Platonic dynamical decoupling sequences for interacting spin systems

Colin Read; Eduardo Serrano-Ensástiga; John Martin

doi:10.22331/q-2025-03-12-1661

Platonic dynamical decoupling sequences for interacting spin systems

Colin Read, Eduardo Serrano-Ensástiga, and John Martin

Institut de Physique Nucléaire, Atomique et de Spectroscopie, CESAM, University of Liège, B-4000 Liège, Belgium

Published:	2025-03-12, volume 9, page 1661
Editor:	Aaron Goldberg
Eprint:	arXiv:2409.04974v2
Doi:	https://doi.org/10.22331/q-2025-03-12-1661
Citation:	Quantum 9, 1661 (2025).

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

Abstract

In the NISQ era, where quantum information processing is hindered by the decoherence and dissipation of elementary quantum systems, developing new protocols to extend the lifetime of quantum states is of considerable practical and theoretical importance. A well-known technique, known as dynamical decoupling, uses a carefully designed sequence of pulses applied to a quantum system, such as a spin-$j$ (which represents a qudit with $d=2j+1$ levels), to suppress the coupling Hamiltonian between the system and its environment, thereby mitigating dissipation. While dynamical decoupling of qubit systems has been widely studied, the decoupling of qudit systems has been far less explored and often involves complex sequences and operations. In this work, we design efficient decoupling sequences composed solely of global $\mathrm{SU}(2)$ rotations and based on tetrahedral, octahedral, and icosahedral point groups, which we call Platonic sequences. We extend the Majorana representation for Hamiltonians to develop a simple framework that establishes the decoupling properties of each Platonic sequence and show its effectiveness on many examples. These sequences are universal in their ability to cancel any type of interaction with the environment for single spin-$j$ with spin quantum number $j\leqslant 5/2$, and they are capable of decoupling up to $5$-body interactions in an ensemble of interacting spin-$1/2$ with only global pulses, provided that the interaction Hamiltonian has no isotropic component, with the exception of the global identity. We also discuss their inherent robustness to finite pulse duration and a wide range of pulse errors, as well as their potential application as building blocks for dynamically corrected gates.

Featured image: (a) A dynamical decoupling sequence can induce some symmetry in the resulting averaged Hamiltonian.
(b) Using the Majorana representation, a Hamiltonian $H$ can be represented by a set of constellations. A $\mathrm{SU}(2)$ transformation on $H$ has the effect of rotating its constellations.
(c) The $\mathrm{SU}(2)$ symmetries of $H$ can be identified with the rotational symmetries of its constellations.

Popular summary

By applying a carefully chosen sequence of pulses to a quantum system—known as a dynamical decoupling sequence— it is possible to modify the Hamiltonian responsible for undesirable dynamics. When this sequence is derived from a symmetry group, it 'symmetrizes' the Hamiltonian, i.e., it transforms it into another with the same symmetry, or even suppresses it if the symmetry is 'inaccessible'. In this work, we study inaccessible SU(2) symmetries based on the Majorana representation, in which any Hamiltonian is represented as a geometrical object (a set of constellations), making rotational symmetries apparent. This approach leads to the design of new robust sequences that mitigate various types of interactions in both single and multipartite quantum systems.

► BibTeX data

@article{Read2025platonicdynamical,
  doi = {10.22331/q-2025-03-12-1661},
  url = {https://doi.org/10.22331/q-2025-03-12-1661},
  title = {Platonic dynamical decoupling sequences for interacting spin systems},
  author = {Read, Colin and Serrano-Ens{\'{a}}stiga, Eduardo and Martin, John},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1661},
  month = mar,
  year = {2025}
}

► References

[1] John Preskill. ``Quantum Computing in the NISQ era and beyond''. Quantum 2, 79 (2018).
https://doi.org/10.22331/q-2018-08-06-79

[2] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik. ``Noisy intermediate-scale quantum algorithms''. Rev. Mod. Phys. 94, 015004 (2022).
https://doi.org/10.1103/RevModPhys.94.015004

[3] Lorenza Viola, Emanuel Knill, and Seth Lloyd. ``Dynamical decoupling of open quantum systems''. Phys. Rev. Lett. 82, 2417 (1999).
https://doi.org/10.1103/PhysRevLett.82.2417

