Quantum Error Mitigation using Symmetry Expansion

Zhenyu Cai

doi:10.22331/q-2021-09-21-548

Quantum Error Mitigation using Symmetry Expansion

Department of Materials, University of Oxford, Oxford, OX1 3PH, United Kingdom
Quantum Motion Technologies Ltd, Nexus, Discovery Way, Leeds, LS2 3AA, United Kingdom

Published:	2021-09-21, volume 5, page 548
Eprint:	arXiv:2101.03151v3
Doi:	https://doi.org/10.22331/q-2021-09-21-548
Citation:	Quantum 5, 548 (2021).

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

Abstract

Even with the recent rapid developments in quantum hardware, noise remains the biggest challenge for the practical applications of any near-term quantum devices. Full quantum error correction cannot be implemented in these devices due to their limited scale. Therefore instead of relying on engineered code symmetry, symmetry verification was developed which uses the inherent symmetry within the physical problem we try to solve. In this article, we develop a general framework named symmetry expansion which provides a wide spectrum of symmetry-based error mitigation schemes beyond symmetry verification, enabling us to achieve different balances between the estimation bias and the sampling cost of the scheme. We show that certain symmetry expansion schemes can achieve a smaller estimation bias than symmetry verification through cancellation between the biases due to the detectable and undetectable noise components. A practical way to search for such a small-bias scheme is introduced. By numerically simulating the Fermi-Hubbard model for energy estimation, the small-bias symmetry expansion we found can achieve an estimation bias 6 to 9 times below what is achievable by symmetry verification when the average number of circuit errors is between 1 to 2. The corresponding sampling cost for random shot noise reduction is just 2 to 6 times higher than symmetry verification. Beyond symmetries inherent to the physical problem, our formalism is also applicable to engineered symmetries. For example, the recent scheme for exponential error suppression using multiple noisy copies of the quantum device is just a special case of symmetry expansion using the permutation symmetry among the copies.

Featured image: Symmetry expansion can achieve a much smaller estimation bias compared to symmetry verification in our simulation.

Popular summary

Mitigating errors using the symmetries of the problem is expected to play key roles in practical near-term quantum applications. A good example is symmetry verification, in which we perform measurements to verify whether the output state is in the right symmetry subspace, so that we can detect certain errors on the state. In this work, we introduce a general framework named symmetry expansion for symmetry-based error-mitigation techniques. It encompasses previous schemes like symmetry verification, symmetry subspace expansion and virtual distillation while also generates a wide range of schemes, offering different balances between the estimation biases and the sampling cost. Furthermore, we devise a way to identify the symmetry expansion schemes that have much smaller estimation biases than symmetry verification. These symmetry expansion schemes allow for cancellation between the biases due to the detectable and undetectable errors, while symmetry verification can only remove the biases due to the detectable errors, leaving behind the biases due to the undetectable errors.

► BibTeX data

@article{Cai2021quantumerror,
  doi = {10.22331/q-2021-09-21-548},
  url = {https://doi.org/10.22331/q-2021-09-21-548},
  title = {Quantum {E}rror {M}itigation using {S}ymmetry {E}xpansion},
  author = {Cai, Zhenyu},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {548},
  month = sep,
  year = {2021}
}

► References

[1] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando G. S. L. Brandao, David A. Buell, Brian Burkett, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Andrew Dunsworth, Edward Farhi, Brooks Foxen, Austin Fowler, Craig Gidney, Marissa Giustina, Rob Graff, Keith Guerin, Steve Habegger, Matthew P. Harrigan, Michael J. Hartmann, Alan Ho, Markus Hoffmann, Trent Huang, Travis S. Humble, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Paul V. Klimov, Sergey Knysh, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Mike Lindmark, Erik Lucero, Dmitry Lyakh, Salvatore Mandrà, Jarrod R. McClean, Matthew McEwen, Anthony Megrant, Xiao Mi, Kristel Michielsen, Masoud Mohseni, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Murphy Yuezhen Niu, Eric Ostby, Andre Petukhov, John C. Platt, Chris Quintana, Eleanor G. Rieffel, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Kevin J. Sung, Matthew D. Trevithick, Amit Vainsencher, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, and John M. Martinis. Quantum supremacy using a programmable superconducting processor. Nature, 574 (7779): 505–510, October 2019. 10.1038/s41586-019-1666-5.
https://doi.org/10.1038/s41586-019-1666-5

