Optimal Twirling Depth for Classical Shadows in the Presence of Noise

Rozon, Pierre-Gabriel; Bao, Ning; Agarwal, Kartiek

doi:10.1103/physrevlett.133.130803

Optimal Twirling Depth for Classical Shadows in the Presence of Noise

Journal Article · Wed Sep 25 00:00:00 EDT 2024 · Physical Review Letters

DOI:https://doi.org/10.1103/physrevlett.133.130803· OSTI ID:2475344

^[1]; ^[2]; ^[3]

McGill University, Montréal, Québec (Canada)
Northeastern University, Boston, MA (United States); Brookhaven National Laboratory (BNL), Upton, NY (United States)
McGill University, Montréal, Québec (Canada); Argonne National Laboratory (ANL), Argonne, IL (United States)

The classical shadows protocol is an efficient strategy for estimating properties of an unknown state p using a small number of state copies and measurements. In its original form, it involves twirling the state with unitaries from some ensemble and measuring the twirled state in a fixed basis. It was recently shown that for computing local properties, optimal sample complexity (copies of the state required) is remarkably achieved for unitaries drawn from shallow depth circuits composed of local entangling gates, as opposed to purely local (zero depth) or global twirling (infinite depth) ensembles. Here, we consider the sample complexity as a function of the depth of the circuit, in the presence of noise. We find that this noise has important implications for determining the optimal twirling ensemble. Under fairly general conditions, we (i) show that any single-site noise can be accounted for using a depolarizing noise channel with an appropriate damping parameter f, (ii) compute thresholds f_th at which optimal twirling reduces to local twirling for Pauli operators, (iii) nth order Renyi entropies (n ≥2), and (iv) provide a meaningful upper bound t_max on the optimal circuit depth for any finite noise strength f, which applies to observables and entanglement entropy measurements. In conclusion, these thresholds strongly constrain the search for optimal strategies to implement shadow tomography and are easily tailored to the experimental system at hand.

View Accepted Manuscript (DOE)

Research Organization:: Brookhaven National Laboratory (BNL), Upton, NY (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES); USDOE Office of Science (SC), Basic Energy Sciences (BES). Materials Sciences & Engineering Division (MSE); USDOE Office of Science (SC), High Energy Physics (HEP)

Grant/Contract Number:: SC0012704

OSTI ID:: 2475344

Alternate ID(s):: OSTI ID: 2473617

Report Number(s):: BNL--226289-2024-JAAM

Journal Information:: Physical Review Letters, Journal Name: Physical Review Letters Journal Issue: 13 Vol. 133; ISSN 0031-9007

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

References (45)

