Gate Set Tomography

Erik Nielsen1, John King Gamble2, Kenneth Rudinger1, Travis Scholten3, Kevin Young1, and Robin Blume-Kohout1

1Quantum Performance Laboratory, Sandia National Laboratories
2Microsoft Research
3IBM Quantum, IBM T.J. Watson Research Center

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


Gate set tomography (GST) is a protocol for detailed, predictive characterization of logic operations (gates) on quantum computing processors. Early versions of GST emerged around 2012-13, and since then it has been refined, demonstrated, and used in a large number of experiments. This paper presents the foundations of GST in comprehensive detail. The most important feature of GST, compared to older state and process tomography protocols, is that it is $\textit{calibration-free}$. GST does not rely on pre-calibrated state preparations and measurements. Instead, it characterizes all the operations in a $\textit{gate set}$ simultaneously and self-consistently, relative to each other. Long sequence GST can estimate gates with very high precision and efficiency, achieving Heisenberg scaling in regimes of practical interest. In this paper, we cover GST's intellectual history, the techniques and experiments used to achieve its intended purpose, data analysis, gauge freedom and fixing, error bars, and the interpretation of gauge-fixed estimates of gate sets. Our focus is fundamental mathematical aspects of GST, rather than implementation details, but we touch on some of the foundational algorithmic tricks used in the $\texttt{pyGSTi}$ implementation.

► BibTeX data

► References

[1] Scott Aaronson. Shadow tomography of quantum states. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 325–338., 2018.

[2] Panos Aliferis and Andrew W. Cross. Subsystem fault tolerance with the bacon-shor code. Phys. Rev. Lett., 98: 220502, May 2007. 10.1103/​PhysRevLett.98.220502.

[3] Panos Aliferis and John Preskill. Fibonacci scheme for fault-tolerant quantum computation. Phys. Rev. A, 79: 012332, Jan 2009. 10.1103/​PhysRevA.79.012332.

[4] Panos Aliferis, Daniel Gottesman, and John Preskill. Quantum accuracy threshold for concatenated distance-3 codes. Quantum Info. Comput., 6 (2): 97–165, March 2006. ISSN 1533-7146.

[5] J B Altepeter, D Branning, E Jeffrey, T C Wei, P G Kwiat, R T Thew, J L O'Brien, M A Nielsen, and A G White. Ancilla-assisted quantum process tomography. Phys. Rev. Lett., 90 (19): 193601, May 2003. ISSN 0031-9007. 10.1103/​PhysRevLett.90.193601.

[6] L M Artiles, R D Gill, and M I Guta. An invitation to quantum tomography. J. R. Stat. Soc. Series B Stat. Methodol., 67 (1): 109–134, February 2005. ISSN 1369-7412, 1467-9868. 10.1111/​j.1467-9868.2005.00491.x.

[7] 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. ISSN 0028-0836, 1476-4687. 10.1038/​s41586-019-1666-5.

[8] Konrad Banaszek, Marcus Cramer, and David Gross. Focus on quantum tomography. New J. Phys., 15 (12): 125020, 2013. ISSN 1367-2630. 10.1088/​1367-2630/​15/​12/​125020. URL https:/​/​​10.1088/​1367-2630/​15/​12/​125020.

[9] R Barends, J Kelly, A Megrant, A Veitia, D Sank, E Jeffrey, T C White, J Mutus, A G Fowler, B Campbell, Y Chen, Z Chen, B Chiaro, A Dunsworth, C Neill, P 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 (7497): 500–503, April 2014. ISSN 0028-0836, 1476-4687. 10.1038/​nature13171.

[10] Ariel Bendersky, Fernando Pastawski, and Juan Pablo Paz. Selective and efficient estimation of parameters for quantum process tomography. Phys. Rev. Lett., 100 (19): 190403, May 2008. ISSN 0031-9007. 10.1103/​PhysRevLett.100.190403.

[11] Ingemar Bengtsson and Karol Zyczkowski. On discrete structures in finite hilbert spaces, 2017.

[12] R C Bialczak, M Ansmann, M Hofheinz, E Lucero, M Neeley, A D O'Connell, D Sank, H Wang, J Wenner, M Steffen, A N Cleland, and J M Martinis. Quantum process tomography of a universal entangling gate implemented with josephson phase qubits. Nat. Phys., 6 (6): 409–413, June 2010. ISSN 1745-2473, 1745-2481. 10.1038/​nphys1639.

[13] Lev S Bishop, Sergey Bravyi, Andrew Cross, Jay M Gambetta, and John Smolin. Quantum volume. Quantum Volume. Technical Report, 2017.

[14] R. Blume-Kohout, J.K. Gamble, E. Nielsen, J. Mizrahi, J.D. Sterk, and P. Maunz. Robust, self-consistent, closed-form tomography of quantum logic gates on a trapped ion qubit. arXiv preprint arXiv:1310.4492, 2013.

[15] Robin Blume-Kohout. Hedged maximum likelihood quantum state estimation. Phys. Rev. Lett., 105 (20): 200504, November 2010a. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.105.200504.

[16] Robin Blume-Kohout. Optimal, reliable estimation of quantum states. New J. Phys., 12 (4): 043034, April 2010b. ISSN 1367-2630. 10.1088/​1367-2630/​12/​4/​043034.

[17] Robin Blume-Kohout. Robust error bars for quantum tomography. arXiv preprint arXiv:1202.5270, February 2012.

[18] Robin Blume-Kohout and Kevin C. Young. A volumetric framework for quantum computer benchmarks. Quantum, 4: 362, November 2020. ISSN 2521-327X. 10.22331/​q-2020-11-15-362. URL https:/​/​​10.22331/​q-2020-11-15-362.

[19] 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. Nat. Commun., 8, February 2017. ISSN 2041-1723. 10.1038/​ncomms14485.

[20] J Z Blumoff, K Chou, C Shen, M Reagor, C Axline, R T Brierley, M P Silveri, C Wang, B Vlastakis, S E Nigg, and Others. Implementing and characterizing precise multiqubit measurements. Physical Review X, 6 (3): 031041, 2016. 10.1103/​PhysRevX.6.031041. URL https:/​/​​doi/​10.1103/​PhysRevX.6.031041.

[21] Sergio Boixo, Sergei V Isakov, Vadim N Smelyanskiy, Ryan Babbush, Nan Ding, Zhang Jiang, Michael J Bremner, John M Martinis, and Hartmut Neven. Characterizing quantum supremacy in near-term devices. Nat. Phys., 14 (6): 595–600, April 2018. ISSN 1745-2473. 10.1038/​s41567-018-0124-x.

[22] S. Boyd and L. Vandenberghe. Convex optimization. Cambridge university press, 2004.

[23] Agata M Brańczyk, Dylan H Mahler, Lee A Rozema, Ardavan Darabi, Aephraim M Steinberg, and Daniel FV James. Self-calibrating quantum state tomography. New Journal of Physics, 14 (8): 085003, 2012. 10.1088/​1367-2630/​14/​8/​085003. URL https:/​/​​10.1088/​1367-2630/​14/​8/​085003.

[24] A Carignan-Dugas, J J Wallman, and J Emerson. Characterizing universal gate sets via dihedral benchmarking. Phys. Rev. A, 2015. ISSN 1050-2947. 10.1103/​PhysRevA.92.060302. URL https:/​/​​doi/​10.1103/​PhysRevA.92.060302.

[25] Pascal Cerfontaine, René Otten, and Hendrik Bluhm. Self-consistent calibration of quantum-gate sets. Phys. Rev. Applied, 13: 044071, Apr 2020. 10.1103/​PhysRevApplied.13.044071. URL https:/​/​​doi/​10.1103/​PhysRevApplied.13.044071.

[26] Robert J. Chapman, Christopher Ferrie, and Alberto Peruzzo. Experimental demonstration of self-guided quantum tomography. Phys. Rev. Lett., 117: 040402, Jul 2016. 10.1103/​PhysRevLett.117.040402. URL https:/​/​​doi/​10.1103/​PhysRevLett.117.040402.

[27] T Chasseur and F K Wilhelm. Complete randomized benchmarking protocol accounting for leakage errors. Phys. Rev. A, 92 (4): 042333, October 2015. ISSN 1050-2947. 10.1103/​PhysRevA.92.042333.

[28] E A Chekhovich, M N Makhonin, A I Tartakovskii, A Yacoby, H Bluhm, K C Nowack, and L M K Vandersypen. Nuclear spin effects in semiconductor quantum dots. Nat. Mater., 12 (6): 494–504, June 2013. 10.1038/​nmat3652. URL https:/​/​​10.1038/​nmat3652.

[29] Yanzhu Chen, Maziar Farahzad, Shinjae Yoo, and Tzu-Chieh Wei. Detector tomography on IBM quantum computers and mitigation of an imperfect measurement. Phys. Rev. A, 100 (5): 052315, 2019. ISSN 1050-2947. 10.1103/​PhysRevA.100.052315. URL https:/​/​​doi/​10.1103/​PhysRevA.100.052315.

[30] Andrew M Childs, Isaac L Chuang, and Debbie W Leung. Realization of quantum process tomography in NMR. Phys. Rev. A, 64 (1): 012314, June 2001. ISSN 1050-2947. 10.1103/​PhysRevA.64.012314.

[31] Man-Duen Choi. Completely positive linear maps on complex matrices. Linear Algebra Appl., 10 (3): 285–290, 1975. ISSN 0024-3795. https:/​/​​10.1016/​0024-3795(75)90075-0. URL https:/​/​​science/​article/​pii/​0024379575900750.

[32] Matthias Christandl and Renato Renner. Reliable quantum state tomography. Phys. Rev. Lett., 109: 120403, Sep 2012. 10.1103/​PhysRevLett.109.120403. URL https:/​/​​doi/​10.1103/​PhysRevLett.109.120403.

[33] Isaac L Chuang and M A Nielsen. Prescription for experimental determination of the dynamics of a quantum black box. J. Mod. Opt., 44 (11-12): 2455–2467, November 1997. ISSN 0950-0340. 10.1080/​09500349708231894.