[2] Han-Sen Zhong, Hui Wang, Yu-Hao Deng, Ming-Cheng Chen, Li-Chao Peng, Yi-Han Luo, Jian Qin, Dian Wu, Xing Ding, Yi Hu, Peng Hu, Xiao-Yan Yang, Wei-Jun Zhang, Hao Li, Yuxuan Li, Xiao Jiang, Lin Gan, Guangwen Yang, Lixing You, Zhen Wang, Li Li, Nai-Le Liu, Chao-Yang Lu, and Jian-Wei Pan. Quantum computational advantage using photons. Science, 370 (6523): 1460–1463, December 2020. 10.1126/science.abe8770.
https://doi.org/10.1126/science.abe8770

[3] Kristan Temme, Sergey Bravyi, and Jay M. Gambetta. Error Mitigation for Short-Depth Quantum Circuits. Physical Review Letters, 119 (18): 180509, November 2017. 10.1103/PhysRevLett.119.180509.
https://doi.org/10.1103/PhysRevLett.119.180509

[4] Suguru Endo, Simon C. Benjamin, and Ying Li. Practical Quantum Error Mitigation for Near-Future Applications. Physical Review X, 8 (3): 031027, July 2018. 10.1103/PhysRevX.8.031027.
https://doi.org/10.1103/PhysRevX.8.031027

[5] Zhenyu Cai. Multi-exponential error extrapolation and combining error mitigation techniques for NISQ applications. npj Quantum Information, 7 (1): 1–12, May 2021. 10.1038/s41534-021-00404-3.
https://doi.org/10.1038/s41534-021-00404-3

[6] Suguru Endo, Zhenyu Cai, Simon C. Benjamin, and Xiao Yuan. Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation. Journal of the Physical Society of Japan, 90 (3): 032001, February 2021. 10.7566/JPSJ.90.032001.
https://doi.org/10.7566/JPSJ.90.032001

[7] Abhinav Kandala, Kristan Temme, Antonio D. Córcoles, Antonio Mezzacapo, Jerry M. Chow, and Jay M. Gambetta. Error mitigation extends the computational reach of a noisy quantum processor. Nature, 567 (7749): 491–495, March 2019. 10.1038/s41586-019-1040-7.
https://doi.org/10.1038/s41586-019-1040-7

[8] T. Giurgica-Tiron, Y. Hindy, R. LaRose, A. Mari, and W. J. Zeng. Digital zero noise extrapolation for quantum error mitigation. In 2020 IEEE International Conference on Quantum Computing and Engineering (QCE), pages 306–316, October 2020. 10.1109/QCE49297.2020.00045.
https://doi.org/10.1109/QCE49297.2020.00045

[9] Ryan LaRose, Andrea Mari, Peter J. Karalekas, Nathan Shammah, and William J. Zeng. Mitiq: A software package for error mitigation on noisy quantum computers. arXiv:2009.04417 [quant-ph], September 2020. URL http://arxiv.org/abs/2009.04417.
arXiv:2009.04417

[10] Google AI Quantum and Collaborators. Hartree-Fock on a superconducting qubit quantum computer. Science, 369 (6507): 1084–1089, August 2020. 10.1126/science.abb9811.
https://doi.org/10.1126/science.abb9811

[11] Shuaining Zhang, Yao Lu, Kuan Zhang, Wentao Chen, Ying Li, Jing-Ning Zhang, and Kihwan Kim. Error-mitigated quantum gates exceeding physical fidelities in a trapped-ion system. Nature Communications, 11 (1): 587, January 2020. 10.1038/s41467-020-14376-z.
https://doi.org/10.1038/s41467-020-14376-z

[12] X. Bonet-Monroig, R. Sagastizabal, M. Singh, and T. E. O'Brien. Low-cost error mitigation by symmetry verification. Physical Review A, 98 (6): 062339, December 2018. 10.1103/PhysRevA.98.062339.
https://doi.org/10.1103/PhysRevA.98.062339

