Classical Shadows With Noise

Dax Enshan Koh1,2 and Sabee Grewal2,3

1Institute of High Performance Computing, Agency for Science, Technology and Research (A*STAR), 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Singapore
2Zapata Computing, Inc., 100 Federal Street, 20th Floor, Boston, Massachusetts 02110, USA
3Department of Computer Science, The University of Texas at Austin, Austin, TX 78712, USA

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

Abstract

The classical shadows protocol, recently introduced by Huang, Kueng, and Preskill [Nat. Phys. 16, 1050 (2020)], is a quantum-classical protocol to estimate properties of an unknown quantum state. Unlike full quantum state tomography, the protocol can be implemented on near-term quantum hardware and requires few quantum measurements to make many predictions with a high success probability.

In this paper, we study the effects of noise on the classical shadows protocol. In particular, we consider the scenario in which the quantum circuits involved in the protocol are subject to various known noise channels and derive an analytical upper bound for the sample complexity in terms of a shadow seminorm for both local and global noise. Additionally, by modifying the classical post-processing step of the noiseless protocol, we define a new estimator that remains unbiased in the presence of noise. As applications, we show that our results can be used to prove rigorous sample complexity upper bounds in the cases of depolarizing noise and amplitude damping.

► BibTeX data

► References

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

[2] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik. Noisy intermediate-scale quantum algorithms. Rev. Mod. Phys., 94:015004, Feb 2022. doi:10.1103/​RevModPhys.94.015004.
https:/​/​doi.org/​10.1103/​RevModPhys.94.015004

[3] Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, et al. Variational quantum algorithms. Nature Reviews Physics, 3(9):625–644, 2021. doi:10.1038/​s42254-021-00348-9.
https:/​/​doi.org/​10.1038/​s42254-021-00348-9

[4] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O’Brien. A variational eigenvalue solver on a photonic quantum processor. Nature communications, 5:4213, 2014. doi:10.1038/​ncomms5213.
https:/​/​doi.org/​10.1038/​ncomms5213

[5] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A Quantum Approximate Optimization Algorithm. arXiv preprint arXiv:1411.4028, 2014. doi:10.48550/​arXiv.1411.4028.
https:/​/​doi.org/​10.48550/​arXiv.1411.4028
arXiv:1411.4028

[6] Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, Mária Kieferová, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P.D. Sawaya, et al. Quantum Chemistry in the Age of Quantum Computing. Chemical reviews, 119(19):10856–10915, 2019. doi:10.1021/​acs.chemrev.8b00803.
https:/​/​doi.org/​10.1021/​acs.chemrev.8b00803

[7] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum metrology. Physical review letters, 96(1):010401, 2006. doi:10.1103/​PhysRevLett.96.010401.
https:/​/​doi.org/​10.1103/​PhysRevLett.96.010401

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

[9] Dave Wecker, Matthew B. Hastings, and Matthias Troyer. Progress towards practical quantum variational algorithms. Physical Review A, 92(4):042303, 2015. doi:10.1103/​PhysRevA.92.042303.
https:/​/​doi.org/​10.1103/​PhysRevA.92.042303

[10] William J. Huggins, Jarrod R. McClean, Nicholas C. Rubin, Zhang Jiang, Nathan Wiebe, K. Birgitta Whaley, and Ryan Babbush. Efficient and noise resilient measurements for quantum chemistry on near-term quantum computers. npj Quantum Information, 7(1):1–9, 2021. doi:10.1038/​s41534-020-00341-7.
https:/​/​doi.org/​10.1038/​s41534-020-00341-7

[11] 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, 2020. doi:10.1038/​s41567-020-0932-7.
https:/​/​doi.org/​10.1038/​s41567-020-0932-7

[12] Jeongwan Haah, Aram Harrow, Zhengfeng Ji, Xiaodi Wu, and Nengkun Yu. Sample-Optimal Tomography of Quantum States. IEEE Transactions on Information Theory, 63(9):5628–5641, 2017. doi:10.1109/​TIT.2017.2719044.
https:/​/​doi.org/​10.1109/​TIT.2017.2719044

[13] Ryan O'Donnell and John Wright. Efficient quantum tomography. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, pages 899–912, 2016. doi:10.1145/​2897518.2897544.
https:/​/​doi.org/​10.1145/​2897518.2897544

[14] Scott Aaronson. Shadow Tomography of Quantum States. SIAM Journal on Computing, 49(5):STOC18–368, 2019. doi:10.1137/​18M120275X.
https:/​/​doi.org/​10.1137/​18M120275X

[15] Mark R. Jerrum, Leslie G. Valiant, and Vijay V. Vazirani. Random Generation of Combinatorial Structures from a Uniform Distribution. Theoretical Computer Science, 43:169–188, 1986. doi:10.1016/​0304-3975(86)90174-X.
https:/​/​doi.org/​10.1016/​0304-3975(86)90174-X

[16] Huangjun Zhu, Richard Kueng, Markus Grassl, and David Gross. The Clifford group fails gracefully to be a unitary 4-design. arXiv preprint arXiv:1609.08172, 2016. doi:10.48550/​arXiv.1609.08172.
https:/​/​doi.org/​10.48550/​arXiv.1609.08172
arXiv:1609.08172