[34] Lukasz Cincio, Kenneth Rudinger, Mohan Sarovar, and Patrick J. Coles. Machine learning of noise-resilient quantum circuits. PRX Quantum, 2: 010324, Feb 2021. 10.1103/​PRXQuantum.2.010324. URL https:/​/​​doi/​10.1103/​PRXQuantum.2.010324.

[35] Jared H Cole. Hamiltonian tomography: the quantum (system) measurement problem. New J. Phys., 17 (10): 101001, September 2015. ISSN 1367-2630. 10.1088/​1367-2630/​17/​10/​101001.

[36] Jared H Cole, Sonia G Schirmer, Andrew D Greentree, Cameron J Wellard, Daniel K L Oi, and Lloyd C L Hollenberg. Identifying an experimental two-state hamiltonian to arbitrary accuracy. Phys. Rev. A, 71 (6): 062312, June 2005. ISSN 1050-2947. 10.1103/​PhysRevA.71.062312.

[37] A D Córcoles, Jay M Gambetta, Jerry M Chow, John A Smolin, Matthew Ware, Joel Strand, B L T Plourde, and M Steffen. Process verification of two-qubit quantum gates by randomized benchmarking. Phys. Rev. A, 87 (3): 030301, March 2013. ISSN 1050-2947. 10.1103/​PhysRevA.87.030301.

[38] Andrew W Cross, Lev S Bishop, Sarah Sheldon, Paul D Nation, and Jay M Gambetta. Validating quantum computers using randomized model circuits. Phys. Rev. A, 100 (3): 032328, September 2019. ISSN 1050-2947. 10.1103/​PhysRevA.100.032328.

[39] Marcus P da Silva, Olivier Landon-Cardinal, and David Poulin. Practical characterization of quantum devices without tomography. Phys. Rev. Lett., 107 (21): 210404, November 2011. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.107.210404.

[40] G M D'Ariano and P Lo Presti. Quantum tomography for measuring experimentally the matrix elements of an arbitrary quantum operation. Phys. Rev. Lett., 86 (19): 4195–4198, May 2001. ISSN 0031-9007. 10.1103/​PhysRevLett.86.4195.

[41] S Debnath, N M Linke, C Figgatt, K A Landsman, K Wright, and C Monroe. Demonstration of a small programmable quantum computer with atomic qubits. Nature, 536 (7614): 63–66, August 2016. 10.1038/​nature18648. URL https:/​/​​10.1038/​nature18648.

[42] Juan P Dehollain, Juha T Muhonen, Robin Blume-Kohout, Kenneth M Rudinger, John King Gamble, Erik Nielsen, Arne Laucht, Stephanie Simmons, Rachpon Kalra, Andrew S Dzurak, and Andrea Morello. Optimization of a solid-state electron spin qubit using gate set tomography. New Journal of Physics, 18 (10): 103018, 2016. 10.1088/​1367-2630/​18/​10/​103018. URL https:/​/​​10.1088/​1367-2630/​18/​10/​103018.

[43] C Di Franco, M Paternostro, and M S Kim. Hamiltonian tomography in an access-limited setting without state initialization. Phys. Rev. Lett., 102 (18): 187203, May 2009. ISSN 0031-9007. 10.1103/​PhysRevLett.102.187203.

[44] Olivia Di Matteo, John Gamble, Chris Granade, Kenneth Rudinger, and Nathan Wiebe. Operational, gauge-free quantum tomography, November 2020. ISSN 2521-327X. URL https:/​/​​10.22331/​q-2020-11-17-364.

[45] David P DiVincenzo. The physical implementation of quantum computation. Fortschritte der Physik: Progress of Physics, 48 (9-11): 771–783, 2000. URL https:/​/​​10.1002/​1521-3978(200009)48:9/​11.

[46] BRADLEY EFRON. Nonparametric estimates of standard error: The jackknife, the bootstrap and other methods. Biometrika, 68 (3): 589–599, 12 1981. ISSN 0006-3444. 10.1093/​biomet/​68.3.589. URL https:/​/​​10.1093/​biomet/​68.3.589.

[47] Artur K Ekert, Carolina Moura Alves, Daniel K L Oi, Michał Horodecki, Paweł Horodecki, and L C Kwek. Direct estimations of linear and nonlinear functionals of a quantum state. Phys. Rev. Lett., 88 (21): 217901, May 2002. ISSN 0031-9007. 10.1103/​PhysRevLett.88.217901.

[48] Joseph Emerson, Marcus Silva, Osama Moussa, Colm Ryan, Martin Laforest, Jonathan Baugh, David G. Cory, and Raymond Laflamme. Symmetrized characterization of noisy quantum processes. Science, 317 (5846): 1893–1896, 2007. ISSN 0036-8075. 10.1126/​science.1145699. URL https:/​/​​content/​317/​5846/​1893.

[49] Suguru Endo, Simon C Benjamin, and Ying Li. Practical quantum error mitigation for Near-Future applications. Phys. Rev. X, 8 (3): 031027, July 2018. 10.1103/​PhysRevX.8.031027.

[50] 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. Nat. Commun., 10 (1): 5347, November 2019. ISSN 2041-1723. 10.1038/​s41467-019-13068-7.

[51] Philippe Faist and Renato Renner. Practical and reliable error bars in quantum tomography. Phys. Rev. Lett., 117 (1): 010404, July 2016. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.117.010404.

[52] Christopher Ferrie. High posterior density ellipsoids of quantum states. New Journal of Physics, 16 (2): 023006, feb 2014. 10.1088/​1367-2630/​16/​2/​023006.

[53] Christopher Ferrie, Christopher E Granade, and D G Cory. How to best sample a periodic probability distribution, or on the accuracy of hamiltonian finding strategies. Quantum Inf. Process., 12 (1): 611–623, January 2013. ISSN 1573-1332. 10.1007/​s11128-012-0407-6.

[54] Jaromír Fiurášek. Maximum-likelihood estimation of quantum measurement. Phys. Rev. A, 64: 024102, Jul 2001. 10.1103/​PhysRevA.64.024102. URL https:/​/​​doi/​10.1103/​PhysRevA.64.024102.

[55] Jaromír Fiurášek and Zdeněk Hradil. Maximum-likelihood estimation of quantum processes. Phys. Rev. A, 63: 020101, Jan 2001. 10.1103/​PhysRevA.63.020101. URL https:/​/​​doi/​10.1103/​PhysRevA.63.020101.

[56] Steven T Flammia and Yi-Kai Liu. Direct fidelity estimation from few pauli measurements. Phys. Rev. Lett., 106 (23): 230501, June 2011. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.106.230501.

[57] 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 (9): 095022, sep 2012. ISSN 1367-2630. 10.1088/​1367-2630/​14/​9/​095022. URL https:/​/​​10.1088/​1367-2630/​14/​9/​095022.

[58] J P Gaebler, A M Meier, T R Tan, R Bowler, Y Lin, D Hanneke, J D Jost, J P Home, E Knill, D Leibfried, and D J Wineland. Randomized benchmarking of multiqubit gates. Phys. Rev. Lett., 108 (26): 260503, June 2012. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.108.260503.

[59] W Gale, E Guth, and G T Trammell. Determination of the quantum state by measurements. Phys. Rev., 165 (5): 1434–1436, January 1968. ISSN 0959-8472. 10.1103/​PhysRev.165.1434.

[60] Jay M Gambetta, A D Córcoles, S T Merkel, B R Johnson, John A Smolin, Jerry M Chow, Colm A Ryan, Chad Rigetti, S Poletto, Thomas A Ohki, Mark B Ketchen, and M Steffen. Characterization of addressability by simultaneous randomized benchmarking. Phys. Rev. Lett., 109 (24): 240504, December 2012. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.109.240504.

[61] Christopher Granade, Joshua Combes, and D G Cory. Practical bayesian tomography. New Journal of Physics, 18 (3): 033024, mar 2016. ISSN 1367-2630. 10.1088/​1367-2630/​18/​3/​033024. URL https:/​/​​10.1088/​1367-2630/​18/​3/​033024.

[62] Christopher Granade, Christopher Ferrie, Ian Hincks, Steven Casagrande, Thomas Alexander, Jonathan Gross, Michal Kononenko, and Yuval Sanders. QInfer: Statistical inference software for quantum applications. Quantum, 1: 5, April 2017. ISSN 2521-327X. 10.22331/​q-2017-04-25-5. URL https:/​/​​10.22331/​q-2017-04-25-5.

[63] Christopher E Granade, Christopher Ferrie, Nathan Wiebe, and D G Cory. Robust online hamiltonian learning. New J. Phys., 14 (10): 103013, October 2012. ISSN 1367-2630. 10.1088/​1367-2630/​14/​10/​103013.

[64] Daniel Greenbaum. Introduction to quantum gate set tomography. arXiv:1509.02921, 2015.

[65] David Gross, Yi-Kai Liu, Steven T Flammia, Stephen Becker, and Jens Eisert. Quantum state tomography via compressed sensing. Phys. Rev. Lett., 105 (15): 150401, October 2010. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.105.150401.

[66] J Haah, A W Harrow, Z Ji, X Wu, and N Yu. Sample-Optimal tomography of quantum states. IEEE Trans. Inf. Theory, 63 (9): 5628–5641, September 2017. ISSN 0018-9448, 1557-9654. 10.1109/​TIT.2017.2719044.

[67] Kathleen E Hamilton, Eugene F Dumitrescu, and Raphael C Pooser. Generative model benchmarks for superconducting qubits. Phys. Rev. A, 99 (6): 062323, June 2019. ISSN 1050-2947. 10.1103/​PhysRevA.99.062323.

[68] Robin Harper and Steven T Flammia. Estimating the fidelity of T gates using standard interleaved randomized benchmarking. Quantum Sci. Technol., 2 (1): 015008, March 2017. ISSN 2058-9565. 10.1088/​2058-9565/​aa5f8d.

[69] Teiko Heinosaari, Luca Mazzarella, and Michael M Wolf. Quantum tomography under prior information. Commun. Math. Phys., 318 (2): 355–374, March 2013. ISSN 1432-0916. 10.1007/​s00220-013-1671-8.

