state manifolds using nonlinear reservoir engineering'' (2024). arXiv:2407.18087.
arXiv:2407.18087

[25] Victor V. Albert, Kyungjoo Noh, Kasper Duivenvoorden, Dylan J. Young, R. T. Brierley, Philip Reinhold, Christophe Vuillot, Linshu Li, Chao Shen, S. M. Girvin, Barbara M. Terhal, and Liang Jiang. ``Performance and structure of single-mode bosonic codes''. Physical Review A 97, 032346 (2018).
https://doi.org/10.1103/PhysRevA.97.032346

[26] Saswata Roy, Alen Senanian, Christopher S. Wang, Owen C. Wetherbee, Luojia Zhang, B. Cole, C. P. Larson, E. Yelton, Kartikeya Arora, Peter L. McMahon, B. L. T. Plourde, Baptiste Royer, and Valla Fatemi. ``Synthetic High Angular Momentum Spin Dynamics in a Microwave Oscillator''. Physical Review X 15, 021009 (2025).
https://doi.org/10.1103/PhysRevX.15.021009

[27] Navin Khaneja, Timo Reiss, Cindie Kehlet, Thomas Schulte-Herbrüggen, and Steffen J. Glaser. ``Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms''. Journal of Magnetic Resonance 172, 296–305 (2005).
https://doi.org/10.1016/j.jmr.2004.11.004

[28] P. de Fouquieres, S. G. Schirmer, S. J. Glaser, and Ilya Kuprov. ``Second order gradient ascent pulse engineering''. Journal of Magnetic Resonance 212, 412–417 (2011).
https://doi.org/10.1016/j.jmr.2011.07.023

[29] Christopher S. Wang, Jacob C. Curtis, Brian J. Lester, Yaxing Zhang, Yvonne Y. Gao, Jessica Freeze, Victor S. Batista, Patrick H. Vaccaro, Isaac L. Chuang, Luigi Frunzio, Liang Jiang, S. M. Girvin, and Robert J. Schoelkopf. ``Efficient Multiphoton Sampling of Molecular Vibronic Spectra on a Superconducting Bosonic Processor''. Physical Review X 10, 021060 (2020).
https://doi.org/10.1103/PhysRevX.10.021060

[30] Axel M. Eriksson, Théo Sépulcre, Mikael Kervinen, Timo Hillmann, Marina Kudra, Simon Dupouy, Yong Lu, Maryam Khanahmadi, Jiaying Yang, Claudia Castillo-Moreno, Per Delsing, and Simone Gasparinetti. ``Universal control of a bosonic mode via drive-activated native cubic interactions''. Nature Communications 15, 2512 (2024).
https://doi.org/10.1038/s41467-024-46507-1

[31] Aron Vanselow, Brieuc Beauseigneur, Louis Lattier, Marius Villiers, Anne Denis, Pascal Morfin, Zaki Leghtas, and Philippe Campagne-Ibarcq. ``Dissipating quartets of excitations in a superconducting circuit'' (2025). arXiv:2501.05960 [quant-ph].
arXiv:2501.05960

[32] Matthew Neeley, Markus Ansmann, Radoslaw C. Bialczak, Max Hofheinz, Erik Lucero, Aaron D. O'Connell, Daniel Sank, Haohua Wang, James Wenner, Andrew N. Cleland, Michael R. Geller, and John M. Martinis. ``Emulation of a Quantum Spin with a Superconducting Phase Qudit''. Science 325, 722–725 (2009).
https://doi.org/10.1126/science.1173440

[33] Xian Wu, S. L. Tomarken, N. Anders Petersson, L. A. Martinez, Yaniv J. Rosen, and Jonathan L. DuBois. ``High-Fidelity Software-Defined Quantum Logic on a Superconducting Qudit''. Physical Review Letters 125, 170502 (2020).
https://doi.org/10.1103/PhysRevLett.125.170502

[34] Laurin E. Fischer, Alessandro Chiesa, Francesco Tacchino, Daniel J. Egger, Stefano Carretta, and Ivano Tavernelli. ``Universal Qudit Gate Synthesis for Transmons''. PRX Quantum 4, 030327 (2023).
https://doi.org/10.1103/PRXQuantum.4.030327

[35] Zihao Wang, Rayleigh W. Parker, Elizabeth Champion, and Machiel S. Blok. ``High-${E_{{J}}|{E_{C}}}$ transmon qudits with up to 12 levels''. Physical Review Applied 23, 034046 (2025).
https://doi.org/10.1103/PhysRevApplied.23.034046

[36] Serwan Asaad, Vincent Mourik, Benjamin Joecker, Mark A. I. Johnson, Andrew D. Baczewski, Hannes R. Firgau, Mateusz T. Mądzik, Vivien Schmitt, Jarryd J. Pla, Fay E. Hudson, Kohei M. Itoh, Jeffrey C. McCallum, Andrew S. Dzurak, Arne Laucht, and Andrea Morello. ``Coherent electrical control of a single high-spin nucleus in silicon''. Nature 579, 205–209 (2020).
https://doi.org/10.1038/s41586-020-2057-7

[37] Pragati Gupta, Arjen Vaartjes, Xi Yu, Andrea Morello, and Barry C. Sanders. ``Robust macroscopic Schrodinger's cat on a nucleus''. Physical Review Research 6, 013101 (2024).
https://doi.org/10.1103/PhysRevResearch.6.013101