[17] Zak Webb. The Clifford group forms a unitary 3-design. Quantum Information & Computation, 16(15&16):1379–1400, 2016. doi:10.26421/​QIC16.15-16-8.
https:/​/​doi.org/​10.26421/​QIC16.15-16-8

[18] Senrui Chen, Wenjun Yu, Pei Zeng, and Steven T. Flammia. Robust Shadow Estimation. PRX Quantum, 2:030348, Sep 2021. doi:10.1103/​PRXQuantum.2.030348.
https:/​/​doi.org/​10.1103/​PRXQuantum.2.030348

[19] Steven T. Flammia and Joel J. Wallman. Efficient Estimation of Pauli Channels. ACM Transactions on Quantum Computing, 1(1):1–32, 2020. doi:10.1145/​3408039.
https:/​/​doi.org/​10.1145/​3408039

[20] Senrui Chen, Sisi Zhou, Alireza Seif, and Liang Jiang. Quantum advantages for Pauli channel estimation. Physical Review A, 105(3):032435, 2022. doi:10.1103/​PhysRevA.105.032435.
https:/​/​doi.org/​10.1103/​PhysRevA.105.032435

[21] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2010. doi:10.1017/​CBO9780511976667.
https:/​/​doi.org/​10.1017/​CBO9780511976667

[22] Zdenek Hradil. Quantum-state estimation. Physical Review A, 55(3):R1561, 1997. doi:10.1103/​PhysRevA.55.R1561.
https:/​/​doi.org/​10.1103/​PhysRevA.55.R1561

[23] Matteo Paris and Jaroslav Rehacek. Quantum State Estimation, volume 649. Springer Science & Business Media, 2004. doi:10.1007/​b98673.
https:/​/​doi.org/​10.1007/​b98673

[24] Robin Blume-Kohout. Optimal, reliable estimation of quantum states. New Journal of Physics, 12(4):043034, apr 2010. doi:10.1088/​1367-2630/​12/​4/​043034.
https:/​/​doi.org/​10.1088/​1367-2630/​12/​4/​043034

[25] K. Banaszek, M. Cramer, and D. Gross. Focus on quantum tomography. New Journal of Physics, 15(12):125020, dec 2013. doi:10.1088/​1367-2630/​15/​12/​125020.
https:/​/​doi.org/​10.1088/​1367-2630/​15/​12/​125020

[26] David Gross, Yi-Kai Liu, Steven T. Flammia, Stephen Becker, and Jens Eisert. Quantum State Tomography via Compressed Sensing. Phys. Rev. Lett., 105:150401, Oct 2010. doi:10.1103/​PhysRevLett.105.150401.
https:/​/​doi.org/​10.1103/​PhysRevLett.105.150401

[27] 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. doi:10.1088/​1367-2630/​14/​9/​095022.
https:/​/​doi.org/​10.1088/​1367-2630/​14/​9/​095022

[28] Takanori Sugiyama, Peter S. Turner, and Mio Murao. Precision-Guaranteed Quantum Tomography. Phys. Rev. Lett., 111:160406, Oct 2013. doi:10.1103/​PhysRevLett.111.160406.
https:/​/​doi.org/​10.1103/​PhysRevLett.111.160406

[29] Richard Kueng, Huangjun Zhu, and David Gross. Low rank matrix recovery from Clifford orbits. arXiv preprint arXiv:1610.08070, 2016. doi:10.48550/​arXiv.1610.08070.
https:/​/​doi.org/​10.48550/​arXiv.1610.08070
arXiv:1610.08070

[30] Richard Kueng, Holger Rauhut, and Ulrich Terstiege. Low rank matrix recovery from rank one measurements. Applied and Computational Harmonic Analysis, 42(1):88–116, 2017. doi:10.1016/​j.acha.2015.07.007.
https:/​/​doi.org/​10.1016/​j.acha.2015.07.007

[31] M Guţă, J. Kahn, R. Kueng, and J. A. Tropp. Fast state tomography with optimal error bounds. Journal of Physics A: Mathematical and Theoretical, 53(20):204001, apr 2020. doi:10.1088/​1751-8121/​ab8111.
https:/​/​doi.org/​10.1088/​1751-8121/​ab8111

[32] Marcus Cramer, Martin B. Plenio, Steven T. Flammia, Rolando Somma, David Gross, Stephen D. Bartlett, Olivier Landon-Cardinal, David Poulin, and Yi-Kai Liu. Efficient quantum state tomography. Nature communications, 1(1):1–7, 2010. doi: 10.1038/​ncomms1147.
https:/​/​doi.org/​10.1038/​ncomms1147

[33] B.P. Lanyon, C. Maier, Milan Holzäpfel, Tillmann Baumgratz, C Hempel, P Jurcevic, Ish Dhand, A.S. Buyskikh, A.J. Daley, Marcus Cramer, et al. Efficient tomography of a quantum many-body system. Nature Physics, 13(12):1158–1162, 2017. doi:10.1038/​nphys4244.
https:/​/​doi.org/​10.1038/​nphys4244