[4] Lorenza Viola and Emanuel Knill. ``Robust Dynamical Decoupling of Quantum Systems with Bounded Controls''. Phys. Rev. Lett. 90, 037901 (2003).
https://doi.org/10.1103/PhysRevLett.90.037901

[5] Götz S. Uhrig. ``Keeping a quantum bit alive by optimized ${\pi}$-pulse sequences''. Phys. Rev. Lett. 98, 100504 (2007).
https://doi.org/10.1103/PhysRevLett.98.100504

[6] Michael J. Biercuk, Hermann Uys, Aaron P. VanDevender, Nobuyasu Shiga, Wayne M. Itano, and John J. Bollinger. ``Optimized dynamical decoupling in a model quantum memory''. Nature 458, 996 (2009).
https://doi.org/10.1038/nature07951

[7] Gregory Quiroz and Daniel A. Lidar. ``Optimized dynamical decoupling via genetic algorithms''. Phys. Rev. A 88, 052306 (2013).
https://doi.org/10.1103/PhysRevA.88.052306

[8] Marcus Stollsteimer and Günter Mahler. ``Suppression of arbitrary internal coupling in a quantum register''. Phys. Rev. A 64, 052301 (2001).
https://doi.org/10.1103/PhysRevA.64.052301

[9] Gerardo A Paz-Silva, Seung-Woo Lee, Todd J Green, and Lorenza Viola. ``Dynamical decoupling sequences for multi-qubit dephasing suppression and long-time quantum memory''. New Journal of Physics 18, 073020 (2016).
https://doi.org/10.1088/1367-2630/18/7/073020

[10] Joonhee Choi, Hengyun Zhou, Helena S. Knowles, Renate Landig, Soonwon Choi, and Mikhail D. Lukin. ``Robust Dynamic Hamiltonian Engineering of Many-Body Spin Systems''. Phys. Rev. X 10, 031002 (2020).
https://doi.org/10.1103/PhysRevX.10.031002

[11] Pai Peng, Xiaoyang Huang, Chao Yin, Linta Joseph, Chandrasekhar Ramanathan, and Paola Cappellaro. ``Deep reinforcement learning for quantum hamiltonian engineering''. Phys. Rev. Appl. 18, 024033 (2022).
https://doi.org/10.1103/PhysRevApplied.18.024033

[12] C. E. Bradley, J. Randall, M. H. Abobeih, R. C. Berrevoets, M. J. Degen, M. A. Bakker, M. Markham, D. J. Twitchen, and T. H. Taminiau. ``A Ten-Qubit Solid-State Spin Register with Quantum Memory up to One Minute''. Phys. Rev. X 9, 031045 (2019).
https://doi.org/10.1103/PhysRevX.9.031045

[13] Jonas Bylander, Simon Gustavsson, Fei Yan, Fumiki Yoshihara, Khalil Harrabi, George Fitch, David G. Cory, Yasunobu Nakamura, Jaw-Shen Tsai, and William D. Oliver. ``Noise spectroscopy through dynamical decoupling with a superconducting flux qubit''. Nature Physics 7, 565 (2011).
https://doi.org/10.1038/nphys1994

[14] Jiangfeng Du, Xing Rong, Nan Zhao, Ya Wang, Jiahui Yang, and RB Liu. ``Preserving electron spin coherence in solids by optimal dynamical decoupling''. Nature 461, 1265 (2009).
https://doi.org/10.1038/nature08470

[15] G. de Lange, Z. H. Wang, D. Ristè, V. V. Dobrovitski, and R. Hanson. ``Universal Dynamical Decoupling of a Single Solid-State Spin from a Spin Bath''. Science 330, 60 (2010).
https://doi.org/10.1126/science.1192739

[16] R. Lo Franco, A. D'Arrigo, G. Falci, G. Compagno, and E. Paladino. ``Preserving entanglement and nonlocality in solid-state qubits by dynamical decoupling''. Phys. Rev. B 90, 054304 (2014).
https://doi.org/10.1103/PhysRevB.90.054304