[13] Sam McArdle, Xiao Yuan, and Simon Benjamin. Error-Mitigated Digital Quantum Simulation. Physical Review Letters, 122 (18): 180501, May 2019. 10.1103/PhysRevLett.122.180501.
https://doi.org/10.1103/PhysRevLett.122.180501

[14] Jarrod R. McClean, Mollie E. Kimchi-Schwartz, Jonathan Carter, and Wibe A. de Jong. Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states. Physical Review A, 95 (4): 042308, April 2017. 10.1103/PhysRevA.95.042308.
https://doi.org/10.1103/PhysRevA.95.042308

[15] Jarrod R. McClean, Zhang Jiang, Nicholas C. Rubin, Ryan Babbush, and Hartmut Neven. Decoding quantum errors with subspace expansions. Nature Communications, 11 (1): 636, January 2020. 10.1038/s41467-020-14341-w.
https://doi.org/10.1038/s41467-020-14341-w

[16] Bálint Koczor. Exponential Error Suppression for Near-Term Quantum Devices. arXiv:2011.05942 [quant-ph], November 2020. URL http://arxiv.org/abs/2011.05942. 10.1103/PhysRevX.11.031057.
https://doi.org/10.1103/PhysRevX.11.031057
arXiv:2011.05942

[17] William J. Huggins, Sam McArdle, Thomas E. O'Brien, Joonho Lee, Nicholas C. Rubin, Sergio Boixo, K. Birgitta Whaley, Ryan Babbush, and Jarrod R. McClean. Virtual Distillation for Quantum Error Mitigation. arXiv:2011.07064 [quant-ph], January 2021a. URL http://arxiv.org/abs/2011.07064.
arXiv:2011.07064

[18] A. Berthiaume, D. Deutsch, and R. Jozsa. The stabilisation of quantum computations. In Proceedings Workshop on Physics and Computation. PhysComp '94, pages 60–62, November 1994. 10.1109/PHYCMP.1994.363698.
https://doi.org/10.1109/PHYCMP.1994.363698

[19] Adriano Barenco, André Berthiaume, David Deutsch, Artur Ekert, Richard Jozsa, and Chiara Macchiavello. Stabilization of Quantum Computations by Symmetrization. SIAM Journal on Computing, 26 (5): 1541–1557, October 1997. 10.1137/S0097539796302452.
https://doi.org/10.1137/S0097539796302452

[20] Asher Peres. Error Symmetrization in Quantum Computers. International Journal of Theoretical Physics, 38 (3): 799–805, March 1999. 10.1023/A:1026648717079.
https://doi.org/10.1023/A:1026648717079

[21] Wu-Ki Tung. Group Theory in Physics: An Introduction to Symmetry Principles, Group Representations, and Special Functions in Classical and Quantum Physics. WORLD SCIENTIFIC, August 1985. 10.1142/0097.
https://doi.org/10.1142/0097

[22] Andrew Jena, Scott Genin, and Michele Mosca. Pauli Partitioning with Respect to Gate Sets. arXiv:1907.07859 [quant-ph], July 2019. URL http://arxiv.org/abs/1907.07859.
arXiv:1907.07859

[23] William J. Huggins, Jarrod R. McClean, Nicholas C. Rubin, Zhang Jiang, Nathan Wiebe, K. Birgitta Whaley, and Ryan Babbush. Efficient and noise resilient measurements for quantum chemistry on near-term quantum computers. npj Quantum Information, 7 (1): 1–9, February 2021b. 10.1038/s41534-020-00341-7.
https://doi.org/10.1038/s41534-020-00341-7

[24] Zhenyu Cai. Resource Estimation for Quantum Variational Simulations of the Hubbard Model. Physical Review Applied, 14 (1): 014059, July 2020. 10.1103/PhysRevApplied.14.014059.
https://doi.org/10.1103/PhysRevApplied.14.014059

[25] Github. https://github.com/CaiQuantum/Symmetry-Expansion-Code, September 2021. URL https://github.com/CaiQuantum/Symmetry-Expansion-Code.
https://github.com/CaiQuantum/Symmetry-Expansion-Code