[70] Jonas Helsen, Francesco Battistel, and Barbara M Terhal. Spectral quantum tomography. npj Quantum Information, 5 (1): 74, September 2019. ISSN 2056-6387. 10.1038/​s41534-019-0189-0.

[71] Cornelius Hempel, Christine Maier, Jonathan Romero, Jarrod McClean, Thomas Monz, Heng Shen, Petar Jurcevic, Ben P Lanyon, Peter Love, Ryan Babbush, Alán Aspuru-Guzik, Rainer Blatt, and Christian F Roos. Quantum chemistry calculations on a Trapped-Ion quantum simulator. Phys. Rev. X, 8 (3): 031022, July 2018. 10.1103/​PhysRevX.8.031022.

[72] Sabrina S Hong, Alexander T Papageorge, Prasahnt Sivarajah, Genya Crossman, Nicolas Didier, Anthony M Polloreno, Eyob A Sete, Stefan W Turkowski, Marcus P da Silva, and Blake R Johnson. Demonstration of a parametrically activated entangling gate protected from flux noise. Phys. Rev. A, 101 (1): 012302, January 2020. ISSN 1050-2947. 10.1103/​PhysRevA.101.012302.

[73] P-Y Hou, L He, F Wang, X-Z Huang, W-G Zhang, X-L Ouyang, X Wang, W-Q Lian, X-Y Chang, and L-M Duan. Experimental hamiltonian learning of an 11-qubit Solid-State quantum spin register. Chin. Physics Lett., 36 (10): 100303, October 2019. ISSN 0256-307X. 10.1088/​0256-307X/​36/​10/​100303.

[74] Zhibo Hou, Huangjun Zhu, Guo-Yong Xiang, Chuan-Feng Li, and Guang-Can Guo. Error-compensation measurements on polarization qubits. J. Opt. Soc. Am. B, 33 (6): 1256–1265, Jun 2016. 10.1364/​JOSAB.33.001256. URL http:/​/​​abstract.cfm?URI=josab-33-6-1256.

[75] Z. Hradil. Quantum-state estimation. Phys. Rev. A, 55: R1561–R1564, Mar 1997. 10.1103/​PhysRevA.55.R1561. URL https:/​/​​doi/​10.1103/​PhysRevA.55.R1561.

[76] Hsin-Yuan Huang, Richard Kueng, and John Preskill. Predicting many properties of a quantum system from very few measurements. Nature Physics, 16 (10): 1050–1057, Oct 2020. ISSN 1745-2481. 10.1038/​s41567-020-0932-7. URL https:/​/​​10.1038/​s41567-020-0932-7.

[77] W Huang, C H Yang, K W Chan, T Tanttu, B Hensen, R C C Leon, M A Fogarty, J C C Hwang, F E Hudson, K M Itoh, A Morello, A Laucht, and A S Dzurak. Fidelity benchmarks for two-qubit gates in silicon. Nature, 569 (7757): 532–536, May 2019. ISSN 0028-0836, 1476-4687. 10.1038/​s41586-019-1197-0.

[78] Christopher Jackson and S. J. van Enk. Detecting correlated errors in state-preparation-and-measurement tomography. Phys. Rev. A, 92: 042312, Oct 2015. 10.1103/​PhysRevA.92.042312. URL https:/​/​​doi/​10.1103/​PhysRevA.92.042312.

[79] Daniel F V James, Paul G Kwiat, William J Munro, and Andrew G White. Measurement of qubits. Phys. Rev. A, 64 (5): 052312, October 2001. ISSN 1050-2947. 10.1103/​PhysRevA.64.052312.

[80] A. Jamiołkowski. Linear transformations which preserve trace and positive semidefiniteness of operators, 1972. ISSN 0034-4877. URL https:/​/​​10.1016/​0034-4877(72)90011-0.

[81] Jun Jing and Lian-Ao Wu. Decoherence and control of a qubit in spin baths: an exact master equation study. Scientific Reports, 8 (1): 1471, January 2018. 10.1038/​s41598-018-19977-9. URL https:/​/​​10.1038/​s41598-018-19977-9.

[82] Manoj K. Joshi, Andreas Elben, Benoı̂t Vermersch, Tiff Brydges, Christine Maier, Peter Zoller, Rainer Blatt, and Christian F. Roos. Quantum information scrambling in a trapped-ion quantum simulator with tunable range interactions. Phys. Rev. Lett., 124: 240505, Jun 2020. 10.1103/​PhysRevLett.124.240505. URL https:/​/​​doi/​10.1103/​PhysRevLett.124.240505.

[83] Adam C Keith, Charles H Baldwin, Scott Glancy, and E Knill. Joint quantum-state and measurement tomography with incomplete measurements. Physical Review A, 98 (4): 042318, 2018. 10.1103/​PhysRevA.98.042318. URL https:/​/​​doi/​10.1103/​PhysRevA.98.042318.

[84] Dohun Kim, Zhan Shi, C B Simmons, D R Ward, J R Prance, Teck Seng Koh, John King Gamble, D E Savage, M G Lagally, Mark Friesen, S N Coppersmith, and Mark A Eriksson. Quantum control and process tomography of a semiconductor quantum dot hybrid qubit. Nature, 511 (7507): 70–74, July 2014. ISSN 0028-0836, 1476-4687. 10.1038/​nature13407.

[85] 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. Nat. Nanotechnol., 10 (3): 243–247, 03 2015. 10.1038/​nnano.2014.336. URL https:/​/​​10.1038/​nnano.2014.336.

[86] 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, Mar 2014. 10.1103/​PhysRevX.4.011050. URL https:/​/​​doi/​10.1103/​PhysRevX.4.011050.

[87] Shelby Kimmel, Guang Hao Low, and Theodore J Yoder. Robust calibration of a universal single-qubit gate set via robust phase estimation. Phys. Rev. A, 92 (6): 062315, December 2015. ISSN 1050-2947. 10.1103/​PhysRevA.92.062315.

[88] E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland. Randomized benchmarking of quantum gates. Phys. Rev. A, 77: 012307, Jan 2008. 10.1103/​PhysRevA.77.012307.

[89] Emanuel Knill. Quantum computing with realistically noisy devices. Nature, 434 (7029): 39–44, 2005. 10.1038/​nature03350. URL https:/​/​​10.1038/​nature03350.

[90] Stefan Krastanov, Sisi Zhou, Steven T Flammia, and Liang Jiang. Stochastic estimation of dynamical variables. Quantum Sci. Technol., 4 (3): 035003, May 2019. ISSN 2058-9565. 10.1088/​2058-9565/​ab18d5.

[91] K Kraus. States, effects, and operations: fundamental notions of quantum theory, volume 190 of Lecture Notes in Physics. Springer-Verlag, 1983.

[92] Benjamin Lévi, Cecilia C López, Joseph Emerson, and D G Cory. Efficient error characterization in quantum information processing. Phys. Rev. A, 75 (2): 022314, February 2007. ISSN 1050-2947. 10.1103/​PhysRevA.75.022314.

[93] Junan Lin, Brandon Buonacorsi, Raymond Laflamme, and Joel J Wallman. On the freedom in representing quantum operations. New J. Phys., 21 (2): 023006, February 2019a. ISSN 1367-2630. 10.1088/​1367-2630/​ab075a.

[94] Junan Lin, Brandon Buonacorsi, Raymond Laflamme, and Joel J Wallman. On the freedom in representing quantum operations. New Journal of Physics, 21 (2): 023006, feb 2019b. 10.1088/​1367-2630/​ab075a. URL https:/​/​​10.1088.

[95] G. Lindblad. On the generators of quantum dynamical semigroups. Comm. Math. Phys., 48 (2): 119–130, 1976. 10.1007/​BF01608499. URL https:/​/​​10.1007/​BF01608499.

[96] Norbert M Linke, Dmitri Maslov, Martin Roetteler, Shantanu Debnath, Caroline Figgatt, Kevin A Landsman, Kenneth Wright, and Christopher Monroe. Experimental comparison of two quantum computing architectures. Proc. Natl. Acad. Sci. U. S. A., 114 (13): 3305–3310, March 2017. ISSN 0027-8424, 1091-6490. 10.1073/​pnas.1618020114.

[97] Mirko Lobino, Dmitry Korystov, Connor Kupchak, Eden Figueroa, Barry C Sanders, and A I Lvovsky. Complete characterization of quantum-optical processes. Science, 322 (5901): 563–566, October 2008. ISSN 0036-8075, 1095-9203. 10.1126/​science.1162086.

[98] A. Luis and L. L. Sánchez-Soto. Complete characterization of arbitrary quantum measurement processes. Phys. Rev. Lett., 83: 3573–3576, Nov 1999. 10.1103/​PhysRevLett.83.3573. URL https:/​/​​doi/​10.1103/​PhysRevLett.83.3573.

[99] J S Lundeen, A Feito, H Coldenstrodt-Ronge, K L Pregnell, Ch Silberhorn, T C Ralph, J Eisert, M B Plenio, and I A Walmsley. Tomography of quantum detectors. Nat. Phys., 5 (1): 27–30, January 2009. ISSN 1745-2473, 1745-2481. 10.1038/​nphys1133.

[100] Easwar Magesan, J M Gambetta, and Joseph Emerson. Scalable and robust randomized benchmarking of quantum processes. Phys. Rev. Lett., 106 (18): 180504, May 2011. ISSN 0031-9007. 10.1103/​PhysRevLett.106.180504.

[101] Easwar Magesan, Jay M Gambetta, and Joseph Emerson. Characterizing quantum gates via randomized benchmarking. Phys. Rev. A, 85 (4): 042311, April 2012a. ISSN 1050-2947. 10.1103/​PhysRevA.85.042311.

[102] 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 (8): 080505, August 2012b. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.109.080505.

[103] D. H. Mahler, Lee A. Rozema, Ardavan Darabi, Christopher Ferrie, Robin Blume-Kohout, and A. M. Steinberg. Adaptive quantum state tomography improves accuracy quadratically. Phys. Rev. Lett., 111: 183601, Oct 2013. 10.1103/​PhysRevLett.111.183601.

[104] 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 (1): 7, February 2018. ISSN 2056-6387. 10.1038/​s41534-017-0052-0.

