Randomized benchmarking with gate-dependent noise

Joel J. Wallman

Institute for Quantum Computing and Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

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

Abstract

We analyze randomized benchmarking for arbitrary gate-dependent noise and prove that the exact impact of gate-dependent noise can be described by a single perturbation term that decays exponentially with the sequence length. That is, the exact behavior of randomized benchmarking under general gate-dependent noise converges exponentially to a true exponential decay of exactly the same form as that predicted by previous analysis for gate-independent noise. Moreover, we show that the operational meaning of the decay parameter for gate-dependent noise is essentially unchanged, that is, we show that it quantifies the average fidelity of the noise between ideal gates. We numerically demonstrate that our analysis is valid for strongly gate-dependent noise models. We also show why alternative analyses do not provide a rigorous justification for the empirical success of randomized benchmarking with gate-dependent noise.

► BibTeX data

► References

[1] Isaac L. Chuang and Michael A. Nielsen, Prescription for experimental determination of the dynamics of a quantum black box, Journal of Modern Optics, 44, 2455 (1997).
https:/​/​doi.org/​10.1080/​09500349708231894

[2] J. F. Poyatos, J. Ignacioi Cirac, and P. Zoller, Complete Characterization of a Quantum Process: The Two-Bit Quantum Gate, Physical Review Letters 78, 390 (1997).
https:/​/​doi.org/​10.1103/​PhysRevLett.78.390

[3] Marcus P. da Silva, Olivier Landon-Cardinal, and David Poulin, Practical Characterization of Quantum Devices without Tomography, Physical Review Letters 107, 210404 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.107.210404

[4] Steven T. Flammia and Yi-Kai Liu, Direct Fidelity Estimation from Few Pauli Measurements, Physical Review Letters 106, 230501 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.106.230501

[5] Steven T. Flammia, David Gross, Yi-Kai Liu, and Jens Eisert, Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators, New Journal of Physics 14, 095022 (2012).
https:/​/​doi.org/​10.1088/​1367-2630/​14/​9/​095022

[6] Daniel M. Reich, Giulia Gualdi, and Christiane P. Koch, Optimal Strategies for Estimating the Average Fidelity of Quantum Gates, Physical Review Letters 111, 200401 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.111.200401

[7] Martin Kliesch, Richard Kueng, Jens Eisert, and David Gross, Guaranteed recovery of quantum processes from few measurements, arXiv:1701.03135 [quant-ph].
arXiv:1701.03135

[8] Joseph Emerson, Robert Alicki, and Karol Życzkowski, Scalable noise estimation with random unitary operators, Journal of Optics B 7, S347 (2005).
https:/​/​doi.org/​10.1088/​1464-4266/​7/​10/​021

[9] Benjamin Lévi, Cecilia C López, Joseph Emerson, and David G. Cory, Efficient error characterization in quantum information processing, Physical Review A 75, 022314 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.75.022314

[10] Emanuel Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and David J. Wineland, Randomized benchmarking of quantum gates, Physical Review A 77, 012307 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.77.012307

[11] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation, Physical Review A 80, 012304 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.80.012304

[12] Easwar Magesan, Jay M. Gambetta, and Joseph Emerson, Scalable and Robust Randomized Benchmarking of Quantum Processes, Physical Review Letters 106, 180504 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.106.180504

[13] Joseph Emerson, Marcus P. da Silva, Osama Moussa, Colm A. Ryan, Martin Laforest, Jonathan Baugh, David G. Cory, and Raymond Laflamme, Symmetrized characterization of noisy quantum processes. Science 317, 1893 (2007).
https:/​/​doi.org/​10.1126/​science.1145699

[14] Easwar Magesan, Jay M. Gambetta, Blake 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 Matthias Steffen, Efficient Measurement of Quantum Gate Error by Interleaved Randomized Benchmarking, Physical Review Letters 109, 080505 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.109.080505

[15] Joel J. Wallman, Christopher 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

[16] Joel J. Wallman, Marie Barnhill, and Joseph Emerson, Robust Characterization of Loss Rates, Physical Review Letters 115, 060501 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.115.060501

[17] Joel J. Wallman, Marie Barnhill, and Joseph Emerson, Robust characterization of leakage errors, New Journal of Physics 18, 043021 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​4/​043021

[18] Arnaud Carignan-Dugas, Joel J. Wallman, and Joseph Emerson, Characterizing universal gate sets via dihedral benchmarking, Physical Review A 92, 060302(R) (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.060302