[34] Olivier Landon-Cardinal and David Poulin. Practical learning method for multi-scale entangled states. New Journal of Physics, 14(8):085004, aug 2012. doi:10.1088/​1367-2630/​14/​8/​085004.
https:/​/​doi.org/​10.1088/​1367-2630/​14/​8/​085004

[35] Juan Carrasquilla, Giacomo Torlai, Roger G. Melko, and Leandro Aolita. Reconstructing quantum states with generative models. Nature Machine Intelligence, 1(3):155–161, 2019. doi:10.1038/​s42256-019-0028-1.
https:/​/​doi.org/​10.1038/​s42256-019-0028-1

[36] Xun Gao and Lu-Ming Duan. Efficient representation of quantum many-body states with deep neural networks. Nature communications, 8(1):1–6, 2017. doi:10.1038/​s41467-017-00705-2.
https:/​/​doi.org/​10.1038/​s41467-017-00705-2

[37] Jordan Cotler and Frank Wilczek. Quantum overlapping tomography. Phys. Rev. Lett., 124:100401, Mar 2020. doi:10.1103/​PhysRevLett.124.100401.
https:/​/​doi.org/​10.1103/​PhysRevLett.124.100401

[38] Scott Aaronson and Guy N. Rothblum. Gentle Measurement of Quantum States and Differential Privacy. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 322–333, 2019. doi:10.1145/​3313276.3316378.
https:/​/​doi.org/​10.1145/​3313276.3316378

[39] Costin Bădescu and Ryan O'Donnell. Improved Quantum Data Analysis. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1398–1411, 2021. doi:10.1145/​3406325.3451109.
https:/​/​doi.org/​10.1145/​3406325.3451109

[40] 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(7671):242–246, 2017. doi:10.1038/​nature23879.
https:/​/​doi.org/​10.1038/​nature23879

[41] Vladyslav Verteletskyi, Tzu-Ching Yen, and Artur F. Izmaylov. Measurement optimization in the variational quantum eigensolver using a minimum clique cover. The Journal of Chemical Physics, 152(12):124114, 2020. doi:10.1063/​1.5141458.
https:/​/​doi.org/​10.1063/​1.5141458

[42] Artur F. Izmaylov, Tzu-Ching Yen, Robert A. Lang, and Vladyslav Verteletskyi. Unitary partitioning approach to the measurement problem in the variational quantum eigensolver method. Journal of Chemical Theory and Computation, 16(1):190–195, 2019. doi:10.1021/​acs.jctc.9b00791.
https:/​/​doi.org/​10.1021/​acs.jctc.9b00791

[43] Andrew Zhao, Andrew Tranter, William M. Kirby, Shu Fay Ung, Akimasa Miyake, and Peter J. Love. Measurement reduction in variational quantum algorithms. Physical Review A, 101(6):062322, 2020. doi:10.1103/​PhysRevA.101.062322.
https:/​/​doi.org/​10.1103/​PhysRevA.101.062322

[44] Guoming Wang, Dax Enshan Koh, Peter D. Johnson, and Yudong Cao. Minimizing Estimation Runtime on Noisy Quantum Computers. PRX Quantum, 2:010346, Mar 2021. doi:10.1103/​PRXQuantum.2.010346.
https:/​/​doi.org/​10.1103/​PRXQuantum.2.010346

[45] Dax Enshan Koh, Guoming Wang, Peter D. Johnson, and Yudong Cao. Foundations for Bayesian inference with engineered likelihood functions for robust amplitude estimation. Journal of Mathematical Physics, 63:052202, 2022. doi:10.1063/​5.0042433.
https:/​/​doi.org/​10.1063/​5.0042433

[46] Jérôme F. Gonthier, Maxwell D. Radin, Corneliu Buda, Eric J. Doskocil, Clena M. Abuan, and Jhonathan Romero. Identifying challenges towards practical quantum advantage through resource estimation: the measurement roadblock in the variational quantum eigensolver. arXiv preprint arXiv:2012.04001, 2020. doi:10.48550/​arXiv.2012.04001.
https:/​/​doi.org/​10.48550/​arXiv.2012.04001
arXiv:2012.04001

[47] Andrew Zhao, Nicholas C. Rubin, and Akimasa Miyake. Fermionic Partial Tomography via Classical Shadows. Phys. Rev. Lett., 127:110504, Sep 2021. doi:10.1103/​PhysRevLett.127.110504.
https:/​/​doi.org/​10.1103/​PhysRevLett.127.110504

[48] Kianna Wan, William J. Huggins, Joonho Lee, and Ryan Babbush. Matchgate Shadows for Fermionic Quantum Simulation. arXiv preprint arXiv:2207.13723, 2022. doi:10.48550/​arXiv.2207.13723.
https:/​/​doi.org/​10.48550/​arXiv.2207.13723
arXiv:2207.13723

[49] Bryan O'Gorman. Fermionic tomography and learning. arXiv preprint arXiv:2207.14787, 2022. doi:10.48550/​arXiv.2207.14787.
https:/​/​doi.org/​10.48550/​arXiv.2207.14787
arXiv:2207.14787