[26] Armands Strikis, Dayue Qin, Yanzhu Chen, Simon C. Benjamin, and Ying Li. Learning-based quantum error mitigation. arXiv:2005.07601 [quant-ph], March 2021. URL http://arxiv.org/abs/2005.07601.
arXiv:2005.07601

[27] Kosuke Mitarai and Keisuke Fujii. Methodology for replacing indirect measurements with direct measurements. Physical Review Research, 1 (1): 013006, August 2019. 10.1103/PhysRevResearch.1.013006.
https://doi.org/10.1103/PhysRevResearch.1.013006

[28] Zhenyu Cai, Xiaosi Xu, and Simon C. Benjamin. Mitigating coherent noise using Pauli conjugation. npj Quantum Information, 6 (1): 1–9, February 2020. 10.1038/s41534-019-0233-0.
https://doi.org/10.1038/s41534-019-0233-0

[29] Victor V. Albert and Liang Jiang. Symmetries and conserved quantities in Lindblad master equations. Physical Review A, 89 (2): 022118, February 2014. 10.1103/PhysRevA.89.022118.
https://doi.org/10.1103/PhysRevA.89.022118

[30] Arne L. Grimsmo, Joshua Combes, and Ben Q. Baragiola. Quantum Computing with Rotation-Symmetric Bosonic Codes. Physical Review X, 10 (1): 011058, March 2020. 10.1103/PhysRevX.10.011058.
https://doi.org/10.1103/PhysRevX.10.011058

[31] Jeffrey M. Gertler, Brian Baker, Juliang Li, Shruti Shirol, Jens Koch, and Chen Wang. Protecting a bosonic qubit with autonomous quantum error correction. Nature, 590 (7845): 243–248, February 2021. 10.1038/s41586-021-03257-0.
https://doi.org/10.1038/s41586-021-03257-0

[32] Tyson Jones and Simon C. Benjamin. QuESTlink – Mathematica embiggened by a hardware-optimised quantum emulator. Quantum Science and Technology, 2020. 10.1088/2058-9565/ab8506.
https://doi.org/10.1088/2058-9565/ab8506

[33] Tyson Jones, Anna Brown, Ian Bush, and Simon C. Benjamin. QuEST and High Performance Simulation of Quantum Computers. Scientific Reports, 9 (1): 1–11, July 2019. 10.1038/s41598-019-47174-9.
https://doi.org/10.1038/s41598-019-47174-9

[34] Joel J. Wallman and Joseph Emerson. Noise tailoring for scalable quantum computation via randomized compiling. Physical Review A, 94 (5): 052325, November 2016. 10.1103/PhysRevA.94.052325.
https://doi.org/10.1103/PhysRevA.94.052325

Cited by

[1] Zhenhuan Liu, Xingjian Zhang, Yue-Yang Fei, and Zhenyu Cai, "Virtual Channel Purification", PRX Quantum 6 2, 020325 (2025).

[2] Ivana Miháliková, Matej Pivoluska, Martin Plesch, Martin Friák, Daniel Nagaj, and Mojmír Šob, "The Cost of Improving the Precision of the Variational Quantum Eigensolver for Quantum Chemistry", Nanomaterials 12 2, 243 (2022).

[3] Zhenyu Cai, Ryan Babbush, Simon C. Benjamin, Suguru Endo, William J. Huggins, Ying Li, Jarrod R. McClean, and Thomas E. O’Brien, "Quantum error mitigation", Reviews of Modern Physics 95 4, 045005 (2023).

[4] Quinn Langfitt, Ji Liu, Benchen Huang, Alvin Gonzales, Kaitlin N. Smith, Nikos Hardavellas, and Zain H. Saleem, "Extrapolating Pauli Checks for Expectation Value Estimation on Noisy Quantum Devices", IEEE Transactions on Quantum Engineering 7, 3103409 (2026).

[5] Ivan Henao, Jader P. Santos, and Raam Uzdin, "Adaptive quantum error mitigation using pulse-based inverse evolutions", npj Quantum Information 9 1, 120 (2023).

[6] Nobuyuki Yoshioka, Hideaki Hakoshima, Yuichiro Matsuzaki, Yuuki Tokunaga, Yasunari Suzuki, and Suguru Endo, "Generalized Quantum Subspace Expansion", Physical Review Letters 129 2, 020502 (2022).

