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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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).