[50] Charles Hadfield, Sergey Bravyi, Rudy Raymond, and Antonio Mezzacapo. Measurements of Quantum Hamiltonians with Locally-Biased Classical Shadows. Communications in Mathematical Physics, 391(3):951–967, 2022. doi:10.1007/​s00220-022-04343-8.
https:/​/​doi.org/​10.1007/​s00220-022-04343-8

[51] Andreas Elben, Richard Kueng, Hsin-Yuan Robert Huang, Rick van Bijnen, Christian Kokail, Marcello Dalmonte, Pasquale Calabrese, Barbara Kraus, John Preskill, Peter Zoller, et al. Mixed-State Entanglement from Local Randomized Measurements. Physical Review Letters, 125(20):200501, 2020. doi:10.1103/​PhysRevLett.125.200501.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.200501

[52] G.I. Struchalin, Ya. A. Zagorovskii, E.V. Kovlakov, S.S. Straupe, and S.P. Kulik. Experimental Estimation of Quantum State Properties from Classical Shadows. PRX Quantum, 2:010307, Jan 2021. doi:10.1103/​PRXQuantum.2.010307.
https:/​/​doi.org/​10.1103/​PRXQuantum.2.010307

[53] Dax Enshan Koh and Sabee Grewal. Classical shadows with noise. arXiv preprint arXiv:2011.11580v1, 2020.
arXiv:2011.11580v1

[54] Robin Harper, Steven T. Flammia, and Joel J. Wallman. Efficient learning of quantum noise. Nature Physics, 16(12):1184–1188, 2020. doi:10.1038/​s41567-020-0992-8.
https:/​/​doi.org/​10.1038/​s41567-020-0992-8

[55] Guangxi Li, Zhixin Song, and Xin Wang. VSQL: Variational shadow quantum learning for classification. Proceedings of the AAAI Conference on Artificial Intelligence, 35(9):8357–8365, May 2021.

[56] Joseph M. Lukens, Kody J. H. Law, and Ryan S. Bennink. A Bayesian analysis of classical shadows. npj Quantum Inf., 7(113):1–10, Jul 2021. doi:10.1038/​s41534-021-00447-6.
https:/​/​doi.org/​10.1038/​s41534-021-00447-6

[57] Roy J. Garcia, You Zhou, and Arthur Jaffe. Quantum scrambling with classical shadows. Phys. Rev. Research, 3:033155, Aug 2021. doi:10.1103/​PhysRevResearch.3.033155.
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.033155

[58] Hong-Ye Hu and Yi-Zhuang You. Hamiltonian-driven shadow tomography of quantum states. Phys. Rev. Research, 4:013054, Jan 2022. doi:10.1103/​PhysRevResearch.4.013054.
https:/​/​doi.org/​10.1103/​PhysRevResearch.4.013054

[59] Antoine Neven, Jose Carrasco, Vittorio Vitale, Christian Kokail, Andreas Elben, Marcello Dalmonte, Pasquale Calabrese, Peter Zoller, Benoı̂t Vermersch, Richard Kueng, et al. Symmetry-resolved entanglement detection using partial transpose moments. npj Quantum Inf., 7(152):1–12, Oct 2021. doi:10.1038/​s41534-021-00487-y.
https:/​/​doi.org/​10.1038/​s41534-021-00487-y

[60] Hsin-Yuan Huang, Richard Kueng, and John Preskill. Efficient estimation of Pauli observables by derandomization. Phys. Rev. Lett., 127:030503, Jul 2021. doi:10.1103/​PhysRevLett.127.030503.
https:/​/​doi.org/​10.1103/​PhysRevLett.127.030503

[61] Atithi Acharya, Siddhartha Saha, and Anirvan M. Sengupta. Shadow tomography based on informationally complete positive operator-valued measure. Phys. Rev. A, 104:052418, Nov 2021. doi:10.1103/​PhysRevA.104.052418.
https:/​/​doi.org/​10.1103/​PhysRevA.104.052418

[62] Stefan Hillmich, Charles Hadfield, Rudy Raymond, Antonio Mezzacapo, and Robert Wille. Decision Diagrams for Quantum Measurements with Shallow Circuits. In 2021 IEEE International Conference on Quantum Computing and Engineering (QCE), pages 24–34. IEEE, 2021. doi:10.1109/​QCE52317.2021.00018.
https:/​/​doi.org/​10.1109/​QCE52317.2021.00018

[63] Charles Hadfield. Adaptive Pauli Shadows for Energy Estimation. arXiv preprint arXiv:2105.12207, 2021. doi:10.48550/​arXiv.2105.12207.
https:/​/​doi.org/​10.48550/​arXiv.2105.12207
arXiv:2105.12207

[64] Bujiao Wu, Jinzhao Sun, Qi Huang, and Xiao Yuan. Overlapped grouping measurement: A unified framework for measuring quantum states. arXiv preprint arXiv:2105.13091, 2021. doi:10.48550/​arXiv.2105.13091.
https:/​/​doi.org/​10.48550/​arXiv.2105.13091
arXiv:2105.13091

