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

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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.

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[1] Kui An, Zilei Liu, Ting Zhang, Siqi Li, You Zhou, Xiao Yuan, Leiran Wang, Wenfu Zhang, Guoxi Wang, and He Lu, "Efficient characterizations of multiphoton states with an ultra-thin optical device", Nature Communications 15 1, 3944 (2024).

[2] Matteo Ippoliti and Vedika Khemani, "Learnability Transitions in Monitored Quantum Dynamics via Eavesdropper’s Classical Shadows", PRX Quantum 5 2, 020304 (2024).

[3] You Zhou and Qing Liu, "Performance analysis of multi-shot shadow estimation", Quantum 7, 1044 (2023).

[4] Yongtao Zhan, Andreas Elben, Hsin-Yuan Huang, and Yu Tong, "Learning Conservation Laws in Unknown Quantum Dynamics", PRX Quantum 5 1, 010350 (2024).

[5] Aniket Rath, Vittorio Vitale, Sara Murciano, Matteo Votto, Jérôme Dubail, Richard Kueng, Cyril Branciard, Pasquale Calabrese, and Benoît Vermersch, "Entanglement Barrier and its Symmetry Resolution: Theory and Experimental Observation", PRX Quantum 4 1, 010318 (2023).

[6] Simone Notarnicola, Andreas Elben, Thierry Lahaye, Antoine Browaeys, Simone Montangero, and Benoît Vermersch, "A randomized measurement toolbox for an interacting Rydberg-atom quantum simulator", New Journal of Physics 25 10, 103006 (2023).

[7] Antonio Anna Mele, "Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial", Quantum 8, 1340 (2024).

[8] Hong-Ye Hu, Soonwon Choi, and Yi-Zhuang You, "Classical shadow tomography with locally scrambled quantum dynamics", Physical Review Research 5 2, 023027 (2023).

[9] Yadong Wu, Juan Yao, Pengfei Zhang, and Xiaopeng Li, "Randomness-Enhanced Expressivity of Quantum Neural Networks", Physical Review Letters 132 1, 010602 (2024).

[10] Jacob Bringewatt, Jonathan Kunjummen, and Niklas Mueller, "Randomized measurement protocols for lattice gauge theories", Quantum 8, 1300 (2024).

[11] Pei Zeng, "Tailoring randomized-measurement schemes for practical devices", Quantum Views 7, 74 (2023).

[12] Daniel Tzu Shiuan Chen, Zain Hamid Saleem, and Michael Alexandrovich Perlin, "Quantum Circuit Cutting for Classical Shadows", ACM Transactions on Quantum Computing 3665335 (2024).

[13] Ryan Levy, Di Luo, and Bryan K. Clark, "Classical shadows for quantum process tomography on near-term quantum computers", Physical Review Research 6 1, 013029 (2024).

[14] N. Renaud, P. Rodríguez-Sánchez, J. Hidding, and P. Chris Broekema, "Quantum radio astronomy: Quantum linear solvers for redundant baseline calibration", Astronomy and Computing 47, 100803 (2024).

[15] Ahmed A. Akhtar, Hong-Ye Hu, and Yi-Zhuang You, "Measurement-induced criticality is tomographically optimal", Physical Review B 109 9, 094209 (2024).

[16] Hamza Jnane, Jonathan Steinberg, Zhenyu Cai, H. Chau Nguyen, and Bálint Koczor, "Quantum Error Mitigated Classical Shadows", PRX Quantum 5 1, 010324 (2024).

[17] 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).

[18] Ahmed A. Akhtar, Hong-Ye Hu, and Yi-Zhuang You, "Scalable and Flexible Classical Shadow Tomography with Tensor Networks", Quantum 7, 1026 (2023).

[19] Matteo Ippoliti, Yaodong Li, Tibor Rakovszky, and Vedika Khemani, "Operator Relaxation and the Optimal Depth of Classical Shadows", Physical Review Letters 130 23, 230403 (2023).

[20] F. Turro, T. Chistolini, A. Hashim, Y. Kim, W. Livingston, J. M. Kreikebaum, K. A. Wendt, J. L. Dubois, F. Pederiva, S. Quaglioni, D. I. Santiago, and I. Siddiqi, "Demonstration of a quantum-classical coprocessing protocol for simulating nuclear reactions", Physical Review A 108 3, 032417 (2023).

[21] Kaifeng Bu, Dax Enshan Koh, Roy J. Garcia, and Arthur Jaffe, "Classical shadows with Pauli-invariant unitary ensembles", npj Quantum Information 10 1, 6 (2024).

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

[23] Marcin Płodzień, Tomasz Wasak, Emilia Witkowska, Maciej Lewenstein, and Jan Chwedeńczuk, "Generation of scalable many-body Bell correlations in spin chains with short-range two-body interactions", Physical Review Research 6 2, 023050 (2024).

[24] Bujiao Wu and Dax Enshan Koh, "Error-mitigated fermionic classical shadows on noisy quantum devices", npj Quantum Information 10 1, 39 (2024).

[25] Daniel McNulty, Filip B. Maciejewski, and Michał Oszmaniec, "Estimating Quantum Hamiltonians via Joint Measurements of Noisy Noncommuting Observables", Physical Review Letters 130 10, 100801 (2023).

[26] Matteo Ippoliti, "Classical shadows based on locally-entangled measurements", Quantum 8, 1293 (2024).

[27] 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).

[28] 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", Physics Reports 986, 1 (2022).

[29] 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).

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

[31] 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, (2021).

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

[33] 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).

[34] Hsin-Yuan Huang, Sitan Chen, and John Preskill, "Learning to Predict Arbitrary Quantum Processes", PRX Quantum 4 4, 040337 (2023).

[35] Hong-Ye Hu, Soonwon Choi, and Yi-Zhuang You, "Classical Shadow Tomography with Locally Scrambled Quantum Dynamics", arXiv:2107.04817, (2021).

[36] Charles Hadfield, "Adaptive Pauli Shadows for Energy Estimation", arXiv:2105.12207, (2021).

[37] Ahmed A. Akhtar, Namit Anand, Jeffrey Marshall, and Yi-Zhuang You, "Dual-Unitary Classical Shadow Tomography", arXiv:2404.01068, (2024).

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

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

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

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

[42] Mirko Arienzo, Markus Heinrich, Ingo Roth, and Martin Kliesch, "Closed-form analytic expressions for shadow estimation with brickwork circuits", arXiv:2211.09835, (2022).

[43] Atithi Acharya, Siddhartha Saha, and Anirvan M. Sengupta, "Informationally complete POVM-based shadow tomography", arXiv:2105.05992, (2021).

[44] 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).

[45] Andrew Arrasmith, Andrew Patterson, Alice Boughton, and Marco Paini, "Development and Demonstration of an Efficient Readout Error Mitigation Technique for use in NISQ Algorithms", arXiv:2303.17741, (2023).

[46] Jose Carrasco, Andreas Elben, Christian Kokail, Barbara Kraus, and Peter Zoller, "Theoretical and Experimental Perspectives of Quantum Verification", arXiv:2102.05927, (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-24 19:13:19) and SAO/NASA ADS (last updated successfully 2024-05-24 19:13:20). The list may be incomplete as not all publishers provide suitable and complete citation data.