[7] Kento Tsubouchi, Takahiro Sagawa, and Nobuyuki Yoshioka, "Universal Cost Bound of Quantum Error Mitigation Based on Quantum Estimation Theory", Physical Review Letters 131 21, 210601 (2023).

[8] Tianyi Li, Yumeng Zeng, Qiming Ding, Zixuan Huo, Xiaosi Xu, Jiajun Ren, Diandong Tang, Xiaoxia Cai, and Xiao Yuan, "Efficient quantum simulation of non-adiabatic molecular dynamics with precise electronic structure", Digital Discovery 5 2, 548 (2026).

[9] Yuri Alexeev, Maximilian Amsler, Marco Antonio Barroca, Sanzio Bassini, Torey Battelle, Daan Camps, David Casanova, Young Jay Choi, Frederic T. Chong, Charles Chung, Christopher Codella, Antonio D. Córcoles, James Cruise, Alberto Di Meglio, Ivan Duran, Thomas Eckl, Sophia Economou, Stephan Eidenbenz, Bruce Elmegreen, Clyde Fare, Ismael Faro, Cristina Sanz Fernández, Rodrigo Neumann Barros Ferreira, Keisuke Fuji, Bryce Fuller, Laura Gagliardi, Giulia Galli, Jennifer R. Glick, Isacco Gobbi, Pranav Gokhale, Salvador de la Puente Gonzalez, Johannes Greiner, Bill Gropp, Michele Grossi, Emanuel Gull, Burns Healy, Matthew R. Hermes, Benchen Huang, Travis S. Humble, Nobuyasu Ito, Artur F. Izmaylov, Ali Javadi-Abhari, Douglas Jennewein, Shantenu Jha, Liang Jiang, Barbara Jones, Wibe Albert de Jong, Petar Jurcevic, William Kirby, Stefan Kister, Masahiro Kitagawa, Joel Klassen, Katherine Klymko, Kwangwon Koh, Masaaki Kondo, Dog̃a Murat Kürkçüog̃lu, Krzysztof Kurowski, Teodoro Laino, Ryan Landfield, Matt Leininger, Vicente Leyton-Ortega, Ang Li, Meifeng Lin, Junyu Liu, Nicolas Lorente, Andre Luckow, Simon Martiel, Francisco Martin-Fernandez, Margaret Martonosi, Claire Marvinney, Arcesio Castaneda Medina, Dirk Merten, Antonio Mezzacapo, Kristel Michielsen, Abhishek Mitra, Tushar Mittal, Kyungsun Moon, Joel Moore, Sarah Mostame, Mario Motta, Young-Hye Na, Yunseong Nam, Prineha Narang, Yu-ya Ohnishi, Daniele Ottaviani, Matthew Otten, Scott Pakin, Vincent R. Pascuzzi, Edwin Pednault, Tomasz Piontek, Jed Pitera, Patrick Rall, Gokul Subramanian Ravi, Niall Robertson, Matteo A.C. Rossi, Piotr Rydlichowski, Hoon Ryu, Georgy Samsonidze, Mitsuhisa Sato, Nishant Saurabh, Vidushi Sharma, Kunal Sharma, Soyoung Shin, George Slessman, Mathias Steiner, Iskandar Sitdikov, In-Saeng Suh, Eric D. Switzer, Wei Tang, Joel Thompson, Synge Todo, Minh C. Tran, Dimitar Trenev, Christian Trott, Huan-Hsin Tseng, Norm M. Tubman, Esin Tureci, David García Valiñas, Sofia Vallecorsa, Christopher Wever, Konrad Wojciechowski, Xiaodi Wu, Shinjae Yoo, Nobuyuki Yoshioka, Victor Wen-zhe Yu, Seiji Yunoki, Sergiy Zhuk, and Dmitry Zubarev, "Quantum-centric supercomputing for materials science: A perspective on challenges and future directions", Future Generation Computer Systems 160, 666 (2024).

[10] Ivana Miháliková, Martin Friák, Matej Pivoluska, Martin Plesch, Martin Saip, and Mojmír Šob, "Best-Practice Aspects of Quantum-Computer Calculations: A Case Study of the Hydrogen Molecule", Molecules 27 3, 597 (2022).