[19] Andrew W. Cross, Easwar Magesan, Lev S. Bishop, John A. Smolin, and Jay M. Gambetta, Scalable randomised benchmarking of non-Clifford gates, npj Quantum Information 2, 16012 (2016).
https:/​/​doi.org/​10.1038/​npjqi.2016.12

[20] Antonio D. Córcoles, Jay M. Gambetta, Jerry M. Chow, John A. Smolin, Matthew Ware, Joel Strand, B. L. T. Plourde, and Matthias Steffen, Process verification of two-qubit quantum gates by randomized benchmarking, Physical Review A 87, 030301(R) (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.87.030301

[21] R. Barends, Julian Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, Austin G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. J. J. O`Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and John M. Martinis, Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature 508, 500 (2014).
https:/​/​doi.org/​10.1038/​nature13171

[22] Julian Kelly, R. Barends, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, Austin G. Fowler, I.-C. Hoi, E. Jeffrey, A. Megrant, J. Mutus, C. Neill, P. J. J. O'Malley, C. Quintana, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, T. C. White, A. N. Cleland, and John M. Martinis, Optimal Quantum Control Using Randomized Benchmarking, Physical Review Letters 112, 240504 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.112.240504

[23] Jeffrey M. Epstein, Andrew W. Cross, Easwar Magesan, and Jay M. Gambetta, Investigating the limits of randomized benchmarking protocols, Physical Review A 89, 062321 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.89.062321

[24] Tobias Chasseur and Frank K. Wilhelm, Complete randomized benchmarking protocol accounting for leakage errors, Physical Review A 92, 042333 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.042333

[25] Harrison Ball, Thomas M. Stace, Steven T. Flammia, and Michael J. Biercuk, Effect of noise correlations on randomized benchmarking, Physical Review A 93, 022303 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.93.022303

[26] Timothy Proctor, Kenneth Rudinger, Kevin Young, Mohan Sarovar, and Robin Blume-kohout, What randomized benchmarking actually measures, Physical Review Letters 119, 130502 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.130502

[27] Joshua Combes, Christopher Granade, Christopher Ferrie, and Steven T. Flammia, Logical Randomized Benchmarking, arXiv:1702.03688 [quant-ph].
arXiv:1702.03688

[28] Easwar Magesan, Jay M. Gambetta, and Joseph Emerson, Characterizing quantum gates via randomized benchmarking, Physical Review A 85, 042311 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.85.042311

[29] Yuval R. Sanders, Joel J. Wallman, and Barry C. Sanders, Bounding quantum gate error rate based on reported average fidelity, New Journal of Physics 18, 012002 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​1/​012002

[30] 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

[31] Christopher Granade, Christopher Ferrie, and David G. Cory, Accelerated randomized benchmarking, New Journal of Physics 17, 013042 (2015).
https:/​/​doi.org/​10.1088/​1367-2630/​17/​1/​013042

[32] Michael A. Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation, Physics Letters A 303, 249 (2002).
https:/​/​doi.org/​10.1016/​S0375-9601(02)01272-0

[33] F. L. Bauer and C. T. Fike, Norms and exclusion theorems, Numerische Mathematik 2, 137 (1960).
https:/​/​doi.org/​10.1007/​BF01386217

[34] David Pérez-García, Michael M. Wolf, Denes Petz, and Mary Beth Ruskai, Contractivity of positive and trace-preserving maps under $L_p$ norms, Journal of Mathematical Physics 47, 083506 (2006).
https:/​/​doi.org/​10.1063/​1.2218675

[35] Robin Blume-Kohout, John King Gamble, Erik Nielsen, Kenneth Rudinger, Jonathan Mizrahi, Kevin Fortier, and Peter Maunz, Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography, Nature Communications 8, 14485 (2017).
https:/​/​doi.org/​10.1038/​ncomms14485

[36] Mark D. Bowdrey, Daniel K. L. Oi, Anthony J. Short, Konrad Banaszek, and Jonathan A. Jones, Fidelity of single qubit maps, Physics Letters A 294, 258 (2002).
https:/​/​doi.org/​10.1016/​S0375-9601(02)00069-5

Cited by

[1] Jiaan Qi and Hui Khoon Ng, "Comparing the randomized benchmarking figure with the average infidelity of a quantum gate-set", International Journal of Quantum Information 17 04, 1950031 (2019).

[2] E. Onorati, A. H. Werner, and J. Eisert, "Randomized Benchmarking for Individual Quantum Gates", Physical Review Letters 123 6, 060501 (2019).