[105] Karl Mayer and Emanuel Knill. Quantum process fidelity bounds from sets of input states. Phys. Rev. A, 98 (5): 052326, November 2018. ISSN 1050-2947. 10.1103/​PhysRevA.98.052326.

[106] Alexander J McCaskey, Zachary P Parks, Jacek Jakowski, Shirley V Moore, Titus D Morris, Travis S Humble, and Raphael C Pooser. Quantum chemistry as a benchmark for near-term quantum computers. npj Quantum Information, 5 (1): 99, November 2019. ISSN 2056-6387. 10.1038/​s41534-019-0209-0.

[107] A. F. McCormick, S. J. van Enk, and M. Beck. Experimental demonstration of loop state-preparation-and-measurement tomography. Phys. Rev. A, 95: 042329, Apr 2017. 10.1103/​PhysRevA.95.042329. URL https:/​/​​doi/​10.1103/​PhysRevA.95.042329.

[108] David C McKay, Sarah Sheldon, John A Smolin, Jerry M Chow, and Jay M Gambetta. Three-Qubit randomized benchmarking. Phys. Rev. Lett., 122 (20): 200502, May 2019. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.122.200502.

[109] J Medford, Johannes Beil, JM Taylor, SD Bartlett, AC Doherty, EI Rashba, DP DiVincenzo, H Lu, AC Gossard, and Charles M Marcus. Self-consistent measurement and state tomography of an exchange-only spin qubit. Nature nanotechnology, 8 (9): 654, 2013. 10.1038/​nnano.2013.168. URL https:/​/​​10.1038/​nnano.2013.168.

[110] 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, Jun 2013. 10.1103/​PhysRevA.87.062119.

[111] Kristel Michielsen, Madita Nocon, Dennis Willsch, Fengping Jin, Thomas Lippert, and Hans De Raedt. Benchmarking gate-based quantum computers. Comput. Phys. Commun., 220: 44–55, November 2017. ISSN 0010-4655. 10.1016/​j.cpc.2017.06.011.

[112] D Mogilevtsev, J Řeháček, and Z Hradil. Self-calibration for self-consistent tomography. New Journal of Physics, 14 (9): 095001, sep 2012. 10.1088/​1367-2630/​14/​9/​095001. URL https:/​/​​10.1088/​1367-2630/​14/​9/​095001.

[113] Mohammadreza Mohammadi and Agata M. Brańczyk. Optimization of quantum state tomography in the presence of experimental constraints. Phys. Rev. A, 89: 012113, Jan 2014. 10.1103/​PhysRevA.89.012113. URL https:/​/​​doi/​10.1103/​PhysRevA.89.012113.

[114] M Mohseni and D A Lidar. Direct characterization of quantum dynamics. Phys. Rev. Lett., 97 (17): 170501, October 2006. ISSN 0031-9007. 10.1103/​PhysRevLett.97.170501.

[115] Osama Moussa, Marcus P da Silva, Colm A Ryan, and Raymond Laflamme. Practical experimental certification of computational quantum gates using a twirling procedure. Phys. Rev. Lett., 109 (7): 070504, August 2012. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.109.070504.

[116] Pranav Mundada, Gengyan Zhang, Thomas Hazard, and Andrew Houck. Suppression of qubit crosstalk in a tunable coupling superconducting circuit. Phys. Rev. Applied, 12: 054023, Nov 2019. 10.1103/​PhysRevApplied.12.054023. URL https:/​/​​doi/​10.1103/​PhysRevApplied.12.054023.

[117] W J Munro, D F V James, A G White, and Paul G Kwiat. Tomography and its role in quantum computation. HP Laboratories Technical Report, 53, 2001.

[118] C Neill, P Roushan, K Kechedzhi, S Boixo, S V Isakov, V Smelyanskiy, A Megrant, B Chiaro, A Dunsworth, K Arya, R Barends, B Burkett, Y Chen, Z Chen, A Fowler, B Foxen, M Giustina, R Graff, E Jeffrey, T Huang, J Kelly, P Klimov, E Lucero, J Mutus, M Neeley, C Quintana, D Sank, A Vainsencher, J Wenner, T C White, H Neven, and J M Martinis. A blueprint for demonstrating quantum supremacy with superconducting qubits. Science, 360 (6385): 195–199, April 2018. ISSN 0036-8075, 1095-9203. 10.1126/​science.aao4309.

[119] Erik Nielsen. Gate set tomography on many qubits. in preparation.

[120] Erik Nielsen, Kenneth Rudinger, John King Gamble, and Robin Blume-Kohout. pyGSTi: A python implementation of gate set tomography, 2016. URL http:/​/​​pyGSTio.

[121] Erik Nielsen, Kenneth Rudinger, Timothy Proctor, Antonio Russo, Kevin Young, and Robin Blume-Kohout. Probing quantum processor performance with pyGSTi. Quantum Science and Technology, 5 (4): 044002, jul 2020. 10.1088/​2058-9565/​ab8aa4. URL https:/​/​​10.1088/​2058-9565/​ab8aa4.

[122] MA Nielsen and Isaac L Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000.

[123] J. L. O'Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White. Quantum process tomography of a controlled-not gate. Phys. Rev. Lett., 93: 080502, Aug 2004. 10.1103/​PhysRevLett.93.080502. URL https:/​/​​doi/​10.1103/​PhysRevLett.93.080502.

[124] T E O'Brien, B Tarasinski, and L DiCarlo. Density-matrix simulation of small surface codes under current and projected experimental noise. npj Quantum Information, 3 (1): 1–8, September 2017. ISSN 2056-6387, 2056-6387. 10.1038/​s41534-017-0039-x.

[125] C Piltz, T Sriarunothai, A F Varón, and C Wunderlich. A trapped-ion-based quantum byte with 10(-5) next-neighbour cross-talk. Nat. Commun., 5: 4679, August 2014. 10.1038/​ncomms5679. URL https:/​/​​10.1038/​ncomms5679.

[126] J F Poyatos, J I Cirac, and P Zoller. Complete characterization of a quantum process: The Two-Bit quantum gate. Phys. Rev. Lett., 78 (2): 390–393, January 1997. ISSN 0031-9007. 10.1103/​PhysRevLett.78.390.

[127] Timothy Proctor, Kenneth Rudinger, Kevin Young, Mohan Sarovar, and Robin Blume-Kohout. What randomized benchmarking actually measures. Phys. Rev. Lett., 119: 130502, Sep 2017. 10.1103/​PhysRevLett.119.130502. URL https:/​/​​doi/​10.1103/​PhysRevLett.119.130502.

[128] Timothy Proctor, Melissa Revelle, Erik Nielsen, Kenneth Rudinger, Daniel Lobser, Peter Maunz, Robin Blume-Kohout, and Kevin Young. Detecting and tracking drift in quantum information processors. Nature Communications, 11 (1): 5396, Oct 2020. ISSN 2041-1723. 10.1038/​s41467-020-19074-4. URL https:/​/​​10.1038/​s41467-020-19074-4.

[129] Timothy J Proctor, Arnaud Carignan-Dugas, Kenneth Rudinger, Erik Nielsen, Robin Blume-Kohout, and Kevin Young. Direct randomized benchmarking for multiqubit devices. Phys. Rev. Lett., 123 (3): 030503, July 2019. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.123.030503.

[130] Zbigniew Puchała, Łukasz Rudnicki, and Karol Życzkowski. Pauli semigroups and unistochastic quantum channels. Phys. Lett. A, 383 (20): 2376–2381, July 2019. ISSN 0375-9601. 10.1016/​j.physleta.2019.04.057.

[131] Nicolás Quesada, Agata M. Brańczyk, and Daniel F. V. James. Self-calibrating tomography for multidimensional systems. Phys. Rev. A, 87: 062118, Jun 2013a. 10.1103/​PhysRevA.87.062118. URL https:/​/​​doi/​10.1103/​PhysRevA.87.062118.

[132] Nicolás Quesada, Agata M. Brańczyk, and Daniel F.V. James. Holistic quantum state and process tomography. In Frontiers in Optics 2013, page FW1C.6. Optical Society of America, 2013b. 10.1364/​FIO.2013.FW1C.6. URL http:/​/​​abstract.cfm?URI=FiO-2013-FW1C.6.

[133] Nicolás Quesada, Agata M. Brańczyk, and Daniel F.V. James. Self-calibrating tomography for non-unitary processes. In The Rochester Conferences on Coherence and Quantum Optics and the Quantum Information and Measurement meeting, page W6.38. Optical Society of America, 2013c. 10.1364/​QIM.2013.W6.38. URL http:/​/​​abstract.cfm?URI=QIM-2013-W6.38.

[134] P. Rebentrost, I. Serban, T. Schulte-Herbrüggen, and F. K. Wilhelm. Optimal control of a qubit coupled to a non-markovian environment. Phys. Rev. Lett., 102: 090401, Mar 2009. 10.1103/​PhysRevLett.102.090401. URL https:/​/​​doi/​10.1103/​PhysRevLett.102.090401.

[135] Daniel M Reich, Giulia Gualdi, and Christiane P Koch. Optimal strategies for estimating the average fidelity of quantum gates. Phys. Rev. Lett., 111 (20): 200401, November 2013. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.111.200401.

[136] M Riebe, K Kim, P Schindler, T Monz, P O Schmidt, T K Körber, W Hänsel, H Häffner, C F Roos, and R Blatt. Process tomography of ion trap quantum gates. Phys. Rev. Lett., 97 (22): 220407, December 2006. ISSN 0031-9007. 10.1103/​PhysRevLett.97.220407.

[137] M A Rol, C C Bultink, T E O'Brien, S R de Jong, L S Theis, X Fu, F Luthi, R F L Vermeulen, J C de Sterke, A Bruno, D Deurloo, R N Schouten, F K Wilhelm, and L DiCarlo. Restless tuneup of High-Fidelity qubit gates. Phys. Rev. Applied, 7 (4): 041001, April 2017. 10.1103/​PhysRevApplied.7.041001.

