Fermionic Averaged Circuit Eigenvalue Sampling
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AbstractFermionic averaged circuit eigenvalue sampling (FACES) is a protocol to simultaneously learn the averaged error rates of many fermionic linear optical (FLO) gates simultaneously and self-consistently from a suitable collection of FLO circuits. It is highly flexible, allowing for the in situ characterization of FLO-averaged gate-dependent noise under natural assumptions on a family of continuously parameterized one- and two-qubit gates. We rigorously show that our protocol has an efficient sampling complexity, owing in-part to useful properties of the Kravchuk transformations that feature in our analysis. We support our conclusions with numerical results. As FLO circuits become universal with access to certain resource states, we expect our results to inform noise characterization and error mitigation techniques on universal quantum computing architectures which naturally admit a fermionic description.Featured image: Given a set of noisy fermionic gates we wish to characterize, we model each by its associated noise channel followed by an application of the ideal gate (left). We compose the gates into a collection of circuits according to a design matrix, with accompanying twirled noise channels (right). By sampling from these circuits and fitting to a log-linear model, we learn many underlying noise parameters simultaneously.Popular summaryQuantum information is extremely fragile, so performing reliable quantum computation requires a complete understanding of the noise affecting our quantum computer. We show that we can achieve such a characterization when our quantum computer is simulating a natural system from the physical world: a gas of noninteracting particles, or fermions. These "free fermion" systems are a natural starting point from which to analyze noise in more powerful models of quantum computation. By taking advantage of the mathematical structure of these systems, we learn a model for the noise of many different free-fermion operations simultaneously. Our work provides a feasible approach for analyzing quantum computing architectures with fermionic structure.► BibTeX data@article{Chapman2026fermionicaveraged, doi = {10.22331/q-2026-04-08-2053}, url = {https://doi.org/10.22331/q-2026-04-08-2053}, title = {Fermionic {A}veraged {C}ircuit {E}igenvalue {S}ampling}, author = {Chapman, Adrian and Flammia, Steven T.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2053}, month = apr, year = {2026} }► 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, et al. ``Quantum supremacy using a programmable superconducting processor''. Nature 574, 505–510 (2019). https://doi.org/10.1038/s41586-019-1666-5 [2] Guido Burkard, Thaddeus D. Ladd, Andrew Pan, John M. Nichol, and Jason R. Petta. ``Semiconductor spin qubits''. Rev. Mod. Phys. 95, 025003 (2023). https://doi.org/10.1103/RevModPhys.95.025003 [3] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan. ``Quantum computational chemistry''. Rev. Mod. Phys. 92, 015003 (2020). https://doi.org/10.1103/RevModPhys.92.015003 [4] E. Knill. ``Quantum computing with realistically noisy devices''. Nature 434, 39–44 (2005). https://doi.org/10.1038/nature03350 [5] 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 [6] 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. https://doi.org/10.48550/arXiv.1310.4492 arXiv:1310.4492 [7] Dohun Kim, D. R. Ward, C. B. Simmons, John King Gamble, Robin Blume-Kohout, Erik Nielsen, D. E. Savage, M. G. Lagally, Mark Friesen, S. N. Coppersmith, and