[3] Ya-Dong Wu, Ge Bai, Giulio Chiribella, and Nana Liu, "Efficient Verification of Continuous-Variable Quantum States and Devices without Assuming Identical and Independent Operations", Physical Review Letters 126 24, 240503 (2021).

[4] Joshua Morris, Felix A. Pollock, and Kavan Modi, "Quantifying non-Markovian Memory in a Superconducting Quantum Computer", Open Systems & Information Dynamics 29 02, 2250007 (2022).

[5] C. H. Baldwin, B. J. Bjork, J. P. Gaebler, D. Hayes, and D. Stack, "Subspace benchmarking high-fidelity entangling operations with trapped ions", Physical Review Research 2 1, 013317 (2020).

[6] Ryan Shaffer, Eli Megidish, Joseph Broz, Wei-Ting Chen, and Hartmut Häffner, "Practical verification protocols for analog quantum simulators", npj Quantum Information 7 1, 46 (2021).

[7] J. P. Gaebler, C. H. Baldwin, S. A. Moses, J. M. Dreiling, C. Figgatt, M. Foss-Feig, D. Hayes, and J. M. Pino, "Suppression of midcircuit measurement crosstalk errors with micromotion", Physical Review A 104 6, 062440 (2021).

[8] Jens Eisert, Dominik Hangleiter, Nathan Walk, Ingo Roth, Damian Markham, Rhea Parekh, Ulysse Chabaud, and Elham Kashefi, "Quantum certification and benchmarking", Nature Reviews Physics 2 7, 382 (2020).

[9] Robin Harper, Ian Hincks, Chris Ferrie, Steven T. Flammia, and Joel J. Wallman, "Statistical analysis of randomized benchmarking", Physical Review A 99 5, 052350 (2019).

[10] Robin Blume-Kohout, Marcus P. da Silva, Erik Nielsen, Timothy Proctor, Kenneth Rudinger, Mohan Sarovar, and Kevin Young, "A Taxonomy of Small Markovian Errors", PRX Quantum 3 2, 020335 (2022).

[11] Arnaud Carignan-Dugas, Joel J Wallman, and Joseph Emerson, "Bounding the average gate fidelity of composite channels using the unitarity", New Journal of Physics 21 5, 053016 (2019).

[12] Robin Harper and Steven T. Flammia, "Fault-Tolerant Logical Gates in the IBM Quantum Experience", Physical Review Letters 122 8, 080504 (2019).

[13] Elena Ferraro and Marco De Michielis, "On the robustness of the hybrid qubit computational gates through simulated randomized benchmarking protocols", Scientific Reports 10 1, 17780 (2020).

[14] Salonik Resch and Ulya R. Karpuzcu, "Benchmarking Quantum Computers and the Impact of Quantum Noise", ACM Computing Surveys 54 7, 1 (2022).

[15] Dax Enshan Koh and Sabee Grewal, "Classical Shadows With Noise", Quantum 6, 776 (2022).

[16] I. Roth, R. Kueng, S. Kimmel, Y.-K. Liu, D. Gross, J. Eisert, and M. Kliesch, "Recovering Quantum Gates from Few Average Gate Fidelities", Physical Review Letters 121 17, 170502 (2018).

[17] Junan Lin, Brandon Buonacorsi, Raymond Laflamme, and Joel J Wallman, "On the freedom in representing quantum operations", New Journal of Physics 21 2, 023006 (2019).

[18] Kristine Boone, Arnaud Carignan-Dugas, Joel J. Wallman, and Joseph Emerson, "Randomized benchmarking under different gate sets", Physical Review A 99 3, 032329 (2019).

[19] J. Helsen, M. Ioannou, J. Kitzinger, E. Onorati, A. H. Werner, J. Eisert, and I. Roth, "Shadow estimation of gate-set properties from random sequences", Nature Communications 14 1, 5039 (2023).

[20] Jonas Helsen and Stephanie Wehner, "A benchmarking procedure for quantum networks", npj Quantum Information 9 1, 17 (2023).

[21] Matthew Girling, Cristina Cîrstoiu, and David Jennings, "Estimation of correlations and nonseparability in quantum channels via unitarity benchmarking", Physical Review Research 4 2, 023041 (2022).

[22] Senrui Chen, Wenjun Yu, Pei Zeng, and Steven T. Flammia, "Robust Shadow Estimation", PRX Quantum 2 3, 030348 (2021).