[138] Kenneth Rudinger and Robert Joynt. Compressed sensing for hamiltonian reconstruction. Phys. Rev. A, 92: 052322, Nov 2015. 10.1103/​PhysRevA.92.052322. URL https:/​/​​doi/​10.1103/​PhysRevA.92.052322.

[139] Kenneth Rudinger, Shelby Kimmel, Daniel Lobser, and Peter Maunz. Experimental demonstration of a cheap and accurate phase estimation. Phys. Rev. Lett., 118 (19): 190502, May 2017a. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.118.190502.

[140] Kenneth Rudinger, Shelby Kimmel, Daniel Lobser, and Peter Maunz. Experimental demonstration of a cheap and accurate phase estimation. Phys. Rev. Lett., 118: 190502, May 2017b. 10.1103/​PhysRevLett.118.190502.

[141] Kenneth Rudinger, Timothy Proctor, Dylan Langharst, Mohan Sarovar, Kevin Young, and Robin Blume-Kohout. Probing context-dependent errors in quantum processors. Phys. Rev. X, 9: 021045, Jun 2019. 10.1103/​PhysRevX.9.021045. URL https:/​/​​doi/​10.1103/​PhysRevX.9.021045.

[142] Łukasz Rudnicki, Zbigniew Puchała, and Karol Zyczkowski. Gauge invariant information concerning quantum channels. Quantum, 2: 60, April 2018. ISSN 2521-327X. 10.22331/​q-2018-04-11-60. URL https:/​/​​10.22331/​q-2018-04-11-60.

[143] Mohan Sarovar, Timothy Proctor, Kenneth Rudinger, Kevin Young, Erik Nielsen, and Robin Blume-Kohout. Detecting crosstalk errors in quantum information processors. Quantum, 4: 321, September 2020. ISSN 2521-327X. 10.22331/​q-2020-09-11-321. URL https:/​/​​10.22331/​q-2020-09-11-321.

[144] S G Schirmer, A Kolli, and D K L Oi. Experimental hamiltonian identification for controlled two-level systems. Phys. Rev. A, 69 (5): 050306, May 2004. ISSN 1050-2947. 10.1103/​PhysRevA.69.050306.

[145] Travis L Scholten and Robin Blume-Kohout. Behavior of the maximum likelihood in quantum state tomography. New Journal of Physics, 20 (2): 023050, feb 2018. 10.1088/​1367-2630/​aaa7e2. URL https:/​/​​10.1088/​1367-2630/​aaa7e2.

[146] Travis L Scholten, Yi-Kai Liu, Kevin Young, and Robin Blume-Kohout. Classifying single-qubit noise using machine learning. arXiv preprint arXiv:1908.11762, August 2019.

[147] Christian Schwemmer, Lukas Knips, Daniel Richart, Harald Weinfurter, Tobias Moroder, Matthias Kleinmann, and Otfried Gühne. Systematic errors in current quantum state tomography tools. Phys. Rev. Lett., 114: 080403, Feb 2015. 10.1103/​PhysRevLett.114.080403. URL https:/​/​​doi/​10.1103/​PhysRevLett.114.080403.

[148] A.J. Scott. Tight informationally complete quantum measurements. Journal of Physics A: Mathematical and General, 39, 10 2006. 10.1088/​0305-4470/​39/​43/​009.

[149] A Shabani, R L Kosut, M Mohseni, H Rabitz, M A Broome, M P Almeida, A Fedrizzi, and A G White. Efficient measurement of quantum dynamics via compressive sensing. Phys. Rev. Lett., 106 (10): 100401, March 2011. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.106.100401.

[150] Jiangwei Shang, Hui Khoon Ng, Arun Sehrawat, Xikun Li, and Berthold-Georg Englert. Optimal error regions for quantum state estimation. New J. Phys., 15 (12): 123026, December 2013. ISSN 1367-2630. 10.1088/​1367-2630/​15/​12/​123026.

[151] Sarah Sheldon, Lev S Bishop, Easwar Magesan, Stefan Filipp, Jerry M Chow, and Jay M Gambetta. Characterizing errors on qubit operations via iterative randomized benchmarking. Phys. Rev. A, 93 (1): 012301, January 2016. ISSN 1050-2947. 10.1103/​PhysRevA.93.012301.

[152] John A. Smolin, Jay M. Gambetta, and Graeme Smith. Efficient method for computing the maximum-likelihood quantum state from measurements with additive gaussian noise. Phys. Rev. Lett., 108: 070502, Feb 2012. 10.1103/​PhysRevLett.108.070502. URL https:/​/​​doi/​10.1103/​PhysRevLett.108.070502.

[153] Chao Song, Jing Cui, H Wang, J Hao, H Feng, and Ying Li. Quantum computation with universal error mitigation on a superconducting quantum processor. Sci Adv, 5 (9): eaaw5686, September 2019. ISSN 2375-2548. 10.1126/​sciadv.aaw5686.

[154] Cyril Stark. Self-consistent tomography of the state-measurement gram matrix. Phys. Rev. A, 89: 052109, May 2014. 10.1103/​PhysRevA.89.052109. URL https:/​/​​doi/​10.1103/​PhysRevA.89.052109.

[155] S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, S. P. Kulik, and D. Mogilevtsev. Self-calibrating tomography for angular schmidt modes in spontaneous parametric down-conversion. Phys. Rev. A, 87: 042109, Apr 2013. 10.1103/​PhysRevA.87.042109. URL https:/​/​​doi/​10.1103/​PhysRevA.87.042109.

[156] Maki Takahashi, Stephen D. Bartlett, and Andrew C. Doherty. Tomography of a spin qubit in a double quantum dot. Phys. Rev. A, 88: 022120, Aug 2013. 10.1103/​PhysRevA.88.022120. URL https:/​/​​doi/​10.1103/​PhysRevA.88.022120.

[157] G Tóth, W Wieczorek, D Gross, R Krischek, C Schwemmer, and H Weinfurter. Permutationally invariant quantum tomography. Phys. Rev. Lett., 105 (25): 250403, December 2010. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.105.250403.

[158] Andrzej Veitia and Steven J. van Enk. Testing the context-independence of quantum gates. arXiv preprint arXiv:1810.05945, 2020.

[159] K Vogel and H Risken. Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A Gen. Phys., 40 (5): 2847–2849, September 1989. ISSN 0556-2791. 10.1103/​physreva.40.2847.

[160] Joel Wallman, Chris Granade, Robin Harper, and Steven T Flammia. Estimating the coherence of noise. New Journal of Physics, 17 (11): 113020, nov 2015a. 10.1088/​1367-2630/​17/​11/​113020. URL https:/​/​​10.1088/​1367-2630/​17/​11/​113020.

[161] Joel J. Wallman. Randomized benchmarking with gate-dependent noise. Quantum, 2: 47, January 2018. ISSN 2521-327X. 10.22331/​q-2018-01-29-47. URL https:/​/​​10.22331/​q-2018-01-29-47.

[162] Joel J. Wallman and Steven T. Flammia. Randomized benchmarking with confidence. New J. Phys., 16 (10): 103032, 2014. ISSN 1367-2630. 10.1088/​1367-2630/​16/​10/​103032.

[163] Joel J Wallman, Marie Barnhill, and Joseph Emerson. Robust characterization of loss rates. Phys. Rev. Lett., 115 (6): 060501, August 2015b. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.115.060501.

[164] Joel J Wallman, Marie Barnhill, and Joseph Emerson. Robust characterization of leakage errors. New J. Phys., 18 (4): 043021, April 2016. ISSN 1367-2630. 10.1088/​1367-2630/​18/​4/​043021.

[165] Jianwei Wang, Stefano Paesani, Raffaele Santagati, Sebastian Knauer, Antonio A Gentile, Nathan Wiebe, Maurangelo Petruzzella, Jeremy L O'Brien, John G Rarity, Anthony Laing, and Mark G Thompson. Experimental quantum hamiltonian learning. Nat. Phys., 13 (6): 551–555, June 2017. ISSN 1745-2473, 1745-2481. 10.1038/​nphys4074.

[166] Sheng-Tao Wang, Dong-Ling Deng, and L-M Duan. Hamiltonian tomography for quantum many-body systems with arbitrary couplings. New J. Phys., 17 (9): 093017, September 2015. ISSN 1367-2630. 10.1088/​1367-2630/​17/​9/​093017.

[167] Matthew Ware, Guilhem Ribeill, Diego Ristè, Colm A. Ryan, Blake Johnson, and Marcus P. da Silva. Experimental pauli-frame randomization on a superconducting qubit. Phys. Rev. A, 103: 042604, Apr 2021. 10.1103/​PhysRevA.103.042604. URL https:/​/​​doi/​10.1103/​PhysRevA.103.042604.

[168] A. E. Webb, S. C. Webster, S. Collingbourne, D. Bretaud, A. M. Lawrence, S. Weidt, F. Mintert, and W. K. Hensinger. Resilient entangling gates for trapped ions. Phys. Rev. Lett., 121: 180501, Nov 2018. 10.1103/​PhysRevLett.121.180501. URL https:/​/​​doi/​10.1103/​PhysRevLett.121.180501.

[169] Yaakov S. Weinstein, Timothy F. Havel, Joseph Emerson, Nicolas Boulant, Marcos Saraceno, Seth Lloyd, and David G. Cory. Quantum process tomography of the quantum fourier transform. The Journal of Chemical Physics, 121 (13): 6117–6133, 2004. 10.1063/​1.1785151. URL https:/​/​​10.1063/​1.1785151.

[170] G.A.L. White, C.D. Hill, and L.C.L. Hollenberg. Performance optimization for drift-robust fidelity improvement of two-qubit gates. Phys. Rev. Applied, 15: 014023, Jan 2021. 10.1103/​PhysRevApplied.15.014023. URL https:/​/​​doi/​10.1103/​PhysRevApplied.15.014023.

[171] S. S. Wilks. The large-sample distribution of the likelihood ratio for testing composite hypotheses. Ann. Math. Statist., 9 (1): 60–62, 03 1938. 10.1214/​aoms/​1177732360. URL https:/​/​​10.1214/​aoms/​1177732360.

