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.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 Google AI Quantum and Collaborators. Hartree-Fock on a superconducting qubit quantum computer. Science, 369 (6507): 1084–1089, August 2020. 10.1126/science.abb9811.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 Asher Peres. Error Symmetrization in Quantum Computers. International Journal of Theoretical Physics, 38 (3): 799–805, March 1999. 10.1023/A:1026648717079.
 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.
 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.
 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.
 Github. https://github.com/CaiQuantum/Symmetry-Expansion-Code, September 2021. URL https://github.com/CaiQuantum/Symmetry-Expansion-Code.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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 (NISQ) algorithms", arXiv:2101.08448.
 Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, and Lukasz Cincio, "Error mitigation with Clifford quantum-circuit data", arXiv:2005.10189.
 Bálint Koczor, "Exponential Error Suppression for Near-Term Quantum Devices", Physical Review X 11 3, 031057 (2021).
 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, Nathan Shammah, and William J. Zeng, "Mitiq: A software package for error mitigation on noisy quantum computers", arXiv:2009.04417.
 Piotr Czarnik, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles, "Qubit-efficient exponential suppression of errors", arXiv:2102.06056.
 Nobuyuki Yoshioka, Hideaki Hakoshima, Yuichiro Matsuzaki, Yuuki Tokunaga, Yasunari Suzuki, and Suguru Endo, "Generalized quantum subspace expansion", arXiv:2107.02611.
 Zhenyu Cai, "Resource-efficient Purification-based Quantum Error Mitigation", arXiv:2107.07279.
 Daniel Bultrini, Max Hunter Gordon, Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, and Lukasz Cincio, "Unifying and benchmarking state-of-the-art quantum error mitigation techniques", arXiv:2107.13470.
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