[65] Aniket Rath, Cyril Branciard, Anna Minguzzi, and Benoı̂t Vermersch. Quantum Fisher information from randomized measurements. Phys. Rev. Lett., 127:260501, Dec 2021. doi:10.1103/​PhysRevLett.127.260501.
https:/​/​doi.org/​10.1103/​PhysRevLett.127.260501

[66] Ting Zhang, Jinzhao Sun, Xiao-Xu Fang, Xiao-Ming Zhang, Xiao Yuan, and He Lu. Experimental quantum state measurement with classical shadows. Phys. Rev. Lett., 127:200501, Nov 2021. doi:10.1103/​PhysRevLett.127.200501.
https:/​/​doi.org/​10.1103/​PhysRevLett.127.200501

[67] Hsin-Yuan Huang, Richard Kueng, Giacomo Torlai, Victor V. Albert, and John Preskill. Provably efficient machine learning for quantum many-body problems. arXiv preprint arXiv:2106.12627, 2021. doi:10.48550/​arXiv.2106.12627.
https:/​/​doi.org/​10.48550/​arXiv.2106.12627
arXiv:2106.12627

[68] William J. Huggins, Bryan A. O'Gorman, Nicholas C. Rubin, David R. Reichman, Ryan Babbush, and Joonho Lee. Unbiasing fermionic quantum Monte Carlo with a quantum computer. Nature, 603(7901):416–420, Mar 2022. doi:10.1038/​s41586-021-04351-z.
https:/​/​doi.org/​10.1038/​s41586-021-04351-z

[69] Hong-Ye Hu, Soonwon Choi, and Yi-Zhuang You. Classical Shadow Tomography with Locally Scrambled Quantum Dynamics. arXiv preprint arXiv:2107.04817, 2021. doi:10.48550/​arXiv.2107.04817.
https:/​/​doi.org/​10.48550/​arXiv.2107.04817
arXiv:2107.04817

[70] Steven T. Flammia. Averaged circuit eigenvalue sampling. arXiv preprint arXiv:2108.05803, 2021. doi:10.48550/​arXiv.2108.05803.
https:/​/​doi.org/​10.48550/​arXiv.2108.05803
arXiv:2108.05803

[71] Ryan Levy, Di Luo, and Bryan K. Clark. Classical Shadows for Quantum Process Tomography on Near-term Quantum Computers. arXiv preprint arXiv:2110.02965, 2021. doi:10.48550/​arXiv.2110.02965.
https:/​/​doi.org/​10.48550/​arXiv.2110.02965
arXiv:2110.02965

[72] Jonathan Kunjummen, Minh C. Tran, Daniel Carney, and Jacob M. Taylor. Shadow process tomography of quantum channels. arXiv preprint arXiv:2110.03629, 2021. doi:10.48550/​arXiv.2110.03629.
https:/​/​doi.org/​10.48550/​arXiv.2110.03629
arXiv:2110.03629

[73] Jonas Helsen, Marios Ioannou, Ingo Roth, Jonas Kitzinger, Emilio Onorati, Albert H. Werner, and Jens Eisert. Estimating gate-set properties from random sequences. arXiv preprint arXiv:2110.13178, 2021. doi:10.48550/​arXiv.2110.13178.
https:/​/​doi.org/​10.48550/​arXiv.2110.13178
arXiv:2110.13178

[74] Sitan Chen, Jordan Cotler, Hsin-Yuan Huang, and Jerry Li. Exponential Separations Between Learning With and Without Quantum Memory. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 574–585, 2022. doi:10.1109/​FOCS52979.2021.00063.
https:/​/​doi.org/​10.1109/​FOCS52979.2021.00063

[75] Simone Notarnicola, Andreas Elben, Thierry Lahaye, Antoine Browaeys, Simone Montangero, and Benoit Vermersch. A randomized measurement toolbox for Rydberg quantum technologies. arXiv preprint arXiv:2112.11046, 2021. doi:10.48550/​arXiv.2112.11046.
https:/​/​doi.org/​10.48550/​arXiv.2112.11046
arXiv:2112.11046

[76] Stefan H. Sack, Raimel A. Medina, Alexios A. Michailidis, Richard Kueng, and Maksym Serbyn. Avoiding barren plateaus using classical shadows. PRX Quantum, 3:020365, Jun 2022. doi:10.1103/​PRXQuantum.3.020365.
https:/​/​doi.org/​10.1103/​PRXQuantum.3.020365

[77] Kaifeng Bu, Dax Enshan Koh, Roy J. Garcia, and Arthur Jaffe. Classical shadows with Pauli-invariant unitary ensembles. arXiv preprint arXiv:2202.03272, 2022. doi:10.48550/​arXiv.2202.03272.
https:/​/​doi.org/​10.48550/​arXiv.2202.03272
arXiv:2202.03272