M. A. Eriksson. ``Microwave-driven coherent operation of a semiconductor quantum dot charge qubit''. Nature Nanotechnology 10, 243–247 (2015). https://doi.org/10.1038/nnano.2014.336 [8] 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 [9] Steven T. Flammia. ``Averaged Circuit Eigenvalue Sampling''. In François Le Gall and Tomoyuki Morimae, editors, 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 [10] Richard Jozsa and Akimasa Miyake. ``Matchgates and classical simulation of quantum circuits''. Proc. R. Soc. A. 464, 3089–3106 (2008). https://doi.org/10.1098/rspa.2008.0189 [11] Senrui Chen, Yunchao Liu, Matthew Otten, Alireza Seif, Bill Fefferman, and Liang Jiang. ``The learnability of Pauli noise''. Nat Commun 14, 52 (2023). https://doi.org/10.1038/s41467-022-35759-4 [12] Senrui Chen, Zhihan Zhang, Liang Jiang, and Steven T. Flammia. ``Efficient self-consistent learning of gate set pauli noise''. PRX Quantum 7, 010305 (2026). https://doi.org/10.1103/1pnv-t9px [13] Evan T. Hockings, Andrew C. Doherty, and Robin Harper. ``Scalable Noise Characterization of Syndrome-Extraction Circuits with Averaged Circuit Eigenvalue Sampling''. PRX Quantum 6, 010334 (2025). https://doi.org/10.1103/PRXQuantum.6.010334 [14] Jahan Claes, Eleanor Rieffel, and Zhihui Wang. ``Character randomized benchmarking for non-multiplicity-free groups with applications to subspace, leakage, and matchgate randomized benchmarking''. PRX Quantum 2, 010351 (2021). https://doi.org/10.1103/PRXQuantum.2.010351 [15] Guoding Liu, Ziyi Xie, Zitai Xu, and Xiongfeng Ma. ``Group twirling and noise tailoring for multiqubit controlled phase gates''. Phys. Rev. Res. 6, 043221 (2024). https://doi.org/10.1103/PhysRevResearch.6.043221 [16] Jadwiga Wilkens, Marios Ioannou, Ellen Derbyshire, Jens Eisert, Dominik Hangleiter, Ingo Roth, and Jonas Haferkamp. ``Benchmarking bosonic and fermionic dynamics'' (2024). arXiv:2408.11105. https://doi.org/10.48550/arXiv.2408.11105 arXiv:2408.11105 [17] Joshua Cudby and Sergii Strelchuk. ``Learning Gaussian Operations and the Matchgate Hierarchy'' (2024). arXiv:2407.12649. https://doi.org/10.48550/arXiv.2407.12649 arXiv:2407.12649 [18] Jonas Helsen, Sepehr Nezami, Matthew Reagor, and Michael Walter. ``Matchgate benchmarking: Scalable benchmarking of a continuous family of many-qubit gates''. Quantum 6, 657 (2022). https://doi.org/10.22331/q-2022-02-21-657 [19] Jędrzej Burkat and Sergii Strelchuk. ``A Lightweight Protocol for Matchgate Fidelity Estimation'' (2024). arXiv:2404.07974. https://doi.org/10.48550/arXiv.2404.07974 arXiv:2404.07974 [20] Christoph Dankert. ``Efficient Simulation of Random Quantum States and Operators'' (2005). arXiv:quant-ph/0512217. https://doi.org/10.48550/arXiv.quant-ph/0512217 arXiv:quant-ph/0512217 [21] Zhang Jiang, Kevin J. Sung, Kostyantyn Kechedzhi, Vadim N. Smelyanskiy, and Sergio Boixo. ``Quantum algorithms to simulate many-body physics of correlated fermions''. Phys. Rev. Appl. 9, 044036 (2018). https://doi.org/10.1103/PhysRevApplied.9.044036 [22] D S França and A K Hashagen. ``Approximate randomized benchmarking for finite groups''. J. Phys. A: Math. Theor. 