Mitigation of readout noise in near-term quantum devices by classical post-processing based on detector tomography

Filip B. Maciejewski1,2,3, Zoltán Zimborás4,5,6, and Michał Oszmaniec2,3

1University of Warsaw, Faculty of Physics, Ludwika Pasteura 5, 02-093 Warszawa, Poland
2International Centre for Theory of Quantum Technologies, University of Gdansk, Wita Stwosza 63, 80-308 Gdansk, Poland
3Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland
4Wigner Research Centre for Physics of the Hungarian Academy of Sciences, H-1525 Budapest, P.O.Box 49, Hungary
5BME-MTA Lendület Quantum Information Theory Research Group, Budapest, Hungary
6Mathematical Institute, Budapest University of Technology and Economics, P.O.Box 91, H-1111, Budapest, Hungary

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Abstract

We propose a simple scheme to reduce readout errors in experiments on quantum systems with finite number of measurement outcomes. Our method relies on performing classical post-processing which is preceded by Quantum Detector Tomography, i.e., the reconstruction of a Positive-Operator Valued Measure (POVM) describing the given quantum measurement device. If the measurement device is affected only by an invertible classical noise, it is possible to correct the outcome statistics of future experiments performed on the same device. To support the practical applicability of this scheme for near-term quantum devices, we characterize measurements implemented in IBM's and Rigetti's quantum processors. We find that for these devices, based on superconducting transmon qubits, classical noise is indeed the dominant source of readout errors. Moreover, we analyze the influence of the presence of coherent errors and finite statistics on the performance of our error-mitigation procedure. Applying our scheme on the IBM's 5-qubit device, we observe a significant improvement of the results of a number of single- and two-qubit tasks including Quantum State Tomography (QST), Quantum Process Tomography (QPT), the implementation of non-projective measurements, and certain quantum algorithms (Grover's search and the Bernstein-Vazirani algorithm). Finally, we present results showing improvement for the implementation of certain probability distributions in the case of five qubits.

Most researchers believe that quantum computing, if ever actually developed, could offer major advances in numerous areas of scientific research. Yet, this technology is currently in its infancy, and the state of the art devices suffer from various problems. One of the most serious obstacles we need to overcome is the noise affecting the qubits. In this context, an important task arises of developing methods to reduce the errors.

In this work, we focus on the noise affecting quantum measurements. We propose a simple procedure to mitigate measurement errors via classical post-processing of the experimental outcome statistics. The procedure works perfectly provided measurement noise is classical and one operates in the infinite-statistics regime. Naturally, neither of those two assumptions is fulfilled exactly in practice, therefore we study the performance of our mitigation scheme in the presence of their violations. Importantly, we show how to validate the model of noise via the procedure known as Quantum Detector Tomography, which allows one to obtain the classical description of the quantum detector.

Our aim is to present a paper exploring the whole procedure of readout error mitigation: from the detailed description of necessary assumptions, through validation of those, finishing at the implementation of presented ideas on the actual quantum hardware from IBM and Rigetti. We believe that such an approach makes the work accessible to readers not necessarily familiar with the formalism of quantum measurements.

To encourage the practical realization of our findings, we developed an open-source GitHub repository implementing the ideas from the paper https://github.com/fbm2718/QREM.

► BibTeX data

► References

[1] John Preskill ``Quantum Computing in the NISQ era and beyond'' Quantum 2, 79 (2018).
https:/​/​doi.org/​https:/​/​doi.org/​10.22331/​q-2018-08-06-79