[17] S. Damodarakurup, M. Lucamarini, G. Di Giuseppe, D. Vitali, and P. Tombesi. ``Experimental inhibition of decoherence on flying qubits via ``bang-bang'' control''. Phys. Rev. Lett. 103, 040502 (2009).
https://doi.org/10.1103/PhysRevLett.103.040502

[18] Bhaskar Roy Bardhan, Petr M. Anisimov, Manish K. Gupta, Katherine L. Brown, N. Cody Jones, Hwang Lee, and Jonathan P. Dowling. ``Dynamical decoupling in optical fibers: Preserving polarization qubits from birefringent dephasing''. Phys. Rev. A 85, 022340 (2012).
https://doi.org/10.1103/PhysRevA.85.022340

[19] Bo-Yuan Ning, Jun Zhuang, J. Q. You, and Wenxian Zhang. ``Enhancement of spin coherence in a spin-1 Bose-Einstein condensate by dynamical decoupling approaches''. Phys. Rev. A 84, 013606 (2011).
https://doi.org/10.1103/PhysRevA.84.013606

[20] Hagai Edri, Boaz Raz, Gavriel Fleurov, Roee Ozeri, and Nir Davidson. ``Observation of nonlinear spin dynamics and squeezing in a BEC using dynamic decoupling''. New Journal of Physics 23, 053005 (2021).
https://doi.org/10.1088/1367-2630/abf703

[21] Rangeet Bhattacharyya, Ipsita Chakraborty, Arnab Chakrabarti, and Swagata Mandal. ``Chapter two - Recent studies on accurate measurements of NMR transverse relaxation times''. In Graham A. Webb, editor, Annual Reports on NMR Spectroscopy. Volume 99 of Annual Reports on NMR Spectroscopy, pages 57–77. Academic Press (2020).
https://doi.org/10.1016/bs.arnmr.2019.09.001

[22] R.M. Goldblatt, A.M. Martin, and A.A. Wood. ``Sensing Coherent Nuclear Spin Dynamics with an Ensemble of Paramagnetic Nitrogen Spins''. PRX Quantum 5, 020334 (2024).
https://doi.org/10.1103/PRXQuantum.5.020334

[23] 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 (2022).
https://doi.org/10.1038/s41567-022-01658-0

[24] 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 (2023).
https://doi.org/10.1038/s41467-023-37375-2

[25] Pei Liu, Ruixia Wang, Jing-Ning Zhang, Yingshan Zhang, Xiaoxia Cai, Huikai Xu, Zhiyuan Li, Jiaxiu Han, Xuegang Li, Guangming Xue, Weiyang Liu, Li You, Yirong Jin, and Haifeng Yu. ``Performing $\mathrm{SU}(d)$ Operations and Rudimentary Algorithms in a Superconducting Transmon Qudit for $d=3$ and $d=4$''. Phys. Rev. X 13, 021028 (2023).
https://doi.org/10.1103/PhysRevX.13.021028

[26] 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

[27] 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'' (2024). arXiv:2405.15695.
arXiv:2405.15695

[28] M. S. Blok, V. V. Ramasesh, T. Schuster, K. O'Brien, J. M. Kreikebaum, D. Dahlen, A. Morvan, B. Yoshida, N. Y. Yao, and I. Siddiqi. ``Quantum Information Scrambling on a Superconducting Qutrit Processor''. Phys. Rev. X 11, 021010 (2021).
https://doi.org/10.1103/PhysRevX.11.021010

[29] Sivaprasad Omanakuttan, Vikas Buchemmavari, Jonathan A. Gross, Ivan H. Deutsch, and Milad Marvian. ``Fault-tolerant quantum computation using large spin-cat codes''. PRX Quantum 5, 020355 (2024).
https://doi.org/10.1103/PRXQuantum.5.020355

[30] Ben Lanyon, Marco Barbieri, Marcelo Almeida, Thomas Jennewein, Timothy Ralph, Kevin Resch, Geoff Pryde, Jeremy O'Brien, Alexei Gilchrist, and Andrew White. ``Simplifying quantum logic using higher-dimensional Hilbert spaces''. Nat. Phys. 5 (2009).
https://doi.org/10.1038/nphys1150