51, 395302 (2018). https://doi.org/10.1088/1751-8121/aad6fa [23] 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 [24] 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 [25] Thomas Schuster, Jonas Haferkamp, and Hsin-Yuan Huang. ``Random unitaries in extremely low depth'' (2024). arXiv:2407.07754. https://doi.org/10.48550/arXiv.2407.07754 arXiv:2407.07754 [26] Nicholas LaRacuente and Felix Leditzky. ``Approximate unitary k-designs from shallow, low-communication circuits''. Commun. Math. Phys. 407 (2026). https://doi.org/10.1007/s00220-025-05542-9 [27] Kianna Wan, William J. Huggins, Joonho Lee, and Ryan Babbush. ``Matchgate shadows for fermionic quantum simulation''. Commun. Math. Phys. 404, 629–700 (2023). https://doi.org/10.1007/s00220-023-04844-0 [28] Valentin Heyraud, Héloise Chomet, and Jules Tilly. ``Unified Framework for Matchgate Classical Shadows'' (2024). arXiv:2409.03836. https://doi.org/10.48550/arXiv.2409.03836 arXiv:2409.03836 [29] Easwar Magesan, Jay M. Gambetta, B. R. Johnson, Colm A. Ryan, Jerry M. Chow, Seth T. Merkel, Marcus P. da Silva, George A. Keefe, Mary B. Rothwell, Thomas A. Ohki, Mark B. Ketchen, and M. Steffen. ``Efficient measurement of quantum gate error by interleaved randomized benchmarking''. Phys. Rev. Lett. 109, 080505 (2012). https://doi.org/10.1103/PhysRevLett.109.080505 [30] Ashley Montanaro and Stasja Stanisic. ``Error mitigation by training with fermionic linear optics'' (2021). arXiv:2102.02120. https://doi.org/10.48550/arXiv.2102.02120 arXiv:2102.02120 [31] Miha Papič, Manuel G. Algaba, Emiliano Godinez-Ramirez, Inés de Vega, Adrian Auer, Fedor Šimkovic IV, and Alessio Calzona. ``Near-Term Fermionic Simulation with Subspace Noise Tailored Quantum Error Mitigation'' (2025). arXiv:2503.11785. https://doi.org/10.48550/arXiv.2503.11785 arXiv:2503.11785 [32] Jahan Claes (2020). code: jahanclaes/Hoffman-Decomposition-and-the-Matchgate-Group.git. https://github.com/jahanclaes/Hoffman-Decomposition-and-the-Matchgate-Group.gitCited byCould not fetch Crossref cited-by data during last attempt 2026-04-08 08:24:31: Could not fetch cited-by data for 10.22331/q-2026-04-08-2053 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-08 08:24:32: 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. AbstractFermionic averaged circuit eigenvalue sampling (FACES) is a protocol to simultaneously learn the averaged error rates of many fermionic linear optical (FLO) gates simultaneously and self-consistently from a suitable collection of FLO circuits. It is highly flexible, allowing for the in situ characterization of FLO-averaged gate-dependent noise under natural assumptions on a family of continuously parameterized one- and two-qubit gates. We rigorously show that our protocol has an efficient sampling complexity, owing in-part to useful properties of the Kravchuk transformations that feature in our analysis. We support our conclusions with numerical results. As FLO circuits become universal with access to certain resource states, we expect our results to inform noise characterization and error mitigation techniques on universal quantum computing architectures which naturally admit a fermionic description.Featured image: Given a set of noisy fermionic gates we wish to characterize, we model each by its associated noise channel followed by an application of the ideal gate (left). We compose the gates into a collection of circuits according to a design matrix, with accompanying twirled noise channels (right). By sampling from these circuits and fitting to a log-linear model, we learn many underlying noise parameters simultaneously.Popular summaryQuantum information is extremely fragile, so performing reliable quantum computation requires a complete understanding of the noise affecting our quantum computer. We show that we can achieve such a characterization when our quantum computer is simulating a natural system from the physical world: a gas of noninteracting particles, or fermions. These "free fermion" systems are a natural starting point from which to analyze noise in more powerful models of quantum computation. By taking advantage of the mathematical structure of these systems, we learn a model for the noise of many different free-fermion operations simultaneously. Our work provides a feasible approach for analyzing quantum computing architectures with fermionic structure.