Theory of quantum error mitigation for non-Clifford gates
Summarize this article with:
AbstractQuantum error mitigation techniques mimic noiseless quantum circuits by running several related noisy circuits and combining their outputs in particular ways. How well such techniques work is thought to depend strongly on how noisy the underlying gates are. Weakly-entangling gates, like $R_{ZZ}(\theta)$ for small angles $\theta$, can be much less noisy than entangling Clifford gates, like CNOT and CZ, and they arise naturally in circuits used to simulate quantum dynamics. However, such weakly-entangling gates are non-Clifford, and are therefore incompatible with two of the most prominent error mitigation techniques to date: probabilistic error cancellation (PEC) and the related form of zero-noise extrapolation (ZNE). This paper generalizes these techniques to non-Clifford gates, and comprises two complementary parts. The first part shows how to effectively transform any given quantum channel into (almost) any desired channel, at the cost of a sampling overhead, by adding random Pauli gates and processing the measurement outcomes. This enables us to cancel or properly amplify noise in non-Clifford gates, provided we can first characterize such gates in detail. The second part therefore introduces techniques to do so for noisy $R_{ZZ}(\theta)$ gates. These techniques are robust to state preparation and measurement (SPAM) errors, and exhibit concentration and sensitivity—crucial statistical properties for many experiments. They are related to randomized benchmarking, and may also be of interest beyond the context of error mitigation. We find that while non-Clifford gates can be less noisy than related Cliffords, their noise is fundamentally more complex, which can lead to surprising and sometimes unwanted effects in error mitigation. Whether this trade-off can be broadly advantageous remains to be seen.Featured image: An illustration of probabilistic error cancellation (PEC) generalized to $R_{ZZ}$ gates with a non-Clifford angle. The gate noise can be accurately learned, even under strong state preparation and measurement (SPAM) errors, and then accurately cancelled in effect.► BibTeX data@article{Layden2026theoryofquantum, doi = {10.22331/q-2026-02-10-2003}, url = {https://doi.org/10.22331/q-2026-02-10-2003}, title = {Theory of quantum error mitigation for non-{C}lifford gates}, author = {Layden, David and Mitchell, Bradley and Siva, Karthik}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2003}, month = feb, year = {2026} }► References [1] 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''. Rev. Mod. Phys. 95, 045005 (2023). https://doi.org/10.1103/RevModPhys.95.045005 [2] Kristan Temme, Sergey Bravyi, and Jay M. Gambetta. ``Error mitigation for short-depth quantum circuits''. Phys. Rev. Lett. 119, 180509 (2017). https://doi.org/10.1103/PhysRevLett.119.180509 [3] Ying Li and Simon C. Benjamin. ``Efficient variational quantum simulator incorporating active error minimization''. Phys. Rev. X 7, 021050 (2017). https://doi.org/10.1103/PhysRevX.7.021050 [4] Sergey Bravyi, Oliver Dial, Jay M. Gambetta, Darío Gil, and Zaira Nazario. ``The future of quantum computing with superconducting qubits''. Journal of Applied Physics 132, 160902 (2022). https://doi.org/10.1063/5.0082975 [5] Seth Lloyd. ``Universal quantum simulators''. Science 273, 1073–1078 (1996). https://doi.org/10.1126/science.273.5278.1073 [6] Andrew M. Childs, Dmitri Maslov, Yunseong Nam, Neil J. Ross, and Yuan Su. ``Toward the first quantum simulation with quantum speedup''. Proceedings of the National Academy of Sciences 115, 9456–9461 (2018). https://doi.org/10.1073/pnas.1801723115 [7] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu. ``Theory of Trotter error with commutator scaling''. Phys. Rev. X 11, 011020 (2021). https://doi.org/10.1103/PhysRevX.11.011020 [8] Laura Clinton, Johannes Bausch, and Toby Cubitt. ``Hamiltonian simulation algorithms for near-term quantum hardware''. Nature Communications 12, 4989 (2021). https://doi.org/10.1038/s41467-021-25196-0 [9] Ewout Van Den Berg, Zlatko K Minev, Abhinav Kandala, and Kristan Temme. ``Probabilistic error cancellation with sparse Pauli-Lindblad models on noisy quantum processors''. Nature Physics (2023). https://doi.org/10.1038/s41567-023-02042-2 [10] Youngseok Kim, Andrew Eddins, Sajant Anand, Ken Xuan Wei, Ewout Van Den Berg, Sami Rosenblatt, Hasan Nayfeh, Yantao Wu, Michael Zaletel, Kristan Temme, et al. ``Evidence for the utility of quantum computing before fault tolerance''. Nature 618, 500–505 (2023). https://doi.org/10.1038/s41586-023-06096-3 [11] Nathan Earnest, Caroline Tornow, and Daniel J. Egger. ``Pulse-efficient circuit transpilation for quantum applications on cross-resonance-based hardware''. Phys. Rev. Res. 3, 043088 (2021). https://doi.org/10.1103/PhysRevResearch.3.043088 [12] John P. T. Stenger, Nicholas T. Bronn, Daniel J. Egger, and David Pekker. ``Simulating the dynamics of braiding of Majorana zero modes using an IBM quantum computer''. Phys. Rev. Res. 3, 033171 (2021). https://doi.org/10.1103/PhysRevResearch.3.033171 [13] Seth T. Merkel, Jay M. Gambetta, John A. Smolin, Stefano Poletto, Antonio D. Córcoles, Blake R. Johnson, Colm A. Ryan, and Matthias Steffen. ``Self-consistent quantum process tomography''. Phys. Rev. A 87, 062119 (2013). https://doi.org/10.1103/PhysRevA.87.062119 [14] Robin Blume-Kohout, John King Gamble, Erik Nielsen, Jonathan Mizrahi, Jonathan D. Sterk, and Peter Maunz. ``Robust, self-consistent, closed-form tomography of quantum logic gates on a trapped ion qubit'' (2013). arXiv:1310.4492. arXiv:1310.4492 [15] Erik Nielsen, John King Gamble, Kenneth Rudinger, Travis Scholten, Kevin Young, and Robin Blume-Kohout. ``Gate Set Tomography''. Quantum 5, 557 (2021). https://doi.org/10.22331/q-2021-10-05-557 [16] Suguru Endo, Simon C. Benjamin, and Ying Li. ``Practical quantum error mitigation for near-future applications''. Phys. Rev. X 8, 031027 (2018). https://doi.org/10.1103/PhysRevX.8.031027 [17] Daniel Greenbaum. ``Introduction to quantum gate set tomography'' (2015). arXiv:1509.02921. arXiv:1509.02921 [18] Steven T Flammia and Joel J Wallman. ``Efficient estimation of Pauli channels''. ACM Transactions on Quantum Computing 1, 1–32 (2020). https://doi.org/10.1145/3408039 [19] W. Dür, M. Hein, J. I. Cirac, and H.-J. Briegel. ``Standard forms of noisy quantum operations via depolarization''. Phys. Rev. A 72, 052326 (2005). https://doi.org/10.1103/PhysRevA.72.052326 [20] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. ``Exact and approximate unitary 2-designs and their application to fidelity estimation''. Phys. Rev. A 80, 012304 (2009). https://doi.org/10.1103/PhysRevA.80.012304 [21] E. Knill. ``Fault-tolerant postselected quantum computation: Threshold analysis'' (2004). arXiv:quant-ph/0404104. arXiv:quant-ph/0404104 [22] Joel J. Wallman and Joseph Emerson. ``Noise tailoring for scalable quantum computation via randomized compiling''. Phys. Rev. A 94, 052325 (2016). https://doi.org/10.1103/PhysRevA.94.052325 [23] Jader P. Santos, Ben Bar, and Raam Uzdin. ``Pseudo twirling mitigation of coherent errors in non-clifford gates''. npj Quantum Information 10, 100 (2024). https://doi.org/10.1038/s41534-024-00889-8 [24] G. Casella and R.L. Berger. ``Statistical inference''. Duxbury advanced series in statistics and decision sciences. Thomson Learning. (2002). https://doi.org/10.1201/9781003456285 [25] Robin Harper, Wenjun Yu, and Steven T. Flammia. ``Fast estimation of sparse quantum noise''. PRX Quantum 2, 010322 (2021). https://doi.org/10.1103/PRXQuantum.2.010322 [26] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver. ``A quantum engineer's guide to superconducting qubits''.