[172] Christopher J. Wood and Jay M. Gambetta. Quantification and characterization of leakage errors. Phys. Rev. A, 97: 032306, Mar 2018. 10.1103/​PhysRevA.97.032306. URL https:/​/​​doi/​10.1103/​PhysRevA.97.032306.

[173] K Wright, K M Beck, S Debnath, J M Amini, Y Nam, N Grzesiak, J-S Chen, N C Pisenti, M Chmielewski, C Collins, K M Hudek, J Mizrahi, J D Wong-Campos, S Allen, J Apisdorf, P Solomon, M Williams, A M Ducore, A Blinov, S M Kreikemeier, V Chaplin, M Keesan, C Monroe, and J Kim. Benchmarking an 11-qubit quantum computer. Nat. Commun., 10 (1): 5464, November 2019. ISSN 2041-1723. 10.1038/​s41467-019-13534-2.

[174] Kübra Yeter-Aydeniz, Eugene F Dumitrescu, Alex J McCaskey, Ryan S Bennink, Raphael C Pooser, and George Siopsis. Scalar quantum field theories as a benchmark for near-term quantum computers. Phys. Rev. A, 99 (3): 032306, March 2019. ISSN 1050-2947. 10.1103/​PhysRevA.99.032306.

[175] Jun Zhang and Mohan Sarovar. Quantum hamiltonian identification from measurement time traces. Phys. Rev. Lett., 113 (8): 080401, August 2014. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.113.080401.

[176] 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. Nat. Commun., 11 (1): 587, January 2020. ISSN 2041-1723. 10.1038/​s41467-020-14376-z.

Cited by

[1] Gabriel O. Samach, Ami Greene, Johannes Borregaard, Matthias Christandl, Joseph Barreto, David K. Kim, Christopher M. McNally, Alexander Melville, Bethany M. Niedzielski, Youngkyu Sung, Danna Rosenberg, Mollie E. Schwartz, Jonilyn L. Yoder, Terry P. Orlando, Joel I-Jan Wang, Simon Gustavsson, Morten Kjaergaard, and William D. Oliver, "Lindblad Tomography of a Superconducting Quantum Processor", Physical Review Applied 18 6, 064056 (2022).

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

[3] Yuxuan Zhang, Daoheng Niu, Alireza Shabani, and Hassan Shapourian, "Quantum Volume for Photonic Quantum Processors", Physical Review Letters 130 11, 110602 (2023).

[4] Simone Roncallo, Lorenzo Maccone, and Chiara Macchiavello, "Pauli transfer matrix direct reconstruction: channel characterization without full process tomography", Quantum Science and Technology 9 1, 015010 (2024).

[5] Senrui Chen, Yunchao Liu, Matthew Otten, Alireza Seif, Bill Fefferman, and Liang Jiang, "The learnability of Pauli noise", Nature Communications 14 1, 52 (2023).

[6] Xiao Xue, Maximilian Russ, Nodar Samkharadze, Brennan Undseth, Amir Sammak, Giordano Scappucci, and Lieven M. K. Vandersypen, "Quantum logic with spin qubits crossing the surface code threshold", Nature 601 7893, 343 (2022).

[7] Mateusz T. Mądzik, Serwan Asaad, Akram Youssry, Benjamin Joecker, Kenneth M. Rudinger, Erik Nielsen, Kevin C. Young, Timothy J. Proctor, Andrew D. Baczewski, Arne Laucht, Vivien Schmitt, Fay E. Hudson, Kohei M. Itoh, Alexander M. Jakob, Brett C. Johnson, David N. Jamieson, Andrew S. Dzurak, Christopher Ferrie, Robin Blume-Kohout, and Andrea Morello, "Precision tomography of a three-qubit donor quantum processor in silicon", Nature 601 7893, 348 (2022).

[8] Bujiao Wu, Xiaoyang Wang, Xiao Yuan, Cupjin Huang, and Jianxin Chen, "Leakage Benchmarking for Universal Gate Sets", Entropy 26 1, 71 (2024).

[9] 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).

[10] Domenico Pomarico, Leonardo Cosmai, Paolo Facchi, Cosmo Lupo, Saverio Pascazio, and Francesco V. Pepe, "Dynamical Quantum Phase Transitions of the Schwinger Model: Real-Time Dynamics on IBM Quantum", Entropy 25 4, 608 (2023).

[11] Zhiyuan Li, Pei Liu, Peng Zhao, Zhenyu Mi, Huikai Xu, Xuehui Liang, Tang Su, Weijie Sun, Guangming Xue, Jing-Ning Zhang, Weiyang Liu, Yirong Jin, and Haifeng Yu, "Error per single-qubit gate below 10−4 in a superconducting qubit", npj Quantum Information 9 1, 111 (2023).

[12] Xin-Yu Chen, Pan Gao, Chu-Dan Qiu, Ya-Nan Lu, Fan Yang, Yuanyuan Zhao, Hang Li, Jiang Zhang, Shijie Wei, Tonghao Xing, Xin-Yu Pan, Dong Ruan, Feihao Zhang, Keren Li, and Guilu Long, "A noise-robust quantum dynamics learning protocol based on Choi–Jamiolkowski isomorphism: theory and experiment", New Journal of Physics 26 3, 033023 (2024).

[13] Y.F. Zolotarev, I.A. Luchnikov, J.A. López-Saldívar, A.K. Fedorov, and E.O. Kiktenko, "Continuous Monitoring for Noisy Intermediate-Scale Quantum Processors", Physical Review Applied 19 1, 014027 (2023).

[14] Adrien Suau, Jon Nelson, Marc Vuffray, Andrey Y. Lokhov, Lukasz Cincio, and Carleton Coffrin, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 1369 (2023) ISBN:979-8-3503-4323-6.

[15] Martin Kliesch, "Randomized benchmarking with a tractable continuously generated group", Quantum Views 6, 64 (2022).

[16] Ezra Bussmann, Robert E. Butera, James H. G. Owen, John N. Randall, Steven M. Rinaldi, Andrew D. Baczewski, and Shashank Misra, "Atomic-precision advanced manufacturing for Si quantum computing", MRS Bulletin 46 7, 607 (2021).

[17] Scott E. Smart, Zixuan Hu, Sabre Kais, and David A. Mazziotti, "Relaxation of stationary states on a quantum computer yields a unique spectroscopic fingerprint of the computer’s noise", Communications Physics 5 1, 28 (2022).

[18] Dripto M. Debroy, Élie Genois, Jonathan A. Gross, Wojciech Mruczkiewicz, Kenny Lee, Sabrina Hong, Zijun Chen, Vadim Smelyanskiy, and Zhang Jiang, "Context-aware fidelity estimation", Physical Review Research 5 4, 043202 (2023).

[19] Tomasz Białecki, Tomasz Rybotycki, Jakub Tworzydło, and Adam Bednorz, "Testing the accuracy of qubit rotations on a public quantum computer", Frontiers in Physics 12, 1360080 (2024).

[20] Thomas Ayral, Pauline Besserve, Denis Lacroix, and Edgar Andres Ruiz Guzman, "Quantum computing with and for many-body physics", The European Physical Journal A 59 10, 227 (2023).

[21] Thomas M. Stace, Jiayin Chen, Li Li, Viktor S. Perunicic, Andre R. R. Carvalho, Michael Hush, Christophe H. Valahu, Ting Rei Tan, and Michael J. Biercuk, "Optimized Bayesian system identification in quantum devices", Physical Review Applied 21 1, 014012 (2024).

[22] Congcong Zheng, Xutao Yu, and Kun Wang, "Cross-platform comparison of arbitrary quantum processes", npj Quantum Information 10 1, 4 (2024).

[23] Daniel Hothem, Jordan Hines, Karthik Nataraj, Robin Blume-Kohout, and Timothy Proctor, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 709 (2023) ISBN:979-8-3503-4323-6.

[24] Kirill Dubovitskii and Yuriy Makhlin, "Partial randomized benchmarking", Scientific Reports 12 1, 10129 (2022).

[25] Simone Roncallo, Lorenzo Maccone, and Chiara Macchiavello, "Multiqubit noise deconvolution and characterization", Physical Review A 107 2, 022419 (2023).

[26] Rostyslav Savytskyy, Tim Botzem, Irene Fernandez de Fuentes, Benjamin Joecker, Jarryd J. Pla, Fay E. Hudson, Kohei M. Itoh, Alexander M. Jakob, Brett C. Johnson, David N. Jamieson, Andrew S. Dzurak, and Andrea Morello, "An electrically driven single-atom “flip-flop” qubit", Science Advances 9 6, eadd9408 (2023).

[27] Nathan Eli Miller, Biswadeep Chakraborty, and Saibal Mukhopadhyay, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 1412 (2023) ISBN:979-8-3503-4323-6.

[28] Guido Burkard, Thaddeus D. Ladd, Andrew Pan, John M. Nichol, and Jason R. Petta, "Semiconductor spin qubits", Reviews of Modern Physics 95 2, 025003 (2023).

[29] Filip B. Maciejewski, Zbigniew Puchała, and Michał Oszmaniec, "Exploring Quantum Average-Case Distances: Proofs, Properties, and Examples", IEEE Transactions on Information Theory 69 7, 4600 (2023).

[30] G. A. L. White, K. Modi, and C. D. Hill, "Filtering Crosstalk from Bath Non-Markovianity via Spacetime Classical Shadows", Physical Review Letters 130 16, 160401 (2023).

[31] David A. Quiroga and Anastasios Kyrillidis, 2023 IEEE International Conference on Rebooting Computing (ICRC) 1 (2023) ISBN:979-8-3503-8204-4.

[32] Shuxiang Cao, Mustafa Bakr, Giulio Campanaro, Simone D Fasciati, James Wills, Deep Lall, Boris Shteynas, Vivek Chidambaram, Ivan Rungger, and Peter Leek, "Emulating two qubits with a four-level transmon qudit for variational quantum algorithms", Quantum Science and Technology 9 3, 035003 (2024).

[33] Gregory A. L. White, "Gate set tomography is not just hyperaccurate, it’s a different way of thinking", Quantum Views 5, 60 (2021).

[34] François Verdeil and Yannick Deville, "Unitary quantum process tomography with unreliable pure input states", Physical Review A 108 6, 062410 (2023).