[78] Max McGinley, Sebastian Leontica, Samuel J. Garratt, Jovan Jovanovic, and Steven H. Simon. Quantifying information scrambling via classical shadow tomography on programmable quantum simulators. arXiv preprint arXiv:2202.05132, 2022. doi:10.48550/​arXiv.2202.05132.
https:/​/​doi.org/​10.48550/​arXiv.2202.05132
arXiv:2202.05132

[79] Lu Liu, Ting Zhang, Xiao Yuan, and He Lu. Experimental Investigation of Quantum Uncertainty Relations With Classical Shadows. Frontiers in Physics, 10, 2022. doi:10.3389/​fphy.2022.873810.
https:/​/​doi.org/​10.3389/​fphy.2022.873810

[80] Joseph M. Lukens, Kody J. H. Law, and Ryan S. Bennink. Classical shadows and Bayesian mean estimation: a comparison. In Conference on Lasers and Electro-Optics, page FW3N.3. Optical Society of America, 2021. doi:10.1364/​CLEO_QELS.2021.FW3N.3.
https:/​/​doi.org/​10.1364/​CLEO_QELS.2021.FW3N.3

[81] Angus Lowe. Learning Quantum States Without Entangled Measurements. Master's thesis, University of Waterloo, 2021.

[82] Hsin-Yuan Huang. Learning quantum states from their classical shadows. Nat. Rev. Phys., 4(2):81, Feb 2022. doi:10.1038/​s42254-021-00411-5.
https:/​/​doi.org/​10.1038/​s42254-021-00411-5

[83] Hong-Ye Hu, Ryan LaRose, Yi-Zhuang You, Eleanor Rieffel, and Zhihui Wang. Logical shadow tomography: Efficient estimation of error-mitigated observables. arXiv preprint arXiv:2203.07263, 2022. doi:10.48550/​arXiv.2203.07263.
https:/​/​doi.org/​10.48550/​arXiv.2203.07263
arXiv:2203.07263

[84] Alireza Seif, Ze-Pei Cian, Sisi Zhou, Senrui Chen, and Liang Jiang. Shadow Distillation: Quantum Error Mitigation with Classical Shadows for Near-Term Quantum Processors. arXiv preprint arXiv:2203.07309, 2022. doi:10.48550/​arXiv.2203.07309.
https:/​/​doi.org/​10.48550/​arXiv.2203.07309
arXiv:2203.07309

[85] Andreas Elben, Steven T. Flammia, Hsin-Yuan Huang, Richard Kueng, John Preskill, Benoı̂t Vermersch, and Peter Zoller. The randomized measurement toolbox. arXiv preprint arXiv:2203.11374, 2022. doi:10.48550/​arXiv.2203.11374.
https:/​/​doi.org/​10.48550/​arXiv.2203.11374
arXiv:2203.11374

[86] Gregory Boyd and Bálint Koczor. Training variational quantum circuits with CoVaR: covariance root finding with classical shadows. arXiv preprint arXiv:2204.08494, 2022. doi:10.48550/​arXiv.2204.08494.
https:/​/​doi.org/​10.48550/​arXiv.2204.08494
arXiv:2204.08494

[87] H. Chau Nguyen, Jan Lennart Bönsel, Jonathan Steinberg, and Otfried Gühne. Optimising shadow tomography with generalised measurements. arXiv preprint arXiv:2205.08990, 2022. doi:10.48550/​arXiv.2205.08990.
https:/​/​doi.org/​10.48550/​arXiv.2205.08990
arXiv:2205.08990

[88] Luuk Coopmans, Yuta Kikuchi, and Marcello Benedetti. Predicting Gibbs State Expectation Values with Pure Thermal Shadows. arXiv preprint arXiv:2206.05302, 2022. doi:10.48550/​arXiv.2206.05302.
https:/​/​doi.org/​10.48550/​arXiv.2206.05302
arXiv:2206.05302

[89] Saumya Shivam, C. W. von Keyserlingk, and S. L. Sondhi. On Classical and Hybrid Shadows of Quantum States. arXiv preprint arXiv:2206.06616, 2022. doi:10.48550/​arXiv.2206.06616.
https:/​/​doi.org/​10.48550/​arXiv.2206.06616
arXiv:2206.06616

[90] Daniel McNulty, Filip B. Maciejewski, and Michał Oszmaniec. Estimating Quantum Hamiltonians via Joint Measurements of Noisy Non-Commuting Observables. arXiv preprint arXiv:2206.08912, 2022. doi:10.48550/​arXiv.2206.08912.
https:/​/​doi.org/​10.48550/​arXiv.2206.08912
arXiv:2206.08912

[91] Suguru Endo, Zhenyu Cai, Simon C. Benjamin, and Xiao Yuan. Hybrid quantum-classical algorithms and quantum error mitigation. Journal of the Physical Society of Japan, 90(3):032001, 2021. doi:10.7566/​JPSJ.90.032001.
https:/​/​doi.org/​10.7566/​JPSJ.90.032001

[92] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. Surface codes: Towards practical large-scale quantum computation. Physical Review A, 86(3):032324, 2012. doi:10.1103/​PhysRevA.86.032324.
https:/​/​doi.org/​10.1103/​PhysRevA.86.032324