[23] Yanwu Gu, Wei-Feng Zhuang, Xudan Chai, and Dong E. Liu, "Benchmarking universal quantum gates via channel spectrum", Nature Communications 14 1, 5880 (2023).

[24] Pedro Figueroa-Romero, Kavan Modi, Robert J. Harris, Thomas M. Stace, and Min-Hsiu Hsieh, "Randomized Benchmarking for Non-Markovian Noise", PRX Quantum 2 4, 040351 (2021).

[25] Winton G. Brown and Bryan Eastin, "Randomized benchmarking with restricted gate sets", Physical Review A 97 6, 062323 (2018).

[26] J. Helsen, I. Roth, E. Onorati, A.H. Werner, and J. Eisert, "General Framework for Randomized Benchmarking", PRX Quantum 3 2, 020357 (2022).

[27] Jahan Claes and Shruti Puri, "Estimating the Bias of CX Gates via Character Randomized Benchmarking", PRX Quantum 4 1, 010307 (2023).

[28] David C. McKay, Sarah Sheldon, John A. Smolin, Jerry M. Chow, and Jay M. Gambetta, "Three-Qubit Randomized Benchmarking", Physical Review Letters 122 20, 200502 (2019).

[29] Mahnaz Jafarzadeh, Ya-Dong Wu, Yuval R Sanders, and Barry C Sanders, "Randomized benchmarking for qudit Clifford gates", New Journal of Physics 22 6, 063014 (2020).

[30] 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 1, 010351 (2021).

[31] Olivia Di Matteo, John Gamble, Chris Granade, Kenneth Rudinger, and Nathan Wiebe, "Operational, gauge-free quantum tomography", Quantum 4, 364 (2020).

[32] C. H. Yang, K. W. Chan, R. Harper, W. Huang, T. Evans, J. C. C. Hwang, B. Hensen, A. Laucht, T. Tanttu, F. E. Hudson, S. T. Flammia, K. M. Itoh, A. Morello, S. D. Bartlett, and A. S. Dzurak, "Silicon qubit fidelities approaching incoherent noise limits via pulse engineering", Nature Electronics 2 4, 151 (2019).

[33] Martin Kliesch and Ingo Roth, "Theory of Quantum System Certification", PRX Quantum 2 1, 010201 (2021).

[34] A. K. Hashagen, S. T. Flammia, D. Gross, and J. J. Wallman, "Real Randomized Benchmarking", Quantum 2, 85 (2018).

[35] 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 1, 5347 (2019).

[36] Timothy Proctor, Stefan Seritan, Kenneth Rudinger, Erik Nielsen, Robin Blume-Kohout, and Kevin Young, "Scalable Randomized Benchmarking of Quantum Computers Using Mirror Circuits", Physical Review Letters 129 15, 150502 (2022).

[37] Jonas Helsen, Xiao Xue, Lieven M. K. Vandersypen, and Stephanie Wehner, "A new class of efficient randomized benchmarking protocols", npj Quantum Information 5 1, 71 (2019).

[38] Hillary Dawkins, Joel Wallman, and Joseph Emerson, "Combining T1 and T2 estimation with randomized benchmarking and bounding the diamond distance", Physical Review A 102 2, 022220 (2020).

[39] M. Kjaergaard, M. E. Schwartz, A. Greene, G. O. Samach, A. Bengtsson, M. O’Keeffe, C. M. McNally, J. Braumüller, D. K. Kim, P. Krantz, M. Marvian, A. Melville, B. M. Niedzielski, Y. Sung, R. Winik, J. Yoder, D. Rosenberg, K. Obenland, S. Lloyd, T. P. Orlando, I. Marvian, S. Gustavsson, and W. D. Oliver, "Demonstration of Density Matrix Exponentiation Using a Superconducting Quantum Processor", Physical Review X 12 1, 011005 (2022).

[40] David Amaro-Alcalá, Barry C. Sanders, and Hubert de Guise, "Benchmarking of universal qutrit gates", Physical Review A 109 1, 012621 (2024).

[41] X. Xue, T. F. Watson, J. Helsen, D. R. Ward, D. E. Savage, M. G. Lagally, S. N. Coppersmith, M. A. Eriksson, S. Wehner, and L. M. K. Vandersypen, "Benchmarking Gate Fidelities in a Si/SiGe Two-Qubit Device", Physical Review X 9 2, 021011 (2019).

[42] Changjun Kim, Kyungdeock Daniel Park, and June-Koo Rhee, "Quantum Error Mitigation With Artificial Neural Network", IEEE Access 8, 188853 (2020).