[35] Ward van der Schoot, Robert Wezeman, Pieter Thijs Eendebak, Niels M. P. Neumann, and Frank Phillipson, "Evaluating three levels of quantum metrics on quantum-inspire hardware", Quantum Information Processing 22 12, 451 (2023).

[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] Jin Ming Koh, Shi-Ning Sun, Mario Motta, and Austin J. Minnich, "Measurement-induced entanglement phase transition on a superconducting quantum processor with mid-circuit readout", Nature Physics 19 9, 1314 (2023).

[38] Melinda Andrews, Thomas Halverson, Joshua Heath, I. Michael Mandelberg, Martin J. McHugh, and Shawn M. Wilder, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 1319 (2023) ISBN:979-8-3503-4323-6.

[39] Marco Cattaneo, Matteo A. C. Rossi, Keijo Korhonen, Elsi-Mari Borrelli, Guillermo García-Pérez, Zoltán Zimborás, and Daniel Cavalcanti, "Self-consistent quantum measurement tomography based on semidefinite programming", Physical Review Research 5 3, 033154 (2023).

[40] Jeongwan Haah, Robin Kothari, Ryan O’Donnell, and Ewin Tang, 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS) 363 (2023) ISBN:979-8-3503-1894-4.

[41] Guillermo García-Pérez, Oskari Kerppo, Matteo A. C. Rossi, and Sabrina Maniscalco, "Experimentally accessible nonseparability criteria for multipartite-entanglement-structure detection", Physical Review Research 5 1, 013226 (2023).

[42] Adam R. Mills, Charles R. Guinn, Michael J. Gullans, Anthony J. Sigillito, Mayer M. Feldman, Erik Nielsen, and Jason R. Petta, "Two-qubit silicon quantum processor with operation fidelity exceeding 99%", Science Advances 8 14, eabn5130 (2022).

[43] Meng Zhang, Jing Wang, Fei Yao, and Junsen Lai, 2023 8th International Conference on Cloud Computing and Big Data Analytics (ICCCBDA) 407 (2023) ISBN:978-1-6654-5533-6.

[44] Violeta N. Ivanova-Rohling, Niklas Rohling, and Guido Burkard, "Optimal quantum state tomography with noisy gates", EPJ Quantum Technology 10 1, 25 (2023).

[45] Nadia Milazzo, Olivier Giraud, Giovanni Gramegna, and Daniel Braun, "Principles of quantum functional testing", Physical Review A 108 2, 022602 (2023).

[46] Colin Kai-Uwe Becker, Nikolay Tcholtchev, Ilie-Daniel Gheorghe-Pop, Sebastian Bock, Raphael Seidel, and Manfred Hauswirth, 2022 IEEE 19th International Conference on Software Architecture Companion (ICSA-C) 160 (2022) ISBN:978-1-6654-9493-9.

[47] Benjamin McDonough, Andrea Mari, Nathan Shammah, Nathaniel T. Stemen, Misty Wahl, William J. Zeng, and Peter P. Orth, 2022 IEEE/ACM Third International Workshop on Quantum Computing Software (QCS) 83 (2022) ISBN:978-1-6654-7536-5.

[48] Irene Fernández de Fuentes, Tim Botzem, Mark A. I. Johnson, Arjen Vaartjes, Serwan Asaad, Vincent Mourik, Fay E. Hudson, Kohei M. Itoh, Brett C. Johnson, Alexander M. Jakob, Jeffrey C. McCallum, David N. Jamieson, Andrew S. Dzurak, and Andrea Morello, "Navigating the 16-dimensional Hilbert space of a high-spin donor qudit with electric and magnetic fields", Nature Communications 15 1, 1380 (2024).

[49] T.J. Evans, W. Huang, J. Yoneda, R. Harper, T. Tanttu, K.W. Chan, F.E. Hudson, K.M. Itoh, A. Saraiva, C.H. Yang, A.S. Dzurak, and S.D. Bartlett, "Fast Bayesian Tomography of a Two-Qubit Gate Set in Silicon", Physical Review Applied 17 2, 024068 (2022).

[50] Yuchen Guo and Shuo Yang, "Quantum Error Mitigation via Matrix Product Operators", PRX Quantum 3 4, 040313 (2022).

[51] Ryan Shaffer, Hang Ren, Emiliia Dyrenkova, Christopher G. Yale, Daniel S. Lobser, Ashlyn D. Burch, Matthew N. H. Chow, Melissa C. Revelle, Susan M. Clark, and Hartmut Häffner, "Sample-efficient verification of continuously-parameterized quantum gates for small quantum processors", Quantum 7, 997 (2023).

[52] Akel Hashim, Stefan Seritan, Timothy Proctor, Kenneth Rudinger, Noah Goss, Ravi K. Naik, John Mark Kreikebaum, David I. Santiago, and Irfan Siddiqi, "Benchmarking quantum logic operations relative to thresholds for fault tolerance", npj Quantum Information 9 1, 109 (2023).

[53] Andreas Elben, Steven T. Flammia, Hsin-Yuan Huang, Richard Kueng, John Preskill, Benoît Vermersch, and Peter Zoller, "The randomized measurement toolbox", Nature Reviews Physics 5 1, 9 (2022).

[54] Lorenzo Buffoni, Stefano Gherardini, Emmanuel Zambrini Cruzeiro, and Yasser Omar, "Third Law of Thermodynamics and the Scaling of Quantum Computers", Physical Review Letters 129 15, 150602 (2022).

[55] Irene López Gutiérrez, Felix Dietrich, and Christian B. Mendl, "Quantum process tomography of unitary maps from time-delayed measurements", Quantum Information Processing 22 6, 251 (2023).

[56] 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).

[57] 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).

[58] Valentin Gebhart, Raffaele Santagati, Antonio Andrea Gentile, Erik M. Gauger, David Craig, Natalia Ares, Leonardo Banchi, Florian Marquardt, Luca Pezzè, and Cristian Bonato, "Learning quantum systems", Nature Reviews Physics 5 3, 141 (2023).

[59] Maximilian Rimbach-Russ, Stephan G J Philips, Xiao Xue, and Lieven M K Vandersypen, "Simple framework for systematic high-fidelity gate operations", Quantum Science and Technology 8 4, 045025 (2023).

[60] Kieran Dalton, Christopher K. Long, Yordan S. Yordanov, Charles G. Smith, Crispin H. W. Barnes, Normann Mertig, and David R. M. Arvidsson-Shukur, "Quantifying the effect of gate errors on variational quantum eigensolvers for quantum chemistry", npj Quantum Information 10 1, 18 (2024).

[61] M.J. Gullans, M. Caranti, A.R. Mills, and J.R. Petta, "Compressed Gate Characterization for Quantum Devices with Time-Correlated Noise", PRX Quantum 5 1, 010306 (2024).

[62] Tian Luan, Zetong Li, Congcong Zheng, Xueheng Kuang, Xutao Yu, and Zaichen Zhang, "Quantum Tomography: From Markovianity to Non-Markovianity", Symmetry 16 2, 180 (2024).

[63] A.R. Mills, C.R. Guinn, M.M. Feldman, A.J. Sigillito, M.J. Gullans, M.T. Rakher, J. Kerckhoff, C.A.C. Jackson, and J.R. Petta, "High-Fidelity State Preparation, Quantum Control, and Readout of an Isotopically Enriched Silicon Spin Qubit", Physical Review Applied 18 6, 064028 (2022).

[64] Grace M. Sommers, David A. Huse, and Michael J. Gullans, "Crystalline Quantum Circuits", PRX Quantum 4 3, 030313 (2023).

[65] J. P. Marceaux and Kevin Young, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 1401 (2023) ISBN:979-8-3503-4323-6.

[66] Raphael Brieger, Ingo Roth, and Martin Kliesch, "Compressive Gate Set Tomography", PRX Quantum 4 1, 010325 (2023).

[67] Christopher W. Warren, Jorge Fernández-Pendás, Shahnawaz Ahmed, Tahereh Abad, Andreas Bengtsson, Janka Biznárová, Kamanasish Debnath, Xiu Gu, Christian Križan, Amr Osman, Anita Fadavi Roudsari, Per Delsing, Göran Johansson, Anton Frisk Kockum, Giovanna Tancredi, and Jonas Bylander, "Extensive characterization and implementation of a family of three-qubit gates at the coherence limit", npj Quantum Information 9 1, 44 (2023).

[68] Yi-Hsien Wu, Leon C. Camenzind, Akito Noiri, Kenta Takeda, Takashi Nakajima, Takashi Kobayashi, Chien-Yuan Chang, Amir Sammak, Giordano Scappucci, Hsi-Sheng Goan, and Seigo Tarucha, "Hamiltonian phase error in resonantly driven CNOT gate above the fault-tolerant threshold", npj Quantum Information 10 1, 8 (2024).

[69] Kenneth Rudinger, Guilhem J. Ribeill, Luke C.G. Govia, Matthew Ware, Erik Nielsen, Kevin Young, Thomas A. Ohki, Robin Blume-Kohout, and Timothy Proctor, "Characterizing Midcircuit Measurements on a Superconducting Qubit Using Gate Set Tomography", Physical Review Applied 17 1, 014014 (2022).

[70] Hsin-Yuan Huang, Yu Tong, Di Fang, and Yuan Su, "Learning Many-Body Hamiltonians with Heisenberg-Limited Scaling", Physical Review Letters 130 20, 200403 (2023).

[71] Corey Ostrove, Kenneth Rudinger, Stefan Seritan, Kevin Young, and Robin Blume-Kohout, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 1422 (2023) ISBN:979-8-3503-4323-6.

[72] L. Pereira, J. J. García-Ripoll, and T. Ramos, "Parallel tomography of quantum non-demolition measurements in multi-qubit devices", npj Quantum Information 9 1, 22 (2023).

[73] Irina Heinz, Adam R. Mills, Jason R. Petta, and Guido Burkard, "Analysis and mitigation of residual exchange coupling in linear spin-qubit arrays", Physical Review Research 6 1, 013153 (2024).

[74] Yuchen Guo and Shuo Yang, "Noise effects on purity and quantum entanglement in terms of physical implementability", npj Quantum Information 9 1, 11 (2023).