[38] Irene Fernández de Fuentes, Tim Botzem, Mark A. I. Johnson, Arjen Vaartjes, Serwan Asaad, Vincent Mourik, Fay E. Hudson, Kohei M. Itoh, Brett C. Johnson, Alexander M. Jakob, Jeffrey C. McCallum, David N. Jamieson, Andrew S. Dzurak, and Andrea Morello. ``Navigating the 16-dimensional Hilbert space of a high-spin donor qudit with electric and magnetic fields''. Nature Communications 15, 1380 (2024).
https://doi.org/10.1038/s41467-024-45368-y

[39] Arjen Vaartjes, Martin Nurizzo, Lin Htoo Zaw, Benjamin Wilhelm, Xi Yu, Danielle Holmes, Daniel Schwienbacher, Anders Kringhøj, Mark R. van Blankenstein, Alexander M. Jakob, Fay E. Hudson, Kohei M. Itoh, Riley J. Murray, Robin Blume-Kohout, Namit Anand, Andrew S. Dzurak, David N. Jamieson, Valerio Scarani, and Andrea Morello. ``Certifying the quantumness of a nuclear spin qudit through its uniform precession''. Newton 1 (2025).
https://doi.org/10.1016/j.newton.2025.100017

[40] Xi Yu, Benjamin Wilhelm, Danielle Holmes, Arjen Vaartjes, Daniel Schwienbacher, Martin Nurizzo, Anders Kringhøj, Mark R. van Blankenstein, Alexander M. Jakob, Pragati Gupta, Fay E. Hudson, Kohei M. Itoh, Riley J. Murray, Robin Blume-Kohout, Thaddeus D. Ladd, Namit Anand, Andrew S. Dzurak, Barry C. Sanders, David N. Jamieson, and Andrea Morello. ``Schrödinger cat states of a nuclear spin qudit in silicon''. Nature Physics 21, 362–367 (2025).
https://doi.org/10.1038/s41567-024-02745-0

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

[42] M. A. Aksenov, I. V. Zalivako, I. A. Semerikov, A. S. Borisenko, N. V. Semenin, P. L. Sidorov, A. K. Fedorov, K. Yu. Khabarova, and N. N. Kolachevsky. ``Realizing quantum gates with optically addressable ${}^{171}{Yb}^{+}$ ion qudits'' Physical Review A 107, 052612 (2023).
https://doi.org/10.1103/PhysRevA.107.052612

[43] Pavel Hrmo, Benjamin Wilhelm, Lukas Gerster, Martin W. van Mourik, Marcus Huber, Rainer Blatt, Philipp Schindler, Thomas Monz, and Martin Ringbauer. ``Native qudit entanglement in a trapped ion quantum processor''. Nature Communications 14, 2242 (2023).
https://doi.org/10.1038/s41467-023-37375-2

[44] Vikas Buchemmavari, Sivaprasad Omanakuttan, Yuan-Yu Jau, and Ivan Deutsch. ``Entangling quantum logic gates in neutral atoms via the microwave-driven spin-flip blockade''. Physical Review A 109, 012615 (2024).
https://doi.org/10.1103/PhysRevA.109.012615

[45] Sivaprasad Omanakuttan, Anupam Mitra, Michael J. Martin, and Ivan H. Deutsch. ``Quantum optimal control of ten-level nuclear spin qudits in ${}^{87}{Sr}$''. Physical Review A 104, L060401 (2021).
https://doi.org/10.1103/PhysRevA.104.L060401

[46] B. E. Anderson, H. Sosa-Martinez, C. A. Riofrío, Ivan H. Deutsch, and Poul S. Jessen. ``Accurate and Robust Unitary Transformations of a High-Dimensional Quantum System''. Physical Review Letters 114, 240401 (2015).
https://doi.org/10.1103/PhysRevLett.114.240401

[47] Qian Xu, Pei Zeng, Daohong Xu, and Liang Jiang. ``Fault-Tolerant Operation of Bosonic Qubits with Discrete-Variable Ancillae''. Physical Review X 14, 031016 (2024).
https://doi.org/10.1103/PhysRevX.14.031016

[48] N. Macon and A. Spitzbart. ``Inverses of Vandermonde Matrices''. The American Mathematical Monthly 65, 95–100 (1958).
https://doi.org/10.1080/00029890.1958.11989147

[49] Akel Hashim. ``Practical Introduction to Benchmarking and Characterization of Quantum Computers''. PRX Quantum 6 (2025).
https://doi.org/10.1103/PRXQuantum.6.030202

[50] Reinier W. Heeres, Brian Vlastakis, Eric Holland, Stefan Krastanov, Victor V. Albert, Luigi Frunzio, Liang Jiang, and Robert J. Schoelkopf. ``Cavity State Manipulation Using Photon-Number Selective Phase Gates''. Physical Review Letters 115, 137002 (2015).
https://doi.org/10.1103/PhysRevLett.115.137002

Cited by

[1] Ivan Rojkov, Matteo Simoni, Elias Zapusek, Florentin Reiter, and Jonathan Home, "Stabilization of Cat-State Manifolds Using Nonlinear Reservoir Engineering", Physical Review X 16 1, 011056 (2026).

[2] Shraddha Singh, "Quantum Computing in Discrete- and Continuous-Variable Architectures", arXiv:2507.01146, (2025).

[3] 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 2026-08-13 22:41:03) and SAO/NASA ADS (last updated successfully 2026-08-13 22:41:04). 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.