[11] Yigal Ilin and Itai Arad, "Dissipative Variational Quantum Algorithms for Gibbs State Preparation", IEEE Transactions on Quantum Engineering 6, 1 (2025).

[12] Peiyi Li, Ji Liu, Alvin Gonzales, Zain Hamid Saleem, Huiyang Zhou, and Paul Hovland, 2024 ACM/IEEE 51st Annual International Symposium on Computer Architecture (ISCA) 103 (2024) ISBN:979-8-3503-2658-1.

[13] Ben Bar, Jader P. Santos, and Raam Uzdin, "Layered KIK quantum error mitigation for dynamic circuits", npj Quantum Information 12 1, 79 (2026).

[14] Smik Patel, Praveen Jayakumar, Tzu-Ching Yen, and Artur F. Izmaylov, "Quantum Measurement for Quantum Chemistry on a Quantum Computer", Chemical Reviews 125 16, 7490 (2025).

[15] Gregory Boyd, Bálint Koczor, and Zhenyu Cai, "High-dimensional subspace expansion using classical shadows", Physical Review A 111 2, 022423 (2025).

[16] Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, and Lukasz Cincio, "Error mitigation with Clifford quantum-circuit data", Quantum 5, 592 (2021).

[17] Xu-Dan Xie, Xiaoming Zhang, Balint Koczor, and Xiao Yuan, "Advances in Quantum Computation in NISQ Era", Entropy 27 10, 1074 (2025).

[18] Maksym Prodius, Piotr Czarnik, Michael McKerns, Andrew T. Sornborger, and Lukasz Cincio, "Robust Design Under Uncertainty in Quantum Error Mitigation", IEEE Transactions on Quantum Engineering 7, 1 (2026).

[19] Edoardo Ballini, Julius Mildenberger, Matteo M. Wauters, and Philipp Hauke, "Symmetry verification for noisy quantum simulations of non-Abelian lattice gauge theories", Quantum 9, 1802 (2025).

[20] Keitaro Anai, Yasunari Suzuki, Yuuki Tokunaga, Yuichiro Matsuzaki, Shuntaro Takeda, and Suguru Endo, "Unitary-transformed projective squeezing: Applications for circuit knitting and state preparation of non-Gaussian states", Physical Review Research 7 4, 043343 (2025).

[21] Xiaosi Xu and Ying Li, "Quantum-assisted Monte Carlo algorithms for fermions", Quantum 7, 1072 (2023).

[22] Xiao-Yue Xu, Chen Ding, Shuo Zhang, Wan-Su Bao, and He-Liang Huang, "Circuit-noise-resilient virtual distillation", Communications Physics 7 1, 325 (2024).

[23] Emiliia Dyrenkova, Raymond Laflamme, and Michael Vasmer, "Scalable simulation of fermionic encoding performance on noisy quantum computers", Physical Review A 113 2, 022607 (2026).

[24] Qing Liu, Zihao Li, Xiao Yuan, Huangjun Zhu, and You Zhou, "Auxiliary-Free Replica Shadows: Efficient Estimation of Multiple Nonlinear Quantum Properties", Physical Review Letters 136 10, 100602 (2026).

[25] Aaron Szasz, Ed Younis, and Wibe Albert de Jong, "Ground state energy and magnetization curve of a frustrated magnetic system from real-time evolution on a digital quantum processor", Quantum 9, 1704 (2025).

[26] Andrew Zhao and Akimasa Miyake, "Group-theoretic error mitigation enabled by classical shadows and symmetries", npj Quantum Information 10 1, 57 (2024).

[27] Michael Krebsbach, Björn Trauzettel, and Alessio Calzona, "Optimization of Richardson extrapolation for quantum error mitigation", Physical Review A 106 6, 062436 (2022).

[28] Seok-Hyung Lee, Lucas H. English, and Stephen D. Bartlett, "Efficient post-selection for general quantum LDPC Codes", npj Quantum Information 12 1, 96 (2026).