[2] Héctor Abraham, Ismail Yunus Akhalwaya, Gadi Aleksandrowicz, Thomas Alexander, Gadi Alexandrowics, Eli Arbel, Abraham Asfaw, Carlos Azaustre, AzizNgoueya, Panagiotis Barkoutsos, George Barron, Luciano Bello, Yael Ben-Haim, Daniel Bevenius, Lev S. Bishop, Samuel Bosch, Sergey Bravyi, David Bucher, Fran Cabrera, Padraic Calpin, Lauren Capelluto, Jorge Carballo, Ginés Carrascal, Adrian Chen, Chun-Fu Chen, Richard Chen, Jerry M. Chow, Christian Claus, Christian Clauss, Abigail J. Cross, Andrew W. Cross, Simon Cross, Juan Cruz-Benito, Chris Culver, Antonio D. Córcoles-Gonzales, Sean Dague, Tareq El Dandachi, Matthieu Dartiailh, DavideFrr, Abdón Rodríguez Davila, Delton Ding, Jun Doi, Eric Drechsler, Drew, Eugene Dumitrescu, Karel Dumon, Ivan Duran, Kareem EL-Safty, Eric Eastman, Pieter Eendebak, Daniel Egger, Mark Everitt, Paco Martín Fernández, Axel Hernández Ferrera, Albert Frisch, Andreas Fuhrer, MELVIN GEORGE, Julien Gacon, Gadi, Borja Godoy Gago, Jay M. Gambetta, Adhisha Gammanpila, Luis Garcia, Shelly Garion, Juan Gomez-Mosquera, Salvador Puente González, Ian Gould, Donny Greenberg, Dmitry Grinko, Wen Guan, John A. Gunnels, Isabel Haide, Ikko Hamamura, Vojtech Havlicek, Joe Hellmers, Łukasz Herok, Stefan Hillmich, Hiroshi Horii, Connor Howington, Shaohan Hu, Wei Hu, Haruki Imai, Takashi Imamichi, Kazuaki Ishizaki, Raban Iten, Toshinari Itoko, Ali Javadi-Abhari, Jessica, Kiran Johns, Tal Kachmann, Naoki Kanazawa, Kang-Bae, Anton Karazeev, Paul Kassebaum, Spencer King, Knabberjoe, Arseny Kovyrshin, Vivek Krishnan, Kevin Krsulich, Gawel Kus, Ryan LaRose, Raphaël Lambert, Joe Latone, Scott Lawrence, Dennis Liu, Peng Liu, Yunho Maeng, Aleksei Malyshev, Jakub Marecek, Manoel Marques, Dolph Mathews, Atsushi Matsuo, Douglas T. McClure, Cameron McGarry, David McKay, Dan McPherson, Srujan Meesala, Martin Mevissen, Antonio Mezzacapo, Rohit Midha, Zlatko Minev, Abby Mitchell, Nikolaj Moll, Michael Duane Mooring, Renier Morales, Niall Moran, Prakash Murali, Jan Müggenburg, David Nadlinger, Giacomo Nannicini, Paul Nation, Yehuda Naveh, Patrick Neuweiler, Pradeep Niroula, Hassi Norlen, Lee James O'Riordan, Oluwatobi Ogunbayo, Pauline Ollitrault, Steven Oud, Dan Padilha, Hanhee Paik, Simone Perriello, Anna Phan, Marco Pistoia, Alejandro Pozas-iKerstjens, Viktor Prutyanov, Daniel Puzzuoli, Jesús Pérez, Quintiii, Rudy Raymond, Rafael Martín-Cuevas Redondo, Max Reuter, Julia Rice, Diego M. Rodríguez, Max Rossmannek, Mingi Ryu, Tharrmashastha SAPV, SamFerracin, Martin Sandberg, Ninad Sathaye, Bruno Schmitt, Chris Schnabel, Zachary Schoenfeld, Travis L. Scholten, Eddie Schoute, Joachim Schwarm, Ismael Faro Sertage, Kanav Setia, Nathan Shammah, Yunong Shi, Adenilton Silva, Andrea Simonetto, Nick Singstock, Yukio Siraichi, Iskandar Sitdikov, Seyon Sivarajah, Magnus Berg Sletfjerding, John A. Smolin, Mathias Soeken, Igor Olegovich Sokolov, SooluThomas, Dominik Steenken, Matt Stypulkoski, Jack Suen, Hitomi Takahashi, Ivano Tavernelli, Charles Taylor, Pete Taylour, Soolu Thomas, Mathieu Tillet, Maddy Tod, Enrique Torre, Kenso Trabing, Matthew Treinish, TrishaPe, Wes Turner, Yotam Vaknin, Carmen Recio Valcarce, Francois Varchon, Almudena Carrera Vazquez, Desiree Vogt-Lee, Christophe Vuillot, James Weaver, Rafal Wieczorek, Jonathan A. Wildstrom, Robert Wille, Erick Winston, Jack J. Woehr, Stefan Woerner, Ryan Woo, Christopher J. Wood, Ryan Wood, Stephen Wood, James Wootton, Daniyar Yeralin, Richard Young, Jessie Yu, Christopher Zachow, Laura Zdanski, Christa Zoufal, Zoufalc, azulehner, bcamorrison, brandhsn, zz, dan1pal, dime10, drholmie, elfrocampeador, faisaldebouni, fanizzamarco, gruu, kanejess, klinvill, kurarrr, lerongil, ma5x, aharoni, ordmoj, sethmerkel, strickroman, sumitpuri, tigerjack, toural, vvilpas, welien, willhbang, yang.luh, yelojakit, and yotamvakninibm, ``Qiskit: An Open-source Framework for Quantum Computing'' (2019).
https:/​/​doi.org/​10.5281/​zenodo.2562110