[31] Yuchen Wang, Zixuan Hu, Barry C. Sanders, and Sabre Kais. ``Qudits and High-Dimensional Quantum Computing''. Frontiers in Physics 8 (2020).
https://doi.org/10.3389/fphy.2020.589504

[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 (2009).
https://doi.org/10.1126/science.1173440

[33] Giuseppe Calajó, Giuseppe Magnifico, Claire Edmunds, Martin Ringbauer, Simone Montangero, and Pietro Silvi. ``Digital Quantum Simulation of a (1+1)D SU(2) Lattice Gauge Theory with Ion Qudits''. PRX Quantum 5, 040309 (2024).
https://doi.org/10.1103/PRXQuantum.5.040309

[34] Pavel P. Popov, Michael Meth, Maciej Lewestein, Philipp Hauke, Martin Ringbauer, Erez Zohar, and Valentin Kasper. ``Variational quantum simulation of U(1) lattice gauge theories with qudit systems''. Phys. Rev. Res. 6, 013202 (2024).
https://doi.org/10.1103/PhysRevResearch.6.013202

[35] Arian Vezvaee, Nathan Earnest-Noble, and Khadijeh Najafi. ``Quantum simulation of Fermi-Hubbard model based on transmon qudit interaction'' (2024). arXiv:2402.01243.
arXiv:2402.01243

[36] John Martin, Stefan Weigert, and Olivier Giraud. ``Optimal Detection of Rotations about Unknown Axes by Coherent and Anticoherent States''. Quantum 4, 285 (2020).
https://doi.org/10.22331/q-2020-06-22-285

[37] Eduardo Serrano-Ensástiga, Chryssomalis Chryssomalakos, and John Martin. ``Quantum metrology of rotations with mixed spin states'' (2024). arXiv:2404.15548.
https://doi.org/10.1103/PhysRevA.111.022435
arXiv:2404.15548

[38] Daniel Gottesman, Alexei Kitaev, and John Preskill. ``Encoding a qubit in an oscillator''. Phys. Rev. A 64, 012310 (2001).
https://doi.org/10.1103/PhysRevA.64.012310

[39] Carlo Cafaro, Federico Maiolini, and Stefano Mancini. ``Quantum stabilizer codes embedding qubits into qudits''. Phys. Rev. A 86, 022308 (2012).
https://doi.org/10.1103/PhysRevA.86.022308

[40] Sreraman Muralidharan, Chang-Ling Zou, Linshu Li, Jianming Wen, and Liang Jiang. ``Overcoming erasure errors with multilevel systems''. New Journal of Physics 19, 013026 (2017).
https://doi.org/10.1088/1367-2630/aa573a

[41] Vinay Tripathi, Noah Goss, Arian Vezvaee, Long B. Nguyen, Irfan Siddiqi, and Daniel A. Lidar. ``Qudit dynamical decoupling on a superconducting quantum processor'' (2024). arXiv:2407.04893.
https://doi.org/10.1103/PhysRevLett.134.050601
arXiv:2407.04893

[42] Lorenza Viola and Seth Lloyd. ``Dynamical suppression of decoherence in two-state quantum systems''. Phys. Rev. A 58, 2733 (1998).
https://doi.org/10.1103/PhysRevA.58.2733

[43] Yutaro Iiyama, Wonho Jang, Naoki Kanazawa, Ryu Sawada, Tamiya Onodera, and Koji Terashi. ``Qudit-Generalization of the Qubit Echo and Its Application to a Qutrit-Based Toffoli Gate'' (2024). arXiv:2405.14752.
arXiv:2405.14752

[44] Pawel Wocjan, Martin Rötteler, Dominik Janzing, and Thomas Beth. ``Simulating hamiltonians in quantum networks: Efficient schemes and complexity bounds''. Phys. Rev. A 65, 042309 (2002).
https://doi.org/10.1103/PhysRevA.65.042309

[45] Pawel Wocjan. ``Efficient decoupling schemes with bounded controls based on eulerian orthogonal arrays''. Phys. Rev. A 73, 062317 (2006).
https://doi.org/10.1103/PhysRevA.73.062317