[29] Yifeng Xiong, Daryus Chandra, Soon Xin Ng, and Lajos Hanzo, "Circuit Symmetry Verification Mitigates Quantum-Domain Impairments", IEEE Transactions on Signal Processing 71, 477 (2023).

[30] Ryan LaRose, Andrea Mari, Sarah Kaiser, Peter J. Karalekas, Andre A. Alves, Piotr Czarnik, Mohamed El Mandouh, Max H. Gordon, Yousef Hindy, Aaron Robertson, Purva Thakre, Misty Wahl, Danny Samuel, Rahul Mistri, Maxime Tremblay, Nick Gardner, Nathaniel T. Stemen, Nathan Shammah, and William J. Zeng, "Mitiq: A software package for error mitigation on noisy quantum computers", Quantum 6, 774 (2022).

[31] Khadija Amadoun, Chaïmaâ Kissi, Idriss Moumen, and Tarik Boujiha, 2025 7th International Symposium on Advanced Electrical and Communication Technologies (ISAECT) 1 (2025) ISBN:979-8-3315-6957-0.

[32] Alvin Gonzales, Ruslan Shaydulin, Zain H. Saleem, and Martin Suchara, "Quantum error mitigation by Pauli check sandwiching", Scientific Reports 13 1, 2122 (2023).

[33] Nishchay Suri, Jason Saied, and Davide Venturelli, "Uniformly decaying subspaces for error-mitigated quantum computation", Physical Review A 110 4, 042621 (2024).

[34] Xiaokai Hou, Guanyu Zhou, Qingyu Li, Shan Jin, and Xiaoting Wang, "A duplication-free quantum neural network for universal approximation", Science China Physics, Mechanics & Astronomy 66 7, 270362 (2023).

[35] Kento Tsubouchi, Yasunari Suzuki, Yuuki Tokunaga, Nobuyuki Yoshioka, and Suguru Endo, "Virtual quantum error detection", Physical Review A 108 4, 042426 (2023).

[36] IlKwon Sohn, Changyeol Lee, Wooyeong Song, Kwangil Bae, and Wonhyuk Lee, "Quantum error mitigation via structural encoding with classical error correction codes", EPJ Quantum Technology 13 1, 25 (2026).

[37] Zhong-Xia Shang, Zi-Han Chen, and Cai-Sheng Cheng, "Decoherence-free quantum error mitigation by density matrix vectorization", Physical Review Research 8 2, 023206 (2026).

[38] Ryuji Takagi, Hiroyasu Tajima, and Mile Gu, "Universal Sampling Lower Bounds for Quantum Error Mitigation", Physical Review Letters 131 21, 210602 (2023).

[39] Unai Aseguinolaza, Nahual Sobrino, Gabriel Sobrino, Joaquim Jornet-Somoza, and Juan Borge, "Error estimation in current noisy quantum computers", Quantum Information Processing 23 5, 181 (2024).

[40] He-Liang Huang, Xiao-Yue Xu, Chu Guo, Guojing Tian, Shi-Jie Wei, Xiaoming Sun, Wan-Su Bao, and Gui-Lu Long, "Near-term quantum computing techniques: Variational quantum algorithms, error mitigation, circuit compilation, benchmarking and classical simulation", Science China Physics, Mechanics & Astronomy 66 5, 250302 (2023).

[41] Samuel Stein, Nathan Wiebe, Yufei Ding, James Ang, and Ang Li, Proceedings of the 50th Annual International Symposium on Computer Architecture 1 (2023) ISBN:9798400700958.

[42] Hideaki Hakoshima, Suguru Endo, Kaoru Yamamoto, Yuichiro Matsuzaki, and Nobuyuki Yoshioka, "Localized Virtual Purification", Physical Review Letters 133 8, 080601 (2024).

[43] Dayue Qin, Xiaosi Xu, and Ying Li, "An overview of quantum error mitigation formulas", Chinese Physics B 31 9, 090306 (2022).

[44] Daniel Bultrini, Max Hunter Gordon, Piotr Czarnik, Andrew Arrasmith, M. Cerezo, Patrick J. Coles, and Lukasz Cincio, "Unifying and benchmarking state-of-the-art quantum error mitigation techniques", Quantum 7, 1034 (2023).