[3] IBM ``https:/​/​quantumexperience.ng.bluemix.net/​qx/​'' (Access: 2018.12.28).
https:/​/​quantumexperience.ng.bluemix.net/​qx/​

[4] Rigetti ``https:/​/​www.rigetti.com/​forest'' (Access: 2018.12.28).
https:/​/​www.rigetti.com/​forest

[5] D-Wave ``https:/​/​cloud.dwavesys.com/​qubist/​'' [Access: 2018.12.28].
https:/​/​cloud.dwavesys.com/​qubist/​

[6] Michael A. Nielsenand Isaac L. Chuang ``Quantum Computation and Quantum Information: 10th Anniversary Edition'' Cambridge University Press (2010).
https:/​/​doi.org/​10.1017/​CBO9780511976667

[7] I. M. Georgescu, S. Ashhab, and Franco Nori, ``Quantum simulation'' Reviews of Modern Physics 86, 153–185 (2014).
https:/​/​doi.org/​10.1103/​RevModPhys.86.153
arXiv:1308.6253

[8] Kentaro Tamuraand Yutaka Shikano ``Quantum Random Numbers generated by the Cloud Superconducting Quantum Computer'' (2019).
arXiv:1906.04410

[9] Ying Liand Simon C. Benjamin ``Efficient variational quantum simulator incorporating active error minimisation'' Phys Rev X 7, 021050 (2017).
https:/​/​doi.org/​https:/​/​doi.org/​10.1103/​PhysRevX.7.021050
arXiv:1611.09301

[10] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, and Jay M. Gambetta, ``Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets'' Nature 549, 242–246 (2017).
https:/​/​doi.org/​10.1038/​nature23879
arXiv:1704.05018

[11] Abhinav Kandala, Kristan Temme, Antonio D. Corcoles, Antonio Mezzacapo, Jerry M. Chow, and Jay M. Gambetta, ``Extending the computational reach of a noisy superconducting quantum processor'' Nature 567, 491 (2019).
https:/​/​doi.org/​https:/​/​doi.org/​10.1038/​s41586-019-1040-7
arXiv:1805.04492

[12] Kristan Temme, Sergey Bravyi, and Jay M. Gambetta, ``Error Mitigation for Short-Depth Quantum Circuits'' Phys. Rev. Lett. 119, 180509 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.180509
arXiv:1612.02058

[13] Suguru Endo, Simon C. Benjamin, and Ying Li, ``Practical Quantum Error Mitigation for Near-Future Applications'' Physical Review X 8, 031027 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.031027
arXiv:1712.09271