[75] Julian Berberich, Daniel Fink, and Christian Holm, "Robustness of quantum algorithms against coherent control errors", Physical Review A 109 1, 012417 (2024).

[76] Ken Xuan Wei, Emily Pritchett, David M. Zajac, David C. McKay, and Seth Merkel, "Characterizing non-Markovian off-resonant errors in quantum gates", Physical Review Applied 21 2, 024018 (2024).

[77] Jordan Hines, Marie Lu, Ravi K. Naik, Akel Hashim, Jean-Loup Ville, Brad Mitchell, John Mark Kriekebaum, David I. Santiago, Stefan Seritan, Erik Nielsen, Robin Blume-Kohout, Kevin Young, Irfan Siddiqi, Birgitta Whaley, and Timothy Proctor, "Demonstrating Scalable Randomized Benchmarking of Universal Gate Sets", Physical Review X 13 4, 041030 (2023).

[78] Vinay Tripathi, Huo Chen, Mostafa Khezri, Ka-Wa Yip, E.M. Levenson-Falk, and Daniel A. Lidar, "Suppression of Crosstalk in Superconducting Qubits Using Dynamical Decoupling", Physical Review Applied 18 2, 024068 (2022).

[79] Liran Shirizly, Grégoire Misguich, and Haggai Landa, "Dissipative Dynamics of Graph-State Stabilizers with Superconducting Qubits", Physical Review Letters 132 1, 010601 (2024).

[80] Will Gilbert, Tuomo Tanttu, Wee Han Lim, MengKe Feng, Jonathan Y. Huang, Jesus D. Cifuentes, Santiago Serrano, Philip Y. Mai, Ross C. C. Leon, Christopher C. Escott, Kohei M. Itoh, Nikolay V. Abrosimov, Hans-Joachim Pohl, Michael L. W. Thewalt, Fay E. Hudson, Andrea Morello, Arne Laucht, Chih Hwan Yang, Andre Saraiva, and Andrew S. Dzurak, "On-demand electrical control of spin qubits", Nature Nanotechnology 18 2, 131 (2023).

[81] Hugo Perrin, Thibault Scoquart, Alexander Shnirman, Jörg Schmalian, and Kyrylo Snizhko, "Mitigating crosstalk errors by randomized compiling: Simulation of the BCS model on a superconducting quantum computer", Physical Review Research 6 1, 013142 (2024).

[82] C. W. Hogle, D. Dominguez, M. Dong, A. Leenheer, H. J. McGuinness, B. P. Ruzic, M. Eichenfield, and D. Stick, "High-fidelity trapped-ion qubit operations with scalable photonic modulators", npj Quantum Information 9 1, 74 (2023).

[83] Vicente Leyton-Ortega, Tyler Kharazi, and Raphael C. Pooser, "Parametrized process characterization with reduced resource requirements", Physical Review A 105 5, 052408 (2022).

[84] Yasunari Suzuki, Suguru Endo, Keisuke Fujii, and Yuuki Tokunaga, "Quantum Error Mitigation as a Universal Error Reduction Technique: Applications from the NISQ to the Fault-Tolerant Quantum Computing Eras", PRX Quantum 3 1, 010345 (2022).

[85] Yasunari Suzuki, Suguru Endo, Keisuke Fujii, and Yuuki Tokunaga, "Quantum error mitigation as a universal error-minimization technique: applications from NISQ to FTQC eras", arXiv:2010.03887, (2020).

[86] R. Au-Yeung, B. Camino, O. Rathore, and V. Kendon, "Quantum algorithms for scientific applications", arXiv:2312.14904, (2023).

[87] G. A. L. White, F. A. Pollock, L. C. L. Hollenberg, K. Modi, and C. D. Hill, "Non-Markovian Quantum Process Tomography", PRX Quantum 3 2, 020344 (2022).

[88] Evan T. Hockings, Andrew C. Doherty, and Robin Harper, "Scalable noise characterisation of syndrome extraction circuits with averaged circuit eigenvalue sampling", arXiv:2404.06545, (2024).

[89] Antonio A. Gentile, Brian Flynn, Sebastian Knauer, Nathan Wiebe, Stefano Paesani, Christopher E. Granade, John G. Rarity, Raffaele Santagati, and Anthony Laing, "Learning models of quantum systems from experiments", Nature Physics 17 7, 837 (2021).

[90] Tuomo Tanttu, Wee Han Lim, Jonathan Y. Huang, Nard Dumoulin Stuyck, Will Gilbert, Rocky Y. Su, MengKe Feng, Jesus D. Cifuentes, Amanda E. Seedhouse, Stefan K. Seritan, Corey I. Ostrove, Kenneth M. Rudinger, Ross C. C. Leon, Wister Huang, Christopher C. Escott, Kohei M. Itoh, Nikolay V. Abrosimov, Hans-Joachim Pohl, Michael L. W. Thewalt, Fay E. Hudson, Robin Blume-Kohout, Stephen D. Bartlett, Andrea Morello, Arne Laucht, Chih Hwan Yang, Andre Saraiva, and Andrew S. Dzurak, "Assessment of error variation in high-fidelity two-qubit gates in silicon", arXiv:2303.04090, (2023).

[91] 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).

[92] Hsin-Yuan Huang, Steven T. Flammia, and John Preskill, "Foundations for learning from noisy quantum experiments", arXiv:2204.13691, (2022).

[93] Kenneth Rudinger, Craig W. Hogle, Ravi K. Naik, Akel Hashim, Daniel Lobser, David I. Santiago, Matthew D. Grace, Erik Nielsen, Timothy Proctor, Stefan Seritan, Susan M. Clark, Robin Blume-Kohout, Irfan Siddiqi, and Kevin C. Young, "Experimental Characterization of Crosstalk Errors with Simultaneous Gate Set Tomography", PRX Quantum 2 4, 040338 (2021).

[94] R. Y. Su, J. Y. Huang, N. Dumoulin. Stuyck, M. K. Feng, W. Gilbert, T. J. Evans, W. H. Lim, F. E. Hudson, K. W. Chan, W. Huang, Kohei M. Itoh, R. Harper, S. D. Bartlett, C. H. Yang, A. Laucht, A. Saraiva, T. Tanttu, and A. S. Dzurak, "Characterizing non-Markovian Quantum Process by Fast Bayesian Tomography", arXiv:2307.12452, (2023).

[95] K. Nestmann, V. Bruch, and M. R. Wegewijs, "How Quantum Evolution with Memory is Generated in a Time-Local Way", Physical Review X 11 2, 021041 (2021).

[96] 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).

[97] Ze-Tong Li, Cong-Cong Zheng, Fan-Xu Meng, Han Zeng, Tian Luan, Zai-Chen Zhang, and Xu-Tao Yu, "Non-Markovian Quantum Gate Set Tomography", arXiv:2307.14696, (2023).

[98] Jeongwan Haah, Robin Kothari, Ryan O'Donnell, and Ewin Tang, "Query-optimal estimation of unitary channels in diamond distance", arXiv:2302.14066, (2023).

[99] Jan Nöller, Nikolai Miklin, Martin Kliesch, and Mariami Gachechiladze, "Device-independent certification of quantum gates under the dimension assumption", arXiv:2401.17006, (2024).

[100] Robin Blume-Kohout, Kenneth Rudinger, Erik Nielsen, Timothy Proctor, and Kevin Young, "Wildcard error: Quantifying unmodeled errors in quantum processors", arXiv:2012.12231, (2020).

[101] Yanwu Gu, Rajesh Mishra, Berthold-Georg Englert, and Hui Khoon Ng, "Randomized Linear Gate-Set Tomography", PRX Quantum 2 3, 030328 (2021).

[102] David A. Quiroga and Anastasios Kyrillidis, "Using non-convex optimization in quantum process tomography: Factored gradient descent is tough to beat", arXiv:2312.01311, (2023).

[103] Salonik Resch and Ulya R. Karpuzcu, "Benchmarking Quantum Computers and the Impact of Quantum Noise", arXiv:1912.00546, (2019).

[104] Corey Ostrove, Kenneth Rudinger, Stefan Seritan, Kevin Young, and Robin Blume-Kohout, "Near-Minimal Gate Set Tomography Experiment Designs", arXiv:2308.08781, (2023).

[105] Arnaud Carignan-Dugas, Shashank Kumar Ranu, and Patrick Dreher, "Estimating Coherent Contributions to the Error Profile Using Cycle Error Reconstruction", arXiv:2303.09945, (2023).

[106] Violeta N. Ivanova-Rohling, Niklas Rohling, and Guido Burkard, "Optimal Quantum State Tomography with Noisy Gates", arXiv:2203.05677, (2022).

[107] Erik Nielsen, Kenneth Rudinger, Timothy Proctor, Kevin Young, and Robin Blume-Kohout, "Efficient flexible characterization of quantum processors with nested error models", New Journal of Physics 23 9, 093020 (2021).

[108] A. Stephens, J. M. Cutshall, T. McPhee, and M. Beck, "Self-consistent state and measurement tomography with fewer measurements", Physical Review A 104 1, 012416 (2021).

[109] Ahmed Abid Moueddene, Nader Khammassi, Sebastian Feld, and Said Hamdioui, "A context-aware gate set tomography characterization of superconducting qubits", arXiv:2103.09922, (2021).

[110] Adrien Suau, Jon Nelson, Marc Vuffray, Andrey Y. Lokhov, Lukasz Cincio, and Carleton Coffrin, "Single-Qubit Cross Platform Comparison of Quantum Computing Hardware", arXiv:2108.11334, (2021).

[111] Violeta N. Ivanova-Rohling, Guido Burkard, and Niklas Rohling, "Quantum state tomography as a numerical optimization problem", New Journal of Physics 23 12, 123034 (2021).

[112] Guedong Park, Yong Siah Teo, and Hyunseok Jeong, "Resource-efficient shadow tomography using equatorial measurements", arXiv:2311.14622, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-04-19 04:18:59) and SAO/NASA ADS (last updated successfully 2024-04-19 04:19:01). The list may be incomplete as not all publishers provide suitable and complete citation data.