[46] M. Rotteler and P. Wocjan. ``Equivalence of decoupling schemes and orthogonal arrays''. IEEE Transactions on Information Theory 52, 4171 (2006).
https://doi.org/10.1109/TIT.2006.880059

[47] Martin Rötteler and Pawel Wocjan. ``Combinatorial approaches to dynamical decoupling''. Page 376–394. Cambridge University Press. (2013).
https://doi.org/10.1017/CBO9781139034807.017

[48] Soonwon Choi, Norman Y. Yao, and Mikhail D. Lukin. ``Dynamical engineering of interactions in qudit ensembles''. Phys. Rev. Lett. 119, 183603 (2017).
https://doi.org/10.1103/PhysRevLett.119.183603

[49] Hengyun Zhou, Haoyang Gao, Nathaniel T. Leitao, Oksana Makarova, Iris Cong, Alexander M. Douglas, Leigh S. Martin, and Mikhail D. Lukin. ``Robust Hamiltonian Engineering for Interacting Qudit Systems''. Phys. Rev. X 14, 031017 (2024).
https://doi.org/10.1103/PhysRevX.14.031017

[50] Paolo Zanardi. ``Symmetrizing evolutions''. Physics Lett. A 258, 77 (1999).
https://doi.org/10.1016/S0375-9601(99)00365-5

[51] E. Majorana. ``Atomi orientati in campo magnetico variabile''. Nuovo Cimento 9, 43 (1932).
https://doi.org/10.1007/BF02960953

[52] Eduardo Serrano-Ensástiga and John Martin. ``Maximum entanglement of mixed symmetric states under unitary transformations''. SciPost Phys. 15, 120 (2023).
https://doi.org/10.21468/SciPostPhys.15.3.120

[53] Nic Ezzell, Bibek Pokharel, Lina Tewala, Gregory Quiroz, and Daniel A. Lidar. ``Dynamical decoupling for superconducting qubits: A performance survey''. Phys. Rev. Appl. 20, 064027 (2023).
https://doi.org/10.1103/PhysRevApplied.20.064027

[54] Kaveh Khodjasteh and Lorenza Viola. ``Dynamically error-corrected gates for universal quantum computation''. Phys. Rev. Lett. 102, 080501 (2009).
https://doi.org/10.1103/PhysRevLett.102.080501

[55] P.C. Moan, University of Cambridge. Department of Applied Mathematics, and Theoretical Physics. ``On backward error analysis and nekhoroshev stability in the numerical analysis of conservative systems of odes''. University of Cambridge. (2002). url: https://books.google.be/books?id=hA2qXwAACAAJ.
https://books.google.be/books?id=hA2qXwAACAAJ

[56] S. Blanes, F. Casas, J.A. Oteo, and J. Ros. ``The magnus expansion and some of its applications''. Physics Reports 470, 151–238 (2009).
https://doi.org/10.1016/j.physrep.2008.11.001

[57] Kaveh Khodjasteh and Lorenza Viola. ``Dynamical quantum error correction of unitary operations with bounded controls''. Phys. Rev. A 80 (2009).
https://doi.org/10.1103/physreva.80.032314

[58] L. M. K. Vandersypen and I. L. Chuang. ``NMR techniques for quantum control and computation''. Rev. Mod. Phys. 76, 1037 (2005).
https://doi.org/10.1103/RevModPhys.76.1037

[59] Hugo Ferretti, Y. Batuhan Yilmaz, Kent Bonsma-Fisher, Aaron Z. Goldberg, Noah Lupu-Gladstein, Arthur O. T. Pang, Lee A. Rozema, and Aephraim M. Steinberg. ``Generating a 4-photon tetrahedron state: toward simultaneous super-sensitivity to non-commuting rotations''. Optica Quantum 2, 91 (2024).
https://doi.org/10.1364/OPTICAQ.510125

[60] David C. Spierings, Joseph H. Thywissen, and Aephraim M. Steinberg. ``Spin Rotations in a Bose-Einstein Condensate Driven by Counterflow and Spin-Independent Interactions''. Phys. Rev. Lett. 132, 173401 (2024).
https://doi.org/10.1103/PhysRevLett.132.173401