[14] Vickram N. Premakumarand Robert Joynt ``Error Mitigation in Quantum Computers subject to Spatially Correlated Noise'' arXiv e-prints arXiv:1812.07076 (2018).
arXiv:1812.07076

[15] X. Bonet-Monroig, R. Sagastizabal, M. Singh, and T. E. O'Brien, ``Low-cost error mitigation by symmetry verification'' Phys. Rev. A 98, 062339 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.062339
arXiv:1807.10050

[16] Joshua Combes, Christopher Granade, Christopher Ferrie, and Steven T. Flammia, ``Logical Randomized Benchmarking'' arXiv e-prints (2017).
arXiv:1702.03688

[17] Mingyu Sunand Michael R. Geller ``Efficient characterization of correlated SPAM errors'' arXiv e-prints (2018).
arXiv:1810.10523

[18] 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'' Nature Physics 5, 27 (2008).
https:/​/​doi.org/​10.1038/​nphys1133

[19] Lijian Zhang, Hendrik B. Coldenstrodt-Ronge, Animesh Datta, Graciana Puentes, Jeff S. Lundeen, Xian-Min Jin, Brian J. Smith, Martin B. Plenio, and Ian A. Walmsley, ``Mapping coherence in measurement via full quantum tomography of a hybrid optical detector'' Nature Photonics 6, 364 (2012).
https:/​/​doi.org/​10.1038/​nphoton.2012.107

[20] Lijian Zhang, Animesh Datta, Hendrik B. Coldenstrodt-Ronge, Xian-Min Jin, Jens Eisert, Martin B. Plenio, and Ian A. Walmsley, ``Recursive quantum detector tomography'' New Journal of Physics 14, 115005 (2012).
https:/​/​doi.org/​10.1088/​1367-2630/​14/​11/​115005
arXiv:1207.3501

[21] J. J. Renema, G. Frucci, Z. Zhou, F. Mattioli, A. Gaggero, R. Leoni, M. J. A. Dood, A. Fiore, and M. P. Exter, ``Modified detector tomography technique applied to a superconducting multiphoton nanodetector'' Opt. Express 20, 2806–2813 (2012).
https:/​/​doi.org/​10.1364/​OE.20.002806
http:/​/​www.opticsexpress.org/​abstract.cfm?URI=oe-20-3-2806

[22] J. Z. Blumoff, K. Chou, C. Shen, M. Reagor, C. Axline, R. T. Brierley, M. P. Silveri, C. Wang, B. Vlastakis, S. E. Nigg, L. Frunzio, M. H. Devoret, L. Jiang, S. M. Girvin, and R. J. Schoelkopf, ``Implementing and Characterizing Precise Multiqubit Measurements'' Phys. Rev. X 6, 031041 (2016).
https:/​/​doi.org/​10.1103/​PhysRevX.6.031041

[23] Jens Koch, Terri M. Yu, Jay Gambetta, A. A. Houck, D. I. Schuster, J. Majer, Alexandre Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, ``Charge-insensitive qubit design derived from the Cooper pair box'' Phys. Rev. A 76, 042319 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.76.042319

[24] Michał Oszmaniec, Filip B. Maciejewski, and Zbigniew Puchała, ``Simulating all quantum measurements using only projective measurements and postselection'' Physical Review A 100 (2019).
https:/​/​doi.org/​10.1103/​physreva.100.012351

[25] Lov K. Grover ``A fast quantum mechanical algorithm for database search'' arXiv e-prints quant–ph/​9605043 (1996).
arXiv:quant-ph/9605043

[26] E. Bernsteinand U. Vazirani ``Quantum complexity theory'' Proc. of the Twenty-Fifth Annual ACM Symposium on Theory of Computing (STOC ’93) 11–20 (1993).
https:/​/​doi.org/​DOI:10.1145/​167088.167097

[27] Asher Peres ``Quantum theory: Concepts and methods'' Springer Science & Business Media (2006).
https:/​/​doi.org/​https:/​/​doi.org/​10.1007/​0-306-47120-5