► BibTeX data@article{Chapman2026fermionicaveraged, doi = {10.22331/q-2026-04-08-2053}, url = {https://doi.org/10.22331/q-2026-04-08-2053}, title = {Fermionic {A}veraged {C}ircuit {E}igenvalue {S}ampling}, author = {Chapman, Adrian and Flammia, Steven T.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2053}, month = apr, year = {2026} }► 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, et al. ``Quantum supremacy using a programmable superconducting processor''. Nature 574, 505–510 (2019). https://doi.org/10.1038/s41586-019-1666-5 [2] Guido Burkard, Thaddeus D. Ladd, Andrew Pan, John M. Nichol, and Jason R. Petta. ``Semiconductor spin qubits''. Rev. Mod. Phys. 95, 025003 (2023). https://doi.org/10.1103/RevModPhys.95.025003 [3] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan. ``Quantum computational chemistry''. Rev. Mod. Phys. 92, 015003 (2020). https://doi.org/10.1103/RevModPhys.92.015003 [4] E. Knill. ``Quantum computing with realistically noisy devices''. Nature 434, 39–44 (2005). https://doi.org/10.1038/nature03350 [5] 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 [6] 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. https://doi.org/10.48550/arXiv.1310.4492 arXiv:1310.4492 [7] Dohun Kim, D. R. Ward, C. B. Simmons, John King Gamble, Robin Blume-Kohout, Erik Nielsen, D. E. Savage, M. G. Lagally, Mark Friesen, S. N. Coppersmith, and M. A. Eriksson. ``Microwave-driven coherent operation of a semiconductor quantum dot charge qubit''. Nature Nanotechnology 10, 243–247 (2015). https://doi.org/10.1038/nnano.2014.336 [8] 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 [9] Steven T. Flammia. ``Averaged Circuit Eigenvalue Sampling''. In François Le Gall and Tomoyuki Morimae, editors, 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 [10] Richard Jozsa and Akimasa Miyake. ``Matchgates and classical simulation of quantum circuits''. Proc. R. Soc. A. 464, 3089–3106 (2008). https://doi.org/10.1098/rspa.2008.0189 [11] Senrui Chen, Yunchao Liu, Matthew Otten, Alireza Seif, Bill Fefferman, and Liang Jiang. ``The learnability of Pauli noise''. Nat Commun 14, 52 (2023). https://doi.org/10.1038/s41467-022-35759-4 [12] Senrui Chen, Zhihan Zhang, Liang Jiang, and Steven T. Flammia. ``Efficient self-consistent learning of gate set pauli noise''. PRX Quantum 7, 010305 (2026). https://doi.org/10.1103/1pnv-t9px [13] Evan T. Hockings, Andrew C. Doherty, and Robin Harper. ``Scalable Noise Characterization of Syndrome-Extraction Circuits with Averaged Circuit Eigenvalue Sampling''. PRX Quantum 6, 010334 (2025). https://doi.org/10.1103/PRXQuantum.6.010334 [14] Jahan Claes, Eleanor Rieffel, and Zhihui Wang. ``Character randomized benchmarking for non-multiplicity-free groups with applications to subspace, leakage, and matchgate randomized benchmarking''. PRX Quantum 2, 010351 (2021). https://doi.org/10.1103/PRXQuantum.2.010351 [15] Guoding Liu, Ziyi Xie, Zitai Xu, and Xiongfeng Ma. ``Group twirling and noise tailoring for multiqubit controlled phase gates''. Phys. Rev. Res. 6, 043221 (2024). https://doi.org/10.1103/PhysRevResearch.6.043221 [16] Jadwiga Wilkens, Marios Ioannou, Ellen Derbyshire, Jens Eisert, Dominik Hangleiter, Ingo Roth, and Jonas