[61] E. Serrano-Ensástiga and D. Braun. ``Majorana representation for mixed states''. Phys. Rev. A 101, 022332 (2020).
https://doi.org/10.1103/PhysRevA.101.022332

[62] Lorenza Viola. ``Introduction to quantum dynamical decoupling''. Page 105–125. Cambridge University Press. (2013).
https://doi.org/10.1017/CBO9781139034807.006

[63] Béla Bollobás. ``Graphs, groups and matrices''. Pages 253–293. Springer New York. New York, NY (1998).
https://doi.org/10.1007/978-1-4612-0619-4_8

[64] Mildred S Dresselhaus, Gene Dresselhaus, and Ado Jorio. ``Group theory: application to the physics of condensed matter''. Springer Science & Business Media. (2007).
https://doi.org/10.1007/978-3-540-32899-5

[65] C. J. Bradley and A. P. Cracknell. ``The mathematical theory of symmetry in solids: Representation theory for point groups and space groups''. Oxford University Press. (2009).
https://doi.org/10.1093/oso/9780199582587.001.0001

[66] U. Fano. ``Geometrical characterization of nuclear states and the theory of angular correlations''. Phys. Rev. 90, 577 (1953).
https://doi.org/10.1103/physrev.90.577

[67] D A Varshalovich, A N Moskalev, and V K Khersonskii. ``Quantum theory of angular momentum''. World Scientific. (1988).
https://doi.org/10.1142/0270

[68] A. V. Shubnikov and N. V. Belov. ``Colored symmetry''. Pergamon Press, Oxford. (1964).

[69] Matthew D Grace, Jason Dominy, Robert L Kosut, Constantin Brif, and Herschel Rabitz. ``Environment-invariant measure of distance between evolutions of an open quantum system''. New Journal of Physics 12, 015001 (2010).
https://doi.org/10.1088/1367-2630/12/1/015001

[70] Sivaprasad Omanakuttan. ``Quantum computation using large spin qudits'' (2024). arXiv:2405.07885.
arXiv:2405.07885

[71] Ivan H. Deutsch and Poul S. Jessen. ``Quantum control and measurement of atomic spins in polarization spectroscopy''. Optics Communications 283, 681 (2010).
https://doi.org/10.1016/j.optcom.2009.10.059

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

[73] Marcis Auzinsh, Dmitry Budker, and Simon Rochester. ``Optically polarized atoms: Understanding light–atom interactions''. Oxford University Press. (2010).

[74] Elliot A. Kearsley and Jeffrey T. Fong. ``Linearly independent sets of isotropic cartesian tensors of ranks up to eight''. Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences 79B, 49 (1975).
https://doi.org/10.6028/jres.079b.005

[75] Maciej Lewenstein, Anna Sanpera, and Verònica Ahufinger. ``Ultracold Atoms in Optical Lattices: Simulating quantum many-body systems''. Oxford University Press. (2012).
https://doi.org/10.1093/acprof:oso/9780199573127.001.0001

[76] Malcolm H Levitt. ``Spin dynamics: basics of nuclear magnetic resonance''. John Wiley & Sons. (2008).

[77] H. P. Büchler, A. Micheli, and P. Zoller. ``Three-body interactions with cold polar molecules''. Nature Physics 3, 726 (2007).
https://doi.org/10.1038/nphys678

[78] D.G Cory, J.B Miller, and A.N Garroway. ``Time-suspension multiple-pulse sequences: applications to solid-state imaging''. Journal of Magnetic Resonance (1969) 90, 205 (1990).
https://doi.org/10.1016/0022-2364(90)90380-R

[79] Thomas Theis, Paul Ganssle, Gwendal Kervern, Svenja Knappe, John Kitching, MP Ledbetter, Dmitry Budker, and Alex Pines. ``Parahydrogen-enhanced zero-field nuclear magnetic resonance''. Nature Physics 7, 571 (2011).
https://doi.org/10.1038/nphys1986

[80] J. S. Waugh, L. M. Huber, and U. Haeberlen. ``Approach to High-Resolution NMR in Solids''. Phys. Rev. Lett. 20, 180 (1968).
https://doi.org/10.1103/physrevlett.20.180

[81] W‐K. Rhim, D. D. Elleman, and R. W. Vaughan. ``Enhanced resolution for solid state NMR''. The Journal of Chemical Physics 58, 1772 (1973).
https://doi.org/10.1063/1.1679423

[82] K. Khodjasteh and D. A. Lidar. ``Fault-tolerant quantum dynamical decoupling''. Phys. Rev. Lett. 95, 180501 (2005).
https://doi.org/10.1103/PhysRevLett.95.180501

[83] Colin Read, Eduardo Serrano-Ensástiga, and John Martin. Work in progress.