Applied Physics Reviews 6, 021318 (2019). https://doi.org/10.1063/1.5089550 [27] S. A. Moses, C. H. Baldwin, M. S. Allman, R. Ancona, L. Ascarrunz, C. Barnes, J. Bartolotta, B. Bjork, P. Blanchard, M. Bohn, et al. ``A race-track trapped-ion quantum processor''. Phys. Rev. X 13, 041052 (2023). https://doi.org/10.1103/PhysRevX.13.041052 [28] Simon J Evered, Dolev Bluvstein, Marcin Kalinowski, Sepehr Ebadi, Tom Manovitz, Hengyun Zhou, Sophie H Li, Alexandra A Geim, Tout T Wang, Nishad Maskara, et al. ``High-fidelity parallel entangling gates on a neutral atom quantum computer''. Nature 622, 268–272 (2023). https://doi.org/10.1038/s41586-023-06481-y [29] Michael A Nielsen. ``A simple formula for the average gate fidelity of a quantum dynamical operation''. Physics Letters A 303, 249–252 (2002). https://doi.org/10.1016/S0375-9601(02)01272-0 [30] Joel Wallman, Chris Granade, Robin Harper, and Steven T Flammia. ``Estimating the coherence of noise''. New Journal of Physics 17, 113020 (2015). https://doi.org/10.1088/1367-2630/17/11/113020 [31] Sergey Bravyi, Sarah Sheldon, Abhinav Kandala, David C. McKay, and Jay M. Gambetta. ``Mitigating measurement errors in multiqubit experiments''. Phys. Rev. A 103, 042605 (2021). https://doi.org/10.1103/PhysRevA.103.042605 [32] Ewout van den Berg, Zlatko K. Minev, and Kristan Temme. ``Model-free readout-error mitigation for quantum expectation values''. Phys. Rev. A 105, 032620 (2022). https://doi.org/10.1103/PhysRevA.105.032620 [33] Alexander Erhard, Joel J Wallman, Lukas Postler, Michael Meth, Roman Stricker, Esteban A Martinez, Philipp Schindler, Thomas Monz, Joseph Emerson, and Rainer Blatt. ``Characterizing large-scale quantum computers via cycle benchmarking''. Nature Communications 10, 5347 (2019). https://doi.org/10.1038/s41467-019-13068-7 [34] Senrui Chen, Sisi Zhou, Alireza Seif, and Liang Jiang. ``Quantum advantages for Pauli channel estimation''. Phys. Rev. A 105, 032435 (2022). https://doi.org/10.1103/PhysRevA.105.032435 [35] Sitan Chen, Weiyuan Gong, and Qi Ye. ``Optimal tradeoffs for estimating Pauli observables''. In 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS). Pages 1086–1105. (2024). https://doi.org/10.1109/FOCS61266.2024.00072 [36] Robbie King, David Gosset, Robin Kothari, and Ryan Babbush. ``Triply efficient shadow tomography''. PRX Quantum 6, 010336 (2025). https://doi.org/10.1103/PRXQuantum.6.010336 [37] Joel J Wallman and Steven T Flammia. ``Randomized benchmarking with confidence''. New Journal of Physics 16, 103032 (2014). https://doi.org/10.1088/1367-2630/16/10/103032 [38] Jonas Helsen, Joel J. Wallman, Steven T. Flammia, and Stephanie Wehner. ``Multiqubit randomized benchmarking using few samples''. Phys. Rev. A 100, 032304 (2019). https://doi.org/10.1103/PhysRevA.100.032304 [39] Andrew Wack, Hanhee Paik, Ali Javadi-Abhari, Petar Jurcevic, Ismael Faro, Jay M. Gambetta, and Blake R. Johnson. ``Quality, speed, and scale: three key attributes to measure the performance of near-term quantum computers'' (2021). arXiv:2110.14108. arXiv:2110.14108 [40] Steven T. Flammia. ``Averaged Circuit Eigenvalue Sampling''. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Volume 232 of Leibniz International Proceedings in Informatics (LIPIcs), pages 4:1–4:10. Dagstuhl, Germany (2022). Schloss Dagstuhl – Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.TQC.2022.4 [41] Senrui Chen, Yunchao Liu, Matthew Otten, Alireza Seif, Bill Fefferman, and Liang Jiang. ``The learnability of Pauli noise''. Nature Communications 14, 52 (2023). https://doi.org/10.1038/s41467-022-35759-4 [42] Shelby Kimmel, Marcus P. da Silva, Colm A. Ryan, Blake R. Johnson, and Thomas Ohki. ``Robust extraction of tomographic information via randomized benchmarking''. Phys. Rev. X 4, 011050 (2014). https://doi.org/10.1103/PhysRevX.4.011050 [43] Jonas Helsen, Francesco Battistel, and Barbara M Terhal. ``Spectral quantum tomography''. npj Quantum Information 5, 74 (2019). https://doi.org/10.1038/s41534-019-0189-0 [44] Earl Campbell. ``Random compiler for fast Hamiltonian simulation''. Phys. Rev. Lett. 123, 070503 (2019). https://doi.org/10.1103/PhysRevLett.123.070503 [45] Bálint Koczor, John J. L. Morton, and Simon C. Benjamin. ``Probabilistic interpolation of quantum rotation angles''. Phys. Rev. Lett. 132, 130602 (2024). https://doi.org/10.1103/PhysRevLett.132.130602 [46] Etienne Granet and Henrik Dreyer. ``Hamiltonian dynamics on digital quantum computers without discretization error''. npj Quantum Information 10, 82 (2024). https://doi.org/10.1038/s41534-024-00877-y [47] Bálint Koczor. ``Sparse probabilistic synthesis of quantum operations''. PRX Quantum 5, 040352 (2024). https://doi.org/10.1103/PRXQuantum.5.040352 [48] Masuo Suzuki. ``General theory of fractal path integrals with applications to many‐body theories and statistical physics''. Journal of Mathematical Physics 32, 400–407 (1991). https://doi.org/10.1063/1.529425Cited byCould not fetch Crossref cited-by data during last attempt 2026-02-10 10:09:27: Could not fetch cited-by data for 10.22331/q-2026-02-10-2003 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-02-10 10:09:27: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractQuantum error mitigation techniques mimic noiseless quantum circuits by running several related noisy circuits and combining their outputs in particular ways. How well such techniques work is thought to depend strongly on how noisy the underlying gates are. Weakly-entangling gates, like $R_{ZZ}(\theta)$ for small angles $\theta$, can be much less noisy than entangling Clifford gates, like CNOT and CZ, and they arise naturally in circuits used to simulate quantum dynamics. However, such weakly-entangling gates are non-Clifford, and are therefore incompatible with two of the most prominent error mitigation techniques to date: probabilistic error cancellation (PEC) and the related form of zero-noise extrapolation (ZNE). This paper generalizes these techniques to non-Clifford gates, and comprises two complementary parts. The first part shows how to effectively transform any given quantum channel into (almost) any desired channel, at the cost of a sampling overhead, by adding random Pauli gates and processing the measurement outcomes. This enables us to cancel or properly amplify noise in non-Clifford gates, provided we can first characterize such gates in detail. The second part therefore introduces techniques to do so for noisy $R_{ZZ}(\theta)$ gates. These techniques are robust to state preparation and measurement (SPAM) errors, and exhibit concentration and sensitivity—crucial statistical properties for many experiments. They are related to randomized benchmarking, and may also be of interest beyond the context of error mitigation. We find that while non-Clifford gates can be less noisy than related Cliffords, their noise is fundamentally more complex, which can lead to surprising and sometimes unwanted effects in error mitigation. Whether this trade-off can be broadly advantageous remains to be seen.Featured image: An illustration of probabilistic error cancellation (PEC) generalized to $R_{ZZ}$ gates with a non-Clifford angle. The gate noise can be accurately learned, even under strong state preparation and measurement (SPAM) errors, and then accurately cancelled in effect.