Efficient Quantum Algorithms for GHZ and W States, and Implementation on the IBM Quantum Computer Cruz, Diogo; Fournier, Romain; Gremion, Fabien Advanced Quantum Technologies, Vol. 2, Issue 5-6 https://doi.org/10.1002/qute.201900015	journal	April 2019
Matchgate Shadows for Fermionic Quantum Simulation Wan, Kianna; Huggins, William J.; Lee, Joonho Communications in Mathematical Physics, Vol. 404, Issue 2 https://doi.org/10.1007/s00220-023-04844-0	journal	October 2023
Benchmarking Advantage and D-Wave 2000Q quantum annealers with exact cover problems Willsch, Dennis; Willsch, Madita; Gonzalez Calaza, Carlos D. Quantum Information Processing, Vol. 21, Issue 4 https://doi.org/10.1007/s11128-022-03476-y	journal	April 2022
Standard form of qudit stabilizer groups Gheorghiu, Vlad Physics Letters A, Vol. 378, Issue 5-6 https://doi.org/10.1016/j.physleta.2013.12.009	journal	January 2014
Quantum computing with trapped ions Haffner, H.; Roos, C.; Blatt, R. Physics Reports, Vol. 469, Issue 4 https://doi.org/10.1016/j.physrep.2008.09.003	journal	December 2008
Efficient quantum state tomography Cramer, Marcus; Plenio, Martin B.; Flammia, Steven T. Nature Communications, Vol. 1, Issue 1 https://doi.org/10.1038/ncomms1147	journal	December 2010
Characterizing large-scale quantum computers via cycle benchmarking Erhard, Alexander; Wallman, Joel J.; Postler, Lukas Nature Communications, Vol. 10, Issue 1 https://doi.org/10.1038/s41467-019-13068-7	journal	November 2019
Error-mitigated fermionic classical shadows on noisy quantum devices Wu, Bujiao; Koh, Dax Enshan npj Quantum Information, Vol. 10, Issue 1 https://doi.org/10.1038/s41534-024-00836-7	journal	April 2024
Group-theoretic error mitigation enabled by classical shadows and symmetries Zhao, Andrew; Miyake, Akimasa npj Quantum Information, Vol. 10, Issue 1 https://doi.org/10.1038/s41534-024-00854-5	journal	June 2024
Predicting many properties of a quantum system from very few measurements Huang, Hsin-Yuan; Kueng, Richard; Preskill, John Nature Physics, Vol. 16, Issue 10 https://doi.org/10.1038/s41567-020-0932-7	journal	June 2020
Quantum supremacy using a programmable superconducting processor Arute, Frank; Arya, Kunal; Babbush, Ryan Nature, Vol. 574, Issue 7779 https://doi.org/10.1038/s41586-019-1666-5	journal	October 2019
Time-crystalline eigenstate order on a quantum processor Mi, Xiao; Ippoliti, Matteo; Quintana, Chris Nature, Vol. 601, Issue 7894 https://doi.org/10.1038/s41586-021-04257-w	journal	November 2021
The randomized measurement toolbox Elben, Andreas; Flammia, Steven T.; Huang, Hsin-Yuan Nature Reviews Physics, Vol. 5, Issue 1 https://doi.org/10.1038/s42254-022-00535-2	journal	December 2022
Experimental Estimation of Quantum State Properties from Classical Shadows Struchalin, G. I.; Zagorovskii, Ya. A.; Kovlakov, E. V. PRX Quantum, Vol. 2, Issue 1 https://doi.org/10.1103/PRXQuantum.2.010307	journal	January 2021
Robust Shadow Estimation Chen, Senrui; Yu, Wenjun; Zeng, Pei PRX Quantum, Vol. 2, Issue 3 https://doi.org/10.1103/PRXQuantum.2.030348	journal	September 2021
Quantum Error Mitigated Classical Shadows Jnane, Hamza; Steinberg, Jonathan; Cai, Zhenyu PRX Quantum, Vol. 5, Issue 1 https://doi.org/10.1103/PRXQuantum.5.010324	journal	February 2024
Learnability Transitions in Monitored Quantum Dynamics via Eavesdropper’s Classical Shadows Ippoliti, Matteo; Khemani, Vedika PRX Quantum, Vol. 5, Issue 2 https://doi.org/10.1103/PRXQuantum.5.020304	journal	April 2024
Shadow process tomography of quantum channels Kunjummen, Jonathan; Tran, Minh C.; Carney, Daniel Physical Review A, Vol. 107, Issue 4 https://doi.org/10.1103/PhysRevA.107.042403	journal	April 2023
Exact and approximate unitary 2-designs and their application to fidelity estimation Dankert, Christoph; Cleve, Richard; Emerson, Joseph Physical Review A, Vol. 80, Issue 1 https://doi.org/10.1103/PhysRevA.80.012304	journal	July 2009
Statistical correlations between locally randomized measurements: A toolbox for probing entanglement in many-body quantum states Elben, A.; Vermersch, B.; Roos, C. F. Physical Review A, Vol. 99, Issue 5 https://doi.org/10.1103/PhysRevA.99.052323	journal	May 2019
Fermionic Partial Tomography via Classical Shadows Zhao, Andrew; Rubin, Nicholas C.; Miyake, Akimasa Physical Review Letters, Vol. 127, Issue 11 https://doi.org/10.1103/PhysRevLett.127.110504	journal	September 2021
Experimental Quantum State Measurement with Classical Shadows Zhang, Ting; Sun, Jinzhao; Fang, Xiao-Xu Physical Review Letters, Vol. 127, Issue 20 https://doi.org/10.1103/PhysRevLett.127.200501	journal	November 2021
Filtering Crosstalk from Bath Non-Markovianity via Spacetime Classical Shadows White, G. A. L.; Modi, K.; Hill, C. D. Physical Review Letters, Vol. 130, Issue 16 https://doi.org/10.1103/PhysRevLett.130.160401	journal	April 2023
Operator Relaxation and the Optimal Depth of Classical Shadows Ippoliti, Matteo; Li, Yaodong; Rakovszky, Tibor Physical Review Letters, Vol. 130, Issue 23 https://doi.org/10.1103/PhysRevLett.130.230403	journal	June 2023