[84] Noah Goss, Alexis Morvan, Brian Marinelli, Bradley K. Mitchell, Long B. Nguyen, Ravi K. Naik, Larry Chen, Christian Jünger, John Mark Kreikebaum, David I. Santiago, Joel J. Wallman, and Irfan Siddiqi. ``High-fidelity qutrit entangling gates for superconducting circuits''. Nature Communications 13 (2022).
https://doi.org/10.1038/s41467-022-34851-z

[85] Kaveh Khodjasteh, Daniel A. Lidar, and Lorenza Viola. ``Arbitrarily accurate dynamical control in open quantum systems''. Phys. Rev. Lett. 104, 090501 (2010).
https://doi.org/10.1103/PhysRevLett.104.090501

[86] Sen Yang, Ya Wang, D. D. Bhaktavatsala Rao, Thai Hien Tran, Ali S. Momenzadeh, M. Markham, D. J. Twitchen, Ping Wang, Wen Yang, Rainer Stöhr, Philipp Neumann, Hideo Kosaka, and Jörg Wrachtrup. ``High-fidelity transfer and storage of photon states in a single nuclear spin''. Nature Photonics 10, 507 (2016).
https://doi.org/10.1038/nphoton.2016.103

[87] N. Kalb, P. C. Humphreys, J. J. Slim, and R. Hanson. ``Dephasing mechanisms of diamond-based nuclear-spin memories for quantum networks''. Phys. Rev. A 97, 062330 (2018).
https://doi.org/10.1103/physreva.97.062330

[88] Yue Fu, Wenquan Liu, Xiangyu Ye, Ya Wang, Chengjie Zhang, Chang-Kui Duan, Xing Rong, and Jiangfeng Du. ``Experimental investigation of quantum correlations in a two-qutrit spin system''. Phys. Rev. Lett. 129, 100501 (2022).
https://doi.org/10.1103/PhysRevLett.129.100501

[89] Dmitry Zakharov, Hans-Albrecht Krug von Nidda, Mikhail Eremin, Joachim Deisenhofer, Rushana Eremina, and Alois Loidl. ``Anisotropic exchange in spin chains''. In Bernard Barbara, Yosef Imry, G. Sawatzky, and P. C. E. Stamp, editors, Quantum Magnetism. Pages 193–238. Dordrecht (2008). Springer Netherlands.
https://doi.org/10.1007/978-1-4020-8512-3_14

[90] Hengyun Zhou, Leigh S. Martin, Matthew Tyler, Oksana Makarova, Nathaniel Leitao, Hongkun Park, and Mikhail D. Lukin. ``Robust higher-order hamiltonian engineering for quantum sensing with strongly interacting systems''. Phys. Rev. Lett. 131, 220803 (2023).
https://doi.org/10.1103/PhysRevLett.131.220803

[91] Sam R. Cohen and Jeff D. Thompson. ``Quantum Computing with Circular Rydberg Atoms''. PRX Quantum 2, 030322 (2021).
https://doi.org/10.1103/PRXQuantum.2.030322

[92] J. Martin, O. Giraud, P. A. Braun, D. Braun, and T. Bastin. ``Multiqubit symmetric states with high geometric entanglement''. Phys. Rev. A 81, 062347 (2010).
https://doi.org/10.1103/PhysRevA.81.062347

[93] C. Chryssomalakos, L. Hanotel, E. Guzmán-González, D. Braun, E. Serrano-Ensástiga, and K. Życzkowski. ``Symmetric multiqudit states: Stars, entanglement, and rotosensors''. Phys. Rev. A 104, 012407 (2021).
https://doi.org/10.1103/PhysRevA.104.012407