► BibTeX data@article{Layden2026theoryofquantum, doi = {10.22331/q-2026-02-10-2003}, url = {https://doi.org/10.22331/q-2026-02-10-2003}, title = {Theory of quantum error mitigation for non-{C}lifford gates}, author = {Layden, David and Mitchell, Bradley and Siva, Karthik}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2003}, month = feb, year = {2026} }► References [1] 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''. Rev. Mod. Phys. 95, 045005 (2023). https://doi.org/10.1103/RevModPhys.95.045005 [2] Kristan Temme, Sergey Bravyi, and Jay M. Gambetta. ``Error mitigation for short-depth quantum circuits''. Phys. Rev. Lett. 119, 180509 (2017). https://doi.org/10.1103/PhysRevLett.119.180509 [3] Ying Li and Simon C. Benjamin. ``Efficient variational quantum simulator incorporating active error minimization''. Phys. Rev. X 7, 021050 (2017). https://doi.org/10.1103/PhysRevX.7.021050 [4] Sergey Bravyi, Oliver Dial, Jay M. Gambetta, Darío Gil, and Zaira Nazario. ``The future of quantum computing with superconducting qubits''. Journal of Applied Physics 132, 160902 (2022). https://doi.org/10.1063/5.0082975 [5] Seth Lloyd. ``Universal quantum simulators''. Science 273, 1073–1078 (1996). https://doi.org/10.1126/science.273.5278.1073 [6] Andrew M. Childs, Dmitri Maslov, Yunseong Nam, Neil J. Ross, and Yuan Su. ``Toward the first quantum simulation with quantum speedup''. Proceedings of the National Academy of Sciences 115, 9456–9461 (2018). https://doi.org/10.1073/pnas.1801723115 [7] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu. ``Theory of Trotter error with commutator scaling''. Phys. Rev. X 11, 011020 (2021). https://doi.org/10.1103/PhysRevX.11.011020 [8] Laura Clinton, Johannes Bausch, and Toby Cubitt. ``Hamiltonian simulation algorithms for near-term quantum hardware''. Nature Communications 12, 4989 (2021). https://doi.org/10.1038/s41467-021-25196-0 [9] Ewout Van Den Berg, Zlatko K Minev, Abhinav Kandala, and Kristan Temme. ``Probabilistic error cancellation with sparse Pauli-Lindblad models on noisy quantum processors''. Nature Physics (2023). https://doi.org/10.1038/s41567-023-02042-2 [10] Youngseok Kim, Andrew Eddins, Sajant Anand, Ken Xuan Wei, Ewout Van Den Berg, Sami Rosenblatt, Hasan Nayfeh, Yantao Wu, Michael Zaletel, Kristan Temme, et al. ``Evidence for the utility of quantum computing before fault tolerance''. Nature 618, 500–505 (2023). https://doi.org/10.1038/s41586-023-06096-3 [11] Nathan Earnest, Caroline Tornow, and Daniel J. Egger. ``Pulse-efficient circuit transpilation for quantum applications on cross-resonance-based hardware''. Phys. Rev. Res. 3, 043088 (2021). https://doi.org/10.1103/PhysRevResearch.3.043088 [12] John P. T. Stenger, Nicholas T. Bronn, Daniel J. Egger, and David Pekker. ``Simulating the dynamics of braiding of Majorana zero modes using an IBM quantum computer''. Phys. Rev. Res. 3, 033171 (2021). https://doi.org/10.1103/PhysRevResearch.3.033171 [13] Seth T. Merkel, Jay M. Gambetta, John A. Smolin, Stefano Poletto, Antonio D. Córcoles, Blake R. Johnson, Colm A. Ryan, and Matthias Steffen. ``Self-consistent quantum process tomography''. Phys. Rev. A 87, 062119 (2013). https://doi.org/10.1103/PhysRevA.87.062119 [14] Robin Blume-Kohout, John King Gamble, Erik Nielsen, Jonathan Mizrahi, Jonathan D. Sterk, and Peter Maunz. ``Robust, self-consistent, closed-form tomography of quantum logic gates on a trapped ion qubit'' (2013). arXiv:1310.4492. arXiv:1310.4492 [15] Erik Nielsen, John King Gamble, Kenneth Rudinger, Travis Scholten, Kevin Young, and Robin Blume-Kohout. ``Gate Set Tomography''. Quantum 5, 557 (2021). https://doi.org/10.22331/q-2021-10-05-557 [16] Suguru Endo, Simon C. Benjamin, and Ying Li. ``Practical quantum error mitigation for near-future applications''. Phys. Rev. X 8, 031027 (2018). https://doi.org/10.1103/PhysRevX.8.031027 [17] Daniel Greenbaum. ``Introduction to quantum gate set tomography'' (2015). arXiv:1509.02921. arXiv:1509.02921 [18] Steven T Flammia and Joel J Wallman. ``Efficient estimation of Pauli channels''. ACM Transactions on Quantum Computing 1, 1–32 (2020). https://doi.org/10.1145/3408039 [19] W. Dür, M. Hein, J. I. Cirac, and H.-J. Briegel. ``Standard forms of noisy quantum operations via depolarization''. Phys. Rev. A 72, 052326 (2005). https://doi.org/10.1103/PhysRevA.72.052326 [20] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. ``Exact and approximate unitary 2-designs and their application to fidelity estimation''. Phys. Rev. A 80, 012304 (2009). https://doi.org/10.1103/PhysRevA.80.012304 [21] E. Knill. ``Fault-tolerant postselected quantum computation: Threshold analysis'' (2004). arXiv:quant-ph/0404104. arXiv:quant-ph/0404104 [22] Joel J. Wallman and Joseph Emerson. ``Noise tailoring for scalable quantum computation via randomized compiling''. Phys. Rev. A 94, 052325 (2016). https://doi.org/10.1103/PhysRevA.94.052325 [23] Jader P. Santos, Ben Bar, and Raam Uzdin. ``Pseudo twirling mitigation of coherent errors in non-clifford gates''. npj Quantum Information 10, 100 (2024). https://doi.org/10.1038/s41534-024-00889-8 [24] G. Casella and R.L. Berger. ``Statistical inference''. Duxbury advanced series in statistics and decision sciences. Thomson Learning. (2002). https://doi.org/10.1201/9781003456285 [25] Robin Harper, Wenjun Yu, and Steven T. Flammia. ``Fast estimation of sparse quantum noise''. PRX Quantum 2, 010322 (2021). https://doi.org/10.1103/PRXQuantum.2.010322 [26] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver. ``A quantum engineer's guide to superconducting qubits''.
Applied Physics Reviews 6, 021318 (2019). https://doi.org/10.1063/1.5089550 [27] S. A. Moses, C. H. Baldwin, M. S. Allman, R. Ancona, L. Ascarrunz, C. Barnes, J. Bartolotta, B. Bjork, P. Blanchard, M. Bohn, et al. ``A race-track trapped-ion quantum processor''. Phys. Rev. X 13, 041052 (2023). https://doi.org/10.1103/PhysRevX.13.041052 [28] Simon J Evered, Dolev Bluvstein, Marcin Kalinowski, Sepehr Ebadi, Tom Manovitz, Hengyun Zhou, Sophie H Li, Alexandra A Geim, Tout T Wang, Nishad Maskara, et al. ``High-fidelity parallel entangling gates on a neutral atom quantum computer''. Nature 622, 268–272 (2023). https://doi.org/10.1038/s41586-023-06481-y [29] Michael A Nielsen. ``A simple formula for the average gate fidelity of a quantum dynamical operation''. Physics Letters A 303, 249–252 (2002). https://doi.org/10.1016/S0375-9601(02)01272-0 [30] Joel Wallman, Chris Granade, Robin Harper, and Steven T Flammia. ``Estimating the coherence of noise''. New Journal of Physics 17, 113020 (2015). https://doi.org/10.1088/1367-2630/17/11/113020 [31] Sergey Bravyi, Sarah Sheldon, Abhinav Kandala, David C. McKay, and Jay M. Gambetta. ``Mitigating measurement errors in multiqubit experiments''. Phys. Rev. A 103, 042605 (2021). https://doi.org/10.1103/PhysRevA.103.042605 [32] Ewout van den Berg, Zlatko K. Minev, and Kristan Temme. ``Model-free readout-error mitigation for quantum expectation values''. Phys. Rev. A 105, 032620 (2022). https://doi.org/10.1103/PhysRevA.105.032620 [33] Alexander Erhard, Joel J Wallman, Lukas Postler, Michael Meth, Roman Stricker, Esteban A Martinez, Philipp