[93] Earl T. Campbell, Barbara M. Terhal, and Christophe Vuillot. Roads towards fault-tolerant universal quantum computation. Nature, 549(7671):172–179, 2017. doi:10.1038/​nature23460.
https:/​/​doi.org/​10.1038/​nature23460

[94] Ying Li and Simon C. Benjamin. Efficient Variational Quantum Simulator Incorporating Active Error Minimization. Phys. Rev. X, 7:021050, Jun 2017. doi:10.1103/​PhysRevX.7.021050.
https:/​/​doi.org/​10.1103/​PhysRevX.7.021050

[95] Kristan Temme, Sergey Bravyi, and Jay M. Gambetta. Error Mitigation for Short-Depth Quantum Circuits. Phys. Rev. Lett., 119:180509, Nov 2017. doi:10.1103/​PhysRevLett.119.180509.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.180509

[96] Tudor Giurgica-Tiron, Yousef Hindy, Ryan LaRose, Andrea Mari, and William J. Zeng. Digital zero noise extrapolation for quantum error mitigation. In 2020 IEEE International Conference on Quantum Computing and Engineering (QCE), pages 306–316, 2020. doi:10.1109/​QCE49297.2020.00045.
https:/​/​doi.org/​10.1109/​QCE49297.2020.00045

[97] Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, and Lukasz Cincio. Error mitigation with Clifford quantum-circuit data. Quantum, 5:592, November 2021. doi:10.22331/​q-2021-11-26-592.
https:/​/​doi.org/​10.22331/​q-2021-11-26-592

[98] Jarrod R. McClean, Mollie E. Kimchi-Schwartz, Jonathan Carter, and Wibe A. de Jong. Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states. Phys. Rev. A, 95:042308, Apr 2017. doi:10.1103/​PhysRevA.95.042308.
https:/​/​doi.org/​10.1103/​PhysRevA.95.042308

[99] Suguru Endo, Simon C. Benjamin, and Ying Li. Practical quantum error mitigation for near-future applications. Phys. Rev. X, 8:031027, Jul 2018. doi:10.1103/​PhysRevX.8.031027.
https:/​/​doi.org/​10.1103/​PhysRevX.8.031027

[100] John Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. doi:10.1017/​9781316848142.
https:/​/​doi.org/​10.1017/​9781316848142

[101] Sepehr Nezami and Michael Walter. Multipartite entanglement in stabilizer tensor networks. Phys. Rev. Lett., 125:241602, Dec 2020. doi:10.1103/​PhysRevLett.125.241602.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.241602

[102] Fernando G. S. L. Brandao and Michal Horodecki. Exponential Quantum Speed-ups are Generic. Quantum Inf. Comput., 13(11&12):901–924, 2013. doi:10.26421/​QIC13.11-12-1.
https:/​/​doi.org/​10.26421/​QIC13.11-12-1

[103] Adam Bouland, Joseph F. Fitzsimons, and Dax Enshan Koh. Complexity Classification of Conjugated Clifford Circuits. In Rocco A. Servedio, editor, 33rd Computational Complexity Conference (CCC 2018), volume 102 of Leibniz International Proceedings in Informatics (LIPIcs), pages 21:1–21:25, Dagstuhl, Germany, 2018. Schloss Dagstuhl–Leibniz-Zentrum für Informatik. doi:10.4230/​LIPIcs.CCC.2018.21.
https:/​/​doi.org/​10.4230/​LIPIcs.CCC.2018.21

[104] Rawad Mezher, Joe Ghalbouni, Joseph Dgheim, and Damian Markham. Efficient approximate unitary t-designs from partially invertible universal sets and their application to quantum speedup. arXiv preprint arXiv:1905.01504, 2019. doi:10.48550/​arXiv.1905.01504.
https:/​/​doi.org/​10.48550/​arXiv.1905.01504
arXiv:1905.01504

[105] Oleg Szehr, Frédéric Dupuis, Marco Tomamichel, and Renato Renner. Decoupling with unitary approximate two-designs. New Journal of Physics, 15(5):053022, 2013. doi:10.1088/​1367-2630/​15/​5/​053022.
https:/​/​doi.org/​10.1088/​1367-2630/​15/​5/​053022

[106] Andris Ambainis, Jan Bouda, and Andreas Winter. Nonmalleable encryption of quantum information. Journal of Mathematical Physics, 50(4):042106, 2009. doi:10.1063/​1.3094756.
https:/​/​doi.org/​10.1063/​1.3094756

[107] Huangjun Zhu. Multiqubit Clifford groups are unitary 3-designs. Physical Review A, 96(6):062336, 2017. doi:10.1103/​PhysRevA.96.062336.
https:/​/​doi.org/​10.1103/​PhysRevA.96.062336

[108] Joel J. Wallman. Randomized benchmarking with gate-dependent noise. Quantum, 2:47, January 2018. doi:10.22331/​q-2018-01-29-47.
https:/​/​doi.org/​10.22331/​q-2018-01-29-47