[43] Jiaan Qi and Hui Khoon Ng, "Randomized benchmarking in the presence of time-correlated dephasing noise", Physical Review A 103 2, 022607 (2021).

[44] Long Ma and Jaron Sanders, Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering 404, 36 (2021) ISBN:978-3-030-92510-9.

[45] Quentin Ficheux, Long B. Nguyen, Aaron Somoroff, Haonan Xiong, Konstantin N. Nesterov, Maxim G. Vavilov, and Vladimir E. Manucharyan, "Fast Logic with Slow Qubits: Microwave-Activated Controlled-Z Gate on Low-Frequency Fluxoniums", Physical Review X 11 2, 021026 (2021).

[46] Lorenzo Leone, Salvatore F. E. Oliviero, and Alioscia Hamma, "Nonstabilizerness determining the hardness of direct fidelity estimation", Physical Review A 107 2, 022429 (2023).

[47] Erik Nielsen, John King Gamble, Kenneth Rudinger, Travis Scholten, Kevin Young, and Robin Blume-Kohout, "Gate Set Tomography", Quantum 5, 557 (2021).

[48] Conrad Strydom and Mark Tame, "Investigating the effect of noise channels on the quality of unitary t -designs", Physical Review A 108 5, 052414 (2023).

[49] Jonas Helsen, Joel J. Wallman, Steven T. Flammia, and Stephanie Wehner, "Multiqubit randomized benchmarking using few samples", Physical Review A 100 3, 032304 (2019).

[50] Timothy J. Proctor, Arnaud Carignan-Dugas, Kenneth Rudinger, Erik Nielsen, Robin Blume-Kohout, and Kevin Young, "Direct Randomized Benchmarking for Multiqubit Devices", Physical Review Letters 123 3, 030503 (2019).

[51] Marco Cattaneo, Matteo A.C. Rossi, Guillermo García-Pérez, Roberta Zambrini, and Sabrina Maniscalco, "Quantum Simulation of Dissipative Collective Effects on Noisy Quantum Computers", PRX Quantum 4 1, 010324 (2023).

[52] Arnaud Carignan-Dugas, Kristine Boone, Joel J Wallman, and Joseph Emerson, "From randomized benchmarking experiments to gate-set circuit fidelity: how to interpret randomized benchmarking decay parameters", New Journal of Physics 20 9, 092001 (2018).

[53] D S França and A K Hashagen, "Approximate randomized benchmarking for finite groups", Journal of Physics A: Mathematical and Theoretical 51 39, 395302 (2018).

[54] Bas Dirkse, Jonas Helsen, and Stephanie Wehner, "Efficient unitarity randomized benchmarking of few-qubit Clifford gates", Physical Review A 99 1, 012315 (2019).

[55] Pedro Figueroa-Romero, Kavan Modi, and Min-Hsiu Hsieh, "Towards a general framework of Randomized Benchmarking incorporating non-Markovian Noise", Quantum 6, 868 (2022).

[56] E Derbyshire, J Yago Malo, A J Daley, E Kashefi, and P Wallden, "Randomized benchmarking in the analogue setting", Quantum Science and Technology 5 3, 034001 (2020).

[57] Ruyu Yang and Ying Li, "Perturbative tomography of small errors in quantum gates", Physical Review A 103 3, 032421 (2021).

[58] Wenjun Yu, Jinzhao Sun, Zeyao Han, and Xiao Yuan, "Robust and Efficient Hamiltonian Learning", Quantum 7, 1045 (2023).

[59] Yihong Zhang, Wenjun Yu, Pei Zeng, Guoding Liu, and Xiongfeng Ma, "Scalable fast benchmarking for individual quantum gates with local twirling", Photonics Research 11 1, 81 (2023).

[60] Jianxin Chen, Dawei Ding, Cupjin Huang, and Linghang Kong, "Linear cross-entropy benchmarking with Clifford circuits", Physical Review A 108 5, 052613 (2023).

[61] Christopher J. Wood and Jay M. Gambetta, "Quantification and characterization of leakage errors", Physical Review A 97 3, 032306 (2018).

[62] Matthew Ware, Guilhem Ribeill, Diego Ristè, Colm A. Ryan, Blake Johnson, and Marcus P. da Silva, "Experimental Pauli-frame randomization on a superconducting qubit", Physical Review A 103 4, 042604 (2021).

[63] Joel J. Wallman and Joseph Emerson, Quantum Information and Measurement (QIM) V: Quantum Technologies S3B.2 (2019) ISBN:978-1-943580-56-9.