Haferkamp. ``Benchmarking bosonic and fermionic dynamics'' (2024). arXiv:2408.11105. https://doi.org/10.48550/arXiv.2408.11105 arXiv:2408.11105 [17] Joshua Cudby and Sergii Strelchuk. ``Learning Gaussian Operations and the Matchgate Hierarchy'' (2024). arXiv:2407.12649. https://doi.org/10.48550/arXiv.2407.12649 arXiv:2407.12649 [18] Jonas Helsen, Sepehr Nezami, Matthew Reagor, and Michael Walter. ``Matchgate benchmarking: Scalable benchmarking of a continuous family of many-qubit gates''. Quantum 6, 657 (2022). https://doi.org/10.22331/q-2022-02-21-657 [19] Jędrzej Burkat and Sergii Strelchuk. ``A Lightweight Protocol for Matchgate Fidelity Estimation'' (2024). arXiv:2404.07974. https://doi.org/10.48550/arXiv.2404.07974 arXiv:2404.07974 [20] Christoph Dankert. ``Efficient Simulation of Random Quantum States and Operators'' (2005). arXiv:quant-ph/0512217. https://doi.org/10.48550/arXiv.quant-ph/0512217 arXiv:quant-ph/0512217 [21] Zhang Jiang, Kevin J. Sung, Kostyantyn Kechedzhi, Vadim N. Smelyanskiy, and Sergio Boixo. ``Quantum algorithms to simulate many-body physics of correlated fermions''. Phys. Rev. Appl. 9, 044036 (2018). https://doi.org/10.1103/PhysRevApplied.9.044036 [22] D S França and A K Hashagen. ``Approximate randomized benchmarking for finite groups''. J. Phys. A: Math. Theor. 51, 395302 (2018). https://doi.org/10.1088/1751-8121/aad6fa [23] 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 [24] 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 [25] Thomas Schuster, Jonas Haferkamp, and Hsin-Yuan Huang. ``Random unitaries in extremely low depth'' (2024). arXiv:2407.07754. https://doi.org/10.48550/arXiv.2407.07754 arXiv:2407.07754 [26] Nicholas LaRacuente and Felix Leditzky. ``Approximate unitary k-designs from shallow, low-communication circuits''. Commun. Math. Phys. 407 (2026). https://doi.org/10.1007/s00220-025-05542-9 [27] Kianna Wan, William J. Huggins, Joonho Lee, and Ryan Babbush. ``Matchgate shadows for fermionic quantum simulation''. Commun. Math. Phys. 404, 629–700 (2023). https://doi.org/10.1007/s00220-023-04844-0 [28] Valentin Heyraud, Héloise Chomet, and Jules Tilly. ``Unified Framework for Matchgate Classical Shadows'' (2024). arXiv:2409.03836. https://doi.org/10.48550/arXiv.2409.03836 arXiv:2409.03836 [29] Easwar Magesan, Jay M. Gambetta, B. R. Johnson, Colm A. Ryan, Jerry M. Chow, Seth T. Merkel, Marcus P. da Silva, George A. Keefe, Mary B. Rothwell, Thomas A. Ohki, Mark B. Ketchen, and M. Steffen. ``Efficient measurement of quantum gate error by interleaved randomized benchmarking''. Phys. Rev. Lett. 109, 080505 (2012). https://doi.org/10.1103/PhysRevLett.109.080505 [30] Ashley Montanaro and Stasja Stanisic. ``Error mitigation by training with fermionic linear optics'' (2021). arXiv:2102.02120. https://doi.org/10.48550/arXiv.2102.02120 arXiv:2102.02120 [31] Miha Papič, Manuel G. Algaba, Emiliano Godinez-Ramirez, Inés de Vega, Adrian Auer, Fedor Šimkovic IV, and Alessio Calzona. ``Near-Term Fermionic Simulation with Subspace Noise Tailored Quantum Error Mitigation'' (2025). arXiv:2503.11785. https://doi.org/10.48550/arXiv.2503.11785 arXiv:2503.11785 [32] Jahan Claes (2020). code: jahanclaes/Hoffman-Decomposition-and-the-Matchgate-Group.git. https://github.com/jahanclaes/Hoffman-Decomposition-and-the-Matchgate-Group.gitCited byCould not fetch Crossref cited-by data during last attempt 2026-04-08 08:24:31: Could not fetch cited-by data for 10.22331/q-2026-04-08-2053 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-08 08:24:32: 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.