Shallow Shadows: Expectation Estimation Using Low-Depth Random Clifford Circuits Bertoni, Christian; Haferkamp, Jonas; Hinsche, Marcel Physical Review Letters, Vol. 133, Issue 2 https://doi.org/10.1103/PhysRevLett.133.020602	journal	July 2024
Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum Smithey, D. T.; Beck, M.; Raymer, M. G. Physical Review Letters, Vol. 70, Issue 9 https://doi.org/10.1103/PhysRevLett.70.1244	journal	March 1993
Experimental Determination of the Quantum-Mechanical State of a Molecular Vibrational Mode Using Fluorescence Tomography Dunn, T. J.; Walmsley, I. A.; Mukamel, S. Physical Review Letters, Vol. 74, Issue 6 https://doi.org/10.1103/PhysRevLett.74.884	journal	February 1995
Quantum Tomography for Measuring Experimentally the Matrix Elements of an Arbitrary Quantum Operation D'Ariano, G. M.; Lo Presti, P. Physical Review Letters, Vol. 86, Issue 19 https://doi.org/10.1103/PhysRevLett.86.4195	journal	May 2001
Quantum scrambling with classical shadows Garcia, Roy J.; Zhou, You; Jaffe, Arthur Physical Review Research, Vol. 3, Issue 3 https://doi.org/10.1103/PhysRevResearch.3.033155	journal	August 2021
Classical shadow tomography with locally scrambled quantum dynamics Hu, Hong-Ye; Choi, Soonwon; You, Yi-Zhuang Physical Review Research, Vol. 5, Issue 2 https://doi.org/10.1103/PhysRevResearch.5.023027	journal	April 2023
Classical shadows for quantum process tomography on near-term quantum computers Levy, Ryan; Luo, Di; Clark, Bryan K. Physical Review Research, Vol. 6, Issue 1 https://doi.org/10.1103/PhysRevResearch.6.013029	journal	January 2024
Continuous-variable optical quantum-state tomography Lvovsky, A. I.; Raymer, M. G. Reviews of Modern Physics, Vol. 81, Issue 1 https://doi.org/10.1103/RevModPhys.81.299	journal	March 2009
Optimal Quantum Tomography Bisio, A.; Chiribella, G.; D'Ariano, G. M. IEEE Journal of Selected Topics in Quantum Electronics, Vol. 15, Issue 6 https://doi.org/10.1109/JSTQE.2009.2029243	journal	January 2009
Sample-optimal tomography of quantum states Haah, Jeongwan; Harrow, Aram W.; Ji, Zhengfeng IEEE Transactions on Information Theory https://doi.org/10.1109/TIT.2017.2719044	journal	January 2017
Classical Shadow Tomography for Continuous Variables Quantum Systems Becker, Simon; Datta, Nilanjana; Lami, Ludovico IEEE Transactions on Information Theory, Vol. 70, Issue 5 https://doi.org/10.1109/TIT.2024.3357972	journal	May 2024
Realizing topologically ordered states on a quantum processor Satzinger, K. J.; Liu, Y. -J; Smith, A. Science, Vol. 374, Issue 6572 https://doi.org/10.1126/science.abi8378	journal	December 2021
Provably efficient machine learning for quantum many-body problems Huang, Hsin-Yuan; Kueng, Richard; Torlai, Giacomo Science, Vol. 377, Issue 6613 https://doi.org/10.1126/science.abk3333	journal	September 2022
Quantum optimization of maximum independent set using Rydberg atom arrays Ebadi, S.; Keesling, A.; Cain, M. Science, Vol. 376, Issue 6598 https://doi.org/10.1126/science.abo6587	journal	June 2022
Shadow Tomography of Quantum States Aaronson, Scott SIAM Journal on Computing, Vol. 49, Issue 5 https://doi.org/10.1137/18m120275x	journal	January 2020
On classical and hybrid shadows of quantum states Shivam, Saumya; von Keyserlingk, Curt W.; Sondhi, Shivaji L. SciPost Physics, Vol. 14, Issue 5 https://doi.org/10.21468/SciPostPhys.14.5.094	journal	May 2023
Classical Shadows With Noise Koh, Dax Enshan; Grewal, Sabee Quantum, Vol. 6 https://doi.org/10.22331/q-2022-08-16-776	journal	August 2022
Scalable and Flexible Classical Shadow Tomography with Tensor Networks Akhtar, Ahmed A.; Hu, Hong-Ye; You, Yi-Zhuang Quantum, Vol. 7 https://doi.org/10.22331/q-2023-06-01-1026	journal	June 2023
The Clifford group forms a unitary 3-design Webb, Zak Quantum Information and Computation, Vol. 16, Issue 15&16 https://doi.org/10.26421/QIC16.15-16-8	journal	November 2016
Closed-form analytic expressions for shadow estimation with brickwork circuits Arienzo, Mirko; Heinrich, Markus; Roth, Ingo Quantum Information and Computation, Vol. 23, Issue 11&12 https://doi.org/10.26421/QIC23.11-12-5	journal	September 2023
Quantum Tomography D'Ariano, G. Mauro; Paris, Matteo G. A.; Sacchi, Massimiliano F. arXiv https://doi.org/10.48550/arXiv.quant-ph/0302028	text	January 2003

Similar Records

Operator Relaxation and the Optimal Depth of Classical Shadows

Journal Article · Thu Jun 08 20:00:00 EDT 2023 · Physical Review Letters · OSTI ID:2421728

Dual-unitary shadow tomography

Journal Article · Mon Jul 28 20:00:00 EDT 2025 · Quantum · OSTI ID:2446948

Practical implementations of twirl operations

Journal Article · Mon Feb 28 23:00:00 EST 2005 · Physical Review. A · OSTI ID:20650365

Related Subjects

97 MATHEMATICS AND COMPUTING
Quantum tomography

Optimal Twirling Depth for Classical Shadows in the Presence of Noise

Citation Formats

References (45)

Similar Records

Related Subjects