[28] Zdeněk Hradil, Jaroslav Řeháček, Jaromír Fiurášek, and Miroslav Ježek, ``3 Maximum-Likelihood Methods in Quantum Mechanics'' Springer Berlin Heidelberg (2004).
https:/​/​doi.org/​10.1007/​978-3-540-44481-7_3

[29] Jaromír Fiurášek ``Maximum-likelihood estimation of quantum measurement'' Physical Review A 64, 024102 (2001).
https:/​/​doi.org/​10.1103/​PhysRevA.64.024102
arXiv:quant-ph/0101027

[30] Aram W. Harrowand Ashley Montanaro ``Quantum computational supremacy'' Nature 549, 203–209 (2017).
https:/​/​doi.org/​10.1038/​nature23458
arXiv:1809.07442

[31] Hakop Pashayan, Stephen D. Bartlett, and David Gross, ``From estimation of quantum probabilities to simulation of quantum circuits'' Quantum 4, 223 (2020).
https:/​/​doi.org/​10.22331/​q-2020-01-13-223

[32] Miguel Navascuésand Sandu Popescu ``How Energy Conservation Limits Our Measurements'' Phys. Rev. Lett. 112, 140502 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.112.140502
arXiv:1211.2101

[33] Z. Puchała, Ł. Pawela, A. Krawiec, and R. Kukulski, ``Strategies for optimal single-shot discrimination of quantum measurements'' Phys. Rev. A 98, 042103 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.042103
arXiv:1804.05856

[34] Zbigniew Puchała, Łukasz Pawela, Aleksand ra Krawiec, Ryszard Kukulski, and Michał Oszmaniec, ``Multiple-shot and unambiguous discrimination of von Neumann measurements'' arXiv e-prints arXiv:1810.05122 (2018).
arXiv:1810.05122

[35] John Watrous ``The Theory of Quantum Information'' Cambridge University Press (2018).
https:/​/​doi.org/​10.1017/​9781316848142

[36] Erkka Haapasalo, Teiko Heinosaari, and Juha-Pekka Pellonpaa, ``Quantum measurements on finite dimensional systems: relabeling and mixing'' Quant. Inf. Process. 11, 1751–1763 (2012).
https:/​/​doi.org/​10.1007/​s11128-011-0330-2

[37] M. Oszmaniec, L. Guerini, P. Wittek, and A. Acín, ``Simulating Positive-Operator-Valued Measures with Projective Measurements'' Physical Review Letters 119, 190501 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.190501
arXiv:1609.06139

[38] Leonardo Guerini, Jessica Bavaresco, Marcelo Terra Cunha, and Antonio Acín, ``Operational framework for quantum measurement simulability'' Journal of Mathematical Physics 58, 092102 (2017).
https:/​/​doi.org/​10.1063/​1.4994303
arXiv:1705.06343

[39] 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'' Physical Review A 77, 012307 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.77.012307
arXiv:0707.0963

[40] 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, 240504 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.109.240504
arXiv:1204.6308

[41] Madalin Guta, Jonas Kahn, Richard Kueng, and Joel A. Tropp, ``Fast state tomography with optimal error bounds'' arXiv e-prints arXiv:1809.11162 (2018).
arXiv:1809.11162

[42] T. Weissman, E. Ordentlich, G. Seroussi, S. Verdul, and M. J. Weinberger, ``Inequalities for the L1 Deviation of the Empirical Distribution'' Technical Report HPL-2003-97R1, Hewlett-Packard Labs (2003).

[43] M. S. Andersen, J. Dahl, and L. Vandenberghe, ``CVXOPT: A Python package for convex optimization, version 1.2'' (2019).
https:/​/​cvxopt.org/​

[44] IBM ``Qiskit Github repository'' (Access: 2019.07.09).