Schindler, Thomas Monz, Joseph Emerson, and Rainer Blatt. ``Characterizing large-scale quantum computers via cycle benchmarking''. Nature Communications 10, 5347 (2019). https://doi.org/10.1038/s41467-019-13068-7 [34] Senrui Chen, Sisi Zhou, Alireza Seif, and Liang Jiang. ``Quantum advantages for Pauli channel estimation''. Phys. Rev. A 105, 032435 (2022). https://doi.org/10.1103/PhysRevA.105.032435 [35] Sitan Chen, Weiyuan Gong, and Qi Ye. ``Optimal tradeoffs for estimating Pauli observables''. In 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS). Pages 1086–1105. (2024). https://doi.org/10.1109/FOCS61266.2024.00072 [36] Robbie King, David Gosset, Robin Kothari, and Ryan Babbush. ``Triply efficient shadow tomography''. PRX Quantum 6, 010336 (2025). https://doi.org/10.1103/PRXQuantum.6.010336 [37] Joel J Wallman and Steven T Flammia. ``Randomized benchmarking with confidence''. New Journal of Physics 16, 103032 (2014). https://doi.org/10.1088/1367-2630/16/10/103032 [38] Jonas Helsen, Joel J. Wallman, Steven T. Flammia, and Stephanie Wehner. ``Multiqubit randomized benchmarking using few samples''. Phys. Rev. A 100, 032304 (2019). https://doi.org/10.1103/PhysRevA.100.032304 [39] Andrew Wack, Hanhee Paik, Ali Javadi-Abhari, Petar Jurcevic, Ismael Faro, Jay M. Gambetta, and Blake R. Johnson. ``Quality, speed, and scale: three key attributes to measure the performance of near-term quantum computers'' (2021). arXiv:2110.14108. arXiv:2110.14108 [40] Steven T. Flammia. ``Averaged Circuit Eigenvalue Sampling''. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Volume 232 of Leibniz International Proceedings in Informatics (LIPIcs), pages 4:1–4:10. Dagstuhl, Germany (2022). Schloss Dagstuhl – Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.TQC.2022.4 [41] Senrui Chen, Yunchao Liu, Matthew Otten, Alireza Seif, Bill Fefferman, and Liang Jiang. ``The learnability of Pauli noise''. Nature Communications 14, 52 (2023). https://doi.org/10.1038/s41467-022-35759-4 [42] Shelby Kimmel, Marcus P. da Silva, Colm A. Ryan, Blake R. Johnson, and Thomas Ohki. ``Robust extraction of tomographic information via randomized benchmarking''. Phys. Rev. X 4, 011050 (2014). https://doi.org/10.1103/PhysRevX.4.011050 [43] Jonas Helsen, Francesco Battistel, and Barbara M Terhal. ``Spectral quantum tomography''. npj Quantum Information 5, 74 (2019). https://doi.org/10.1038/s41534-019-0189-0 [44] Earl Campbell. ``Random compiler for fast Hamiltonian simulation''. Phys. Rev. Lett. 123, 070503 (2019). https://doi.org/10.1103/PhysRevLett.123.070503 [45] Bálint Koczor, John J. L. Morton, and Simon C. Benjamin. ``Probabilistic interpolation of quantum rotation angles''. Phys. Rev. Lett. 132, 130602 (2024). https://doi.org/10.1103/PhysRevLett.132.130602 [46] Etienne Granet and Henrik Dreyer. ``Hamiltonian dynamics on digital quantum computers without discretization error''. npj Quantum Information 10, 82 (2024). https://doi.org/10.1038/s41534-024-00877-y [47] Bálint Koczor. ``Sparse probabilistic synthesis of quantum operations''. PRX Quantum 5, 040352 (2024). https://doi.org/10.1103/PRXQuantum.5.040352 [48] Masuo Suzuki. ``General theory of fractal path integrals with applications to many‐body theories and statistical physics''. Journal of Mathematical Physics 32, 400–407 (1991). https://doi.org/10.1063/1.529425Cited byCould not fetch Crossref cited-by data during last attempt 2026-02-10 10:09:27: Could not fetch cited-by data for 10.22331/q-2026-02-10-2003 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-02-10 10:09:27: No response from ADS or unable to decode the received json data when getting the list of citing works.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.