[109] Kevin Young, Stephen Bartlett, Robin J. Blume-Kohout, John King Gamble, Daniel Lobser, Peter Maunz, Erik Nielsen, Timothy James Proctor, Melissa Revelle, and Kenneth Michael Rudinger. Diagnosing and destroying non-Markovian noise. Technical report, Sandia National Lab. (SNL-CA), Livermore, CA (United States), 2020. doi:10.2172/​1671379.
https:/​/​doi.org/​10.2172/​1671379

[110] Tilo Eggeling and Reinhard F. Werner. Separability properties of tripartite states with $U\otimes U\otimes U$ symmetry. Physical Review A, 63(4):042111, 2001. doi:10.1103/​PhysRevA.63.042111.
https:/​/​doi.org/​10.1103/​PhysRevA.63.042111

[111] Peter D. Johnson and Lorenza Viola. Compatible quantum correlations: Extension problems for Werner and isotropic states. Physical Review A, 88(3):032323, 2013. doi:10.1103/​PhysRevA.88.032323.
https:/​/​doi.org/​10.1103/​PhysRevA.88.032323

Cited by

[1] Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H. Booth, and Jonathan Tennyson, "The Variational Quantum Eigensolver: a review of methods and best practices", arXiv:2111.05176.

[2] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, "Noisy intermediate-scale quantum algorithms", Reviews of Modern Physics 94 1, 015004 (2022).

[3] Hsin-Yuan Huang, Richard Kueng, Giacomo Torlai, Victor V. Albert, and John Preskill, "Provably efficient machine learning for quantum many-body problems", arXiv:2106.12627.

[4] Antoine Neven, Jose Carrasco, Vittorio Vitale, Christian Kokail, Andreas Elben, Marcello Dalmonte, Pasquale Calabrese, Peter Zoller, Benoît Vermersch, Richard Kueng, and Barbara Kraus, "Symmetry-resolved entanglement detection using partial transpose moments", npj Quantum Information 7, 152 (2021).

[5] Stefan H. Sack, Raimel A. Medina, Alexios A. Michailidis, Richard Kueng, and Maksym Serbyn, "Avoiding Barren Plateaus Using Classical Shadows", PRX Quantum 3 2, 020365 (2022).

[6] Hsin-Yuan Huang, Richard Kueng, and John Preskill, "Efficient Estimation of Pauli Observables by Derandomization", Physical Review Letters 127 3, 030503 (2021).

[7] Hong-Ye Hu, Soonwon Choi, and Yi-Zhuang You, "Classical Shadow Tomography with Locally Scrambled Quantum Dynamics", arXiv:2107.04817.

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

[9] Andreas Elben, Steven T. Flammia, Hsin-Yuan Huang, Richard Kueng, John Preskill, Benoît Vermersch, and Peter Zoller, "The randomized measurement toolbox", arXiv:2203.11374.

[10] Hong-Ye Hu and Yi-Zhuang You, "Hamiltonian-driven shadow tomography of quantum states", Physical Review Research 4 1, 013054 (2022).

[11] Roy J. Garcia, You Zhou, and Arthur Jaffe, "Quantum scrambling with classical shadows", Physical Review Research 3 3, 033155 (2021).

[12] Ryan Levy, Di Luo, and Bryan K. Clark, "Classical Shadows for Quantum Process Tomography on Near-term Quantum Computers", arXiv:2110.02965.

[13] Daniel McNulty, Filip B. Maciejewski, and Michał Oszmaniec, "Estimating Quantum Hamiltonians via Joint Measurements of Noisy Non-Commuting Observables", arXiv:2206.08912.

[14] Aniket Rath, Cyril Branciard, Anna Minguzzi, and Benoît Vermersch, "Quantum Fisher Information from Randomized Measurements", Physical Review Letters 127 26, 260501 (2021).

[15] Charles Hadfield, "Adaptive Pauli Shadows for Energy Estimation", arXiv:2105.12207.

[16] Jose Carrasco, Andreas Elben, Christian Kokail, Barbara Kraus, and Peter Zoller, "Theoretical and Experimental Perspectives of Quantum Verification", arXiv:2102.05927.

[17] Lorenzo Leone, Salvatore F. E. Oliviero, and Alioscia Hamma, "Magic hinders quantum certification", arXiv:2204.02995.

[18] Atithi Acharya, Siddhartha Saha, and Anirvan M. Sengupta, "Informationally complete POVM-based shadow tomography", arXiv:2105.05992.

[19] Kaifeng Bu, Dax Enshan Koh, Roy J. Garcia, and Arthur Jaffe, "Classical shadows with Pauli-invariant unitary ensembles", arXiv:2202.03272.

[20] Simone Notarnicola, Andreas Elben, Thierry Lahaye, Antoine Browaeys, Simone Montangero, and Benoit Vermersch, "A randomized measurement toolbox for Rydberg quantum technologies", arXiv:2112.11046.

[21] Atithi Acharya, Siddhartha Saha, and Anirvan M. Sengupta, "Shadow tomography based on informationally complete positive operator-valued measure", Physical Review A 104 5, 052418 (2021).

The above citations are from SAO/NASA ADS (last updated successfully 2022-10-02 01:19:41). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2022-10-02 01:19:39).