[45] Robin Blume-Kohout, John King Gamble, Erik Nielsen, Jonathan Mizrahi, Jonathan D. Sterk, and Peter Maunz, ``Robust, self-consistent, closed-form tomography of quantum logic gates on a trapped ion qubit'' arXiv e-prints arXiv:1310.4492 (2013).
arXiv:1310.4492

[46] Seth T. Merkel, Jay M. Gambetta, John A. Smolin, Stefano Poletto, Antonio D. Córcoles, Blake R. Johnson, Colm A. Ryan, and Matthias Steffen, ``Self-consistent quantum process tomography'' Phys. Rev. A 87, 062119 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.87.062119
arXiv:1211.0322

[47] Michael D. Mazurek, Matthew F. Pusey, Kevin J. Resch, and Robert W. Spekkens, ``Experimentally bounding deviations from quantum theory in the landscape of generalized probabilistic theories'' arXiv e-prints arXiv:1710.05948 (2017).
arXiv:1710.05948

[48] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann, ``A Quantum Approximate Optimization Algorithm'' arXiv e-prints arXiv:1411.4028 (2014).
arXiv:1411.4028

[49] John A. Smolin, Jay M. Gambetta, and Graeme Smith, ``Maximum Likelihood, Minimum Effort'' Phys. Rev. Lett 108, 070502 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.108.070502
arXiv:1106.5458

[50] Benjamin Schumacher ``Sending quantum entanglement through noisy channels'' arXiv e-prints quant–ph/​9604023 (1996).
arXiv:quant-ph/9604023

[51] Pawel Horodecki, Michal Horodecki, and Ryszard Horodecki, ``General teleportation channel, singlet fraction and quasi-distillation'' arXiv e-prints quant–ph/​9807091 (1998).
arXiv:quant-ph/9807091

[52] Anthony Chefles ``Unambiguous discrimination between linearly independent quantum states'' Physics Letters A 239, 339–347 (1998).
https:/​/​doi.org/​10.1016/​S0375-9601(98)00064-4
arXiv:quant-ph/9807022

[53] Stephen M. Barnettand Sarah Croke ``Quantum state discrimination'' Advances in Optics and Photonics 1, 238 (2009).
https:/​/​doi.org/​10.1364/​AOP.1.000238
arXiv:0810.1970

[54] Joseph M. Renes, Robin Blume-Kohout, A. J. Scott, and Carlton M. Caves, ``Symmetric informationally complete quantum measurements'' Journal of Mathematical Physics 45, 2171–2180 (2004).
https:/​/​doi.org/​10.1063/​1.1737053
arXiv:quant-ph/0310075

[55] Satoshi Ishizakaand Tohya Hiroshima ``Asymptotic Teleportation Scheme as a Universal Programmable Quantum Processor'' Phys. Rev. Lett. 101, 240501 (2008).
https:/​/​doi.org/​10.1103/​PhysRevLett.101.240501
arXiv:0807.4568

[56] Andrew M. Childsand Wim Dam ``Quantum algorithms for algebraic problems'' Rev. Mod. Phys. 82, 1–52 (2010).
https:/​/​doi.org/​10.1103/​RevModPhys.82.1

[57] John Preskill ``Quantum computing and the entanglement frontier'' arXiv e-prints (2012).
arXiv:1203.5813

[58] Patrick J. Coles, Stephan Eidenbenz, Scott Pakin, Adetokunbo Adedoyin, John Ambrosiano, Petr Anisimov, William Casper, Gopinath Chennupati, Carleton Coffrin, Hristo Djidjev, David Gunter, Satish Karra, Nathan Lemons, Shizeng Lin, Andrey Lokhov, Alexander Malyzhenkov, David Mascarenas, Susan Mniszewski, Balu Nadiga, Dan O'Malley, Diane Oyen, Lakshman Prasad, Randy Roberts, Phil Romero, Nandakishore Santhi, Nikolai Sinitsyn, Pieter Swart, Marc Vuffray, Jim Wendelberger, Boram Yoon, Richard Zamora, and Wei Zhu, ``Quantum Algorithm Implementations for Beginners'' arXiv e-prints (2018).
arXiv:1804.03719

[59] Additional experimental data, such as the exact form of the implemented and reconstructed operators, is accessible online in the github repository – https:/​/​github.com/​fbm2718/​mitigation_paper2019. For the Python code implementing the ideas from the work, see GitHub repository: https:/​/​github.com/​fbm2718/​QREM.