[64] Samuele Ferracin, Theodoros Kapourniotis, and Animesh Datta, "Accrediting outputs of noisy intermediate-scale quantum computing devices", New Journal of Physics 21 11, 113038 (2019).

[65] Conrad Strydom and Mark Tame, "Measurement-based interleaved randomised benchmarking using IBM processors", Physica Scripta 98 2, 025106 (2023).

[66] Seth T. Merkel, Emily J. Pritchett, and Bryan H. Fong, "Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors", Quantum 5, 581 (2021).

[67] Jianxin Chen, Dawei Ding, and Cupjin Huang, "Randomized Benchmarking beyond Groups", PRX Quantum 3 3, 030320 (2022).

[68] Timothy Proctor, Kenneth Rudinger, Kevin Young, Mohan Sarovar, and Robin Blume-Kohout, "What Randomized Benchmarking Actually Measures", Physical Review Letters 119 13, 130502 (2017).

[69] S. Mavadia, C. L. Edmunds, C. Hempel, H. Ball, F. Roy, T. M. Stace, and M. J. Biercuk, "Experimental quantum verification in the presence of temporally correlated noise", npj Quantum Information 4, 7 (2018).

[70] D. Willsch, M. Nocon, F. Jin, H. De Raedt, and K. Michielsen, "Gate-error analysis in simulations of quantum computers with transmon qubits", Physical Review A 96 6, 062302 (2017).

[71] Steven T. Flammia and Joel J. Wallman, "Efficient estimation of Pauli channels", arXiv:1907.12976, (2019).

[72] Markus Heinrich, Martin Kliesch, and Ingo Roth, "Randomized benchmarking with random quantum circuits", arXiv:2212.06181, (2022).

[73] Martin Kliesch and Ingo Roth, "Theory of quantum system certification: a tutorial", arXiv:2010.05925, (2020).

[74] Arnaud Carignan-Dugas, Dar Dahlen, Ian Hincks, Egor Ospadov, Stefanie J. Beale, Samuele Ferracin, Joshua Skanes-Norman, Joseph Emerson, and Joel J. Wallman, "The Error Reconstruction and Compiled Calibration of Quantum Computing Cycles", arXiv:2303.17714, (2023).

[75] Robin Harper, Ian Hincks, Chris Ferrie, Steven T. Flammia, and Joel J. Wallman, "Statistical analysis of randomized benchmarking", arXiv:1901.00535, (2019).

[76] David C. McKay, Sarah Sheldon, John A. Smolin, Jerry M. Chow, and Jay M. Gambetta, "Three Qubit Randomized Benchmarking", arXiv:1712.06550, (2017).

[77] C. H. Yang, K. W. Chan, R. Harper, W. Huang, T. Evans, J. C. C. Hwang, B. Hensen, A. Laucht, T. Tanttu, F. E. Hudson, S. T. Flammia, K. M. Itoh, A. Morello, S. D. Bartlett, and A. S. Dzurak, "Silicon qubit fidelities approaching incoherent noise limits via pulse engineering", arXiv:1807.09500, (2018).

[78] Kevin Slagle, "Testing Quantum Mechanics using Noisy Quantum Computers", arXiv:2108.02201, (2021).

[79] Ian Hincks, Joel J. Wallman, Chris Ferrie, Chris Granade, and David G. Cory, "Bayesian Inference for Randomized Benchmarking Protocols", arXiv:1802.00401, (2018).

[80] C. L. Edmunds, C. Hempel, R. Harris, H. Ball, V. Frey, T. M. Stace, and M. J. Biercuk, "Measuring and Suppressing Error Correlations in Quantum Circuits", arXiv:1712.04954, (2017).

[81] Stefanie J. Beale and Joel J. Wallman, "Randomized compiling for subsystem measurements", arXiv:2304.06599, (2023).

[82] Emilio Onorati, Tamara Kohler, and Toby S. Cubitt, "Fitting time-dependent Markovian dynamics to noisy quantum channels", arXiv:2303.08936, (2023).

[83] John Gamble, Chris Granade, and Nathan Wiebe, "Bayesian ACRONYM Tuning", arXiv:1902.05940, (2019).

The above citations are from Crossref's cited-by service (last updated successfully 2024-03-28 08:14:47) and SAO/NASA ADS (last updated successfully 2024-03-28 08:14:49). The list may be incomplete as not all publishers provide suitable and complete citation data.