[45] Yuma Nakamura, Yoshichika Yano, and Nobuyuki Yoshioka, "Adaptive measurement strategy for quantum subspace methods", New Journal of Physics 26 3, 033028 (2024).

[46] Tian‐Ren Jin, Yun‐Hao Shi, Zheng‐An Wang, Tian‐Ming Li, Kai Xu, and Heng Fan, "Purity‐Assisted Zero‐Noise Extrapolation for Quantum Error Mitigation", Advanced Quantum Technologies 7 12, 2400150 (2024).

[47] Tim Weaving, Alexis Ralli, William M. Kirby, Peter J. Love, Sauro Succi, and Peter V. Coveney, "Benchmarking noisy intermediate scale quantum error mitigation strategies for ground state preparation of the HCl molecule", Physical Review Research 5 4, 043054 (2023).

[48] Danila Babukhin, "Echo-evolution data generation for quantum error mitigation via neural networks", Quantum Information Processing 23 12, 405 (2024).

[49] Kento Tsubouchi, Yosuke Mitsuhashi, Ryuji Takagi, and Nobuyuki Yoshioka, "Symmetric Channel Verification for Purifying Noisy Quantum Channels", PRX Quantum 6 4, 040310 (2025).

[50] Lajos Hanzo, Zunaira Babar, Zhenyu Cai, Daryus Chandra, Ivan B. Djordjevic, Balint Koczor, Soon Xin Ng, Mohsen Razavi, and Osvaldo Simeone, "Quantum Information Processing, Sensing, and Communications: Their Myths, Realities, and Futures", Proceedings of the IEEE 113 9, 1024 (2025).

[51] Tim Weaving, Alexis Ralli, Peter J. Love, Sauro Succi, and Peter V. Coveney, "Contextual subspace variational quantum eigensolver calculation of the dissociation curve of molecular nitrogen on a superconducting quantum computer", npj Quantum Information 11 1, 25 (2025).

[52] Rajni Bala, Sooryansh Asthana, and V. Ravishankar, "Combating errors in quantum communication: an integrated approach", Scientific Reports 13 1, 2979 (2023).

[53] Suguru Endo, Yasunari Suzuki, Kento Tsubouchi, Rui Asaoka, Kaoru Yamamoto, Yuichiro Matsuzaki, and Yuuki Tokunaga, "Quantum error mitigation for rotation-symmetric bosonic codes with symmetry expansion", Physical Review A 111 6, 062402 (2025).

[54] Piotr Czarnik, Michael McKerns, Andrew T. Sornborger, and Lukasz Cincio, "Improving the efficiency of learning-based error mitigation", Quantum 9, 1727 (2025).

[55] 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", Reviews of Modern Physics 94 1, 015004 (2022).

[56] Bálint Koczor, "Exponential Error Suppression for Near-Term Quantum Devices", Physical Review X 11 3, 031057 (2021).

[57] Piotr Czarnik, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles, "Qubit-efficient exponential suppression of errors", arXiv:2102.06056, (2021).

[58] Zhenyu Cai, "Resource-efficient Purification-based Quantum Error Mitigation", arXiv:2107.07279, (2021).

[59] Peiyi Li, Ji Liu, Alvin Gonzales, Zain Hamid Saleem, Huiyang Zhou, and Paul Hovland, "QuTracer: Mitigating Quantum Gate and Measurement Errors by Tracing Subsets of Qubits", arXiv:2404.19712, (2024).

[60] Laurin E. Fischer, "Enabling large-scale digital quantum simulations with superconducting qubits", arXiv:2602.04719, (2026).

[61] Yifeng Xiong, Daryus Chandra, Soon Xin Ng, and Lajos Hanzo, "Circuit Symmetry Verification Mitigates Quantum-Domain Impairments", arXiv:2112.13904, (2021).

[62] Vicenzo Scavino Alfaro, "Certified Finite-Shot Operating Windows for Virtual Distillation and Symmetry Verification", arXiv:2606.15464, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-12 05:04:51) and SAO/NASA ADS (last updated successfully 2026-08-11 16:38:58). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch ADS cited-by data during last attempt 2026-08-12 05:04:51: Cannot retrieve data from ADS due to rate limitations.

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.