[60] Nikolaj Moll, Panagiotis Barkoutsos, Lev S. Bishop, Jerry M. Chow, Andrew Cross, Daniel J. Egger, Stefan Filipp, Andreas Fuhrer, Jay M. Gambetta, and Marc Ganzhorn, ``Quantum optimization using variational algorithms on near-term quantum devices'' Quantum Science and Technology 3, 030503 (2018).
https:/​/​doi.org/​10.1088/​2058-9565/​aab822
arXiv:1710.01022

[61] Jinzhao Wang, Volkher B. Scholz, and Renato Renner, ``Confidence Polytopes in Quantum State Tomography'' Phys. Rev. Lett. 122, 190401 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.190401
arXiv:1808.09988

[62] Yanzhu Chen, Maziar Farahzad, Shinjae Yoo, and Tzu-Chieh Wei, ``Detector tomography on IBM quantum computers and mitigation of an imperfect measurement'' Physical Review A 100 (2019).
https:/​/​doi.org/​10.1103/​physreva.100.052315

Cited by

[1] Benjamin Nachman, Miroslav Urbanek, Wibe A. de Jong, and Christian W. Bauer, "Unfolding quantum computer readout noise", arXiv:1910.01969, npj Quantum Information 6 1, 84 (2020).

[2] Chufan Lyu, Victor Montenegro, and Abolfazl Bayat, "Accelerated variational algorithms for digital quantum simulation of many-body ground states", Quantum 4, 324 (2020).

[3] Frank Leymann and Johanna Barzen, "The bitter truth about gate-based quantum algorithms in the NISQ era", Quantum Science and Technology 5 4, 044007 (2020).

[4] Mohan Sarovar, Timothy Proctor, Kenneth Rudinger, Kevin Young, Erik Nielsen, and Robin Blume-Kohout, "Detecting crosstalk errors in quantum information processors", Quantum 4, 321 (2020).

[5] E. O. Kiktenko, A. O. Malyshev, A. S. Mastiukova, V. I. Man'ko, A. K. Fedorov, and D. Chruściński, "Probability representation of quantum dynamics using pseudostochastic maps", Physical Review A 101 5, 052320 (2020).

[6] Sergey Bravyi, Sarah Sheldon, Abhinav Kandala, David C. Mckay, and Jay M. Gambetta, "Mitigating measurement errors in multi-qubit experiments", arXiv:2006.14044.

[7] Pranav Gokhale, Ali Javadi-Abhari, Nathan Earnest, Yunong Shi, and Frederic T. Chong, "Optimized Quantum Compilation for Near-Term Algorithms with OpenPulse", arXiv:2004.11205.

[8] Michael R. Geller and Mingyu Sun, "Efficient correction of multiqubit measurement errors", arXiv:2001.09980.

[9] Hyeokjea Kwon and Joonwoo Bae, "A hybrid quantum-classical approach to mitigating measurement errors", arXiv:2003.12314.

[10] Ashley Montanaro and Stasja Stanisic, "Compressed variational quantum eigensolver for the Fermi-Hubbard model", arXiv:2006.01179.

[11] Megan N. Lilly and Travis S. Humble, "Modeling Noisy Quantum Circuits Using Experimental Characterization", arXiv:2001.08653.

[12] Rebecca Hicks, Christian W. Bauer, and Benjamin Nachman, "Readout Rebalancing for Near Term Quantum Computers", arXiv:2010.07496.

The above citations are from Crossref's cited-by service (last updated successfully 2020-10-20 11:11:36) and SAO/NASA ADS (last updated successfully 2020-10-20 11:11:37). The list may be incomplete as not all publishers provide suitable and complete citation data.