[94] D. Morachis Galindo and Jesús A. Maytorena. ``Entangling power of symmetric two-qubit quantum gates and three-level operations''. Phys. Rev. A 105, 012601 (2022).
https://doi.org/10.1103/PhysRevA.105.012601

[95] Danisch & Krumbiegel. ``Makie.jl: Flexible high-performance data visualization for julia''. Journal of Open Source Software 6, 3349 (2021).
https://doi.org/10.21105/joss.03349

[96] Joseph J Rotman. ``An introduction to the theory of groups''. Volume 148. Springer Science & Business Media. (2012).
https://doi.org/10.1007/978-1-4612-4176-8

[97] Harold Scott Macdonald Coxeter. ``Regular polytopes''. Dover, New York. (1973). 3rd edition.

[98] Igor Pak and Radoš Radoičić. ``Hamiltonian paths in cayley graphs''. Discrete Mathematics 309, 5501 (2009).
https://doi.org/10.1016/j.disc.2009.02.018

[99] Carl Hierholzer. ``Ueber die möglichkeit, einen linienzug ohne wiederholung und ohne unterbrechung zu umfahren''. Mathematische Annalen 6, 30 (1873). url: http://eudml.org/doc/156599.
http://eudml.org/doc/156599

[100] C. Chryssomalakos, E. Guzmán-González, and E. Serrano-Ensástiga. ``Geometry of spin coherent states''. J. Phys. A: Math. Theor. 51, 165202 (2018).
https://doi.org/10.1088/1751-8121/aab349

[101] P. G. Appleby, B. R. Duffy, and R. W. Ogden. ``On the classification of isotropic tensors''. Glasgow Mathematical Journal 29, 185 (1987).
https://doi.org/10.1017/S0017089500006832

[102] Harold Jeffreys. ``On isotropic tensors''. Mathematical Proceedings of the Cambridge Philosophical Society 73, 173 (1973).
https://doi.org/10.1017/S0305004100047587

[103] Philip G. Hodge. ``On isotropic cartesian tensors''. The American Mathematical Monthly 68, 793 (1961).
https://doi.org/10.2307/2311997

Cited by

[1] Wenqu Xu, Tianshi Zhou, Kaixuan Zhang, Zishuo Li, Tingting Guo, Quan Zuo, Mingyuan Yu, Junliang Jiang, Jiazheng Pan, Jinpeng Li, Lingxiao Jing, Huashi Ma, Shunhong Zhou, Binglin Li, Shiyao Yang, Yongyang Yu, Junzhou Zhang, Jiyuan Zhu, Ziru Hou, Haifeng Song, Chunhai Cao, Guozhu Sun, and Peiheng Wu, "Multilevel dynamical decoupling for a coupled superconducting two-qubit system", Physical Review A 112 2, 022435 (2025).

[2] Jérôme Denis, Colin Read, and John Martin, "Coherent generation and protection of anticoherent spin states", SciPost Physics Core 9 1, 001 (2026).

[3] Colin Read, Eduardo Serrano-Ensástiga, and John Martin, "Dynamical decoupling of interacting spins through group factorization", Physical Review A 112 4, 042601 (2025).

[4] O.M. Konovalenko and Z.A. Maizelis, "Multiple Joint Weak Measurements as a Way to Suppress the Decoherence", Ukrainian Journal of Physics 70 8, 516 (2025).

[5] Michael C. D. Tayler and Mohamed Sabba, "Cyclicity of interaction-frame transformations", Physical Review A 112 5, 052432 (2025).

[6] Minh T.P. Nguyen, Maximilian Rimbach-Russ, and Stefano Bosco, "Color It, Code It, Cancel It: k -Local Dynamical Decoupling from Classical Additive Codes", PRX Quantum 7 2, 020338 (2026).

[7] Jérôme Denis, Colin Read, and John Martin, "Coherent Generation and Protection of Anticoherent Spin States", arXiv:2505.06154, (2025).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-12 00:51:54) and SAO/NASA ADS (last updated successfully 2026-08-12 00:51:55). 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.