Classical shadows based on locally-entangled measurements

Matteo Ippoliti

Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA
Department of Physics, Stanford University, Stanford, CA 94305, USA

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


We study classical shadows protocols based on randomized measurements in $n$-qubit entangled bases, generalizing the random Pauli measurement protocol ($n = 1$). We show that entangled measurements ($n\geq 2$) enable nontrivial and potentially advantageous trade-offs in the sample complexity of learning Pauli expectation values. This is sharply illustrated by shadows based on two-qubit Bell measurements: the scaling of sample complexity with Pauli weight $k$ improves quadratically (from $\sim 3^k$ down to $\sim 3^{k/2}$) for many operators, while others become impossible to learn. Tuning the amount of entanglement in the measurement bases defines a family of protocols that interpolate between Pauli and Bell shadows, retaining some of the benefits of both. For large $n$, we show that randomized measurements in $n$-qubit GHZ bases further improve the best scaling to $\sim (3/2)^k$, albeit on an increasingly restricted set of operators. Despite their simplicity and lower hardware requirements, these protocols can match or outperform recently-introduced "shallow shadows" in some practically-relevant Pauli estimation tasks.

► BibTeX data

► References

[1] Hsin-Yuan Huang, Richard Kueng, and John Preskill. ``Predicting many properties of a quantum system from very few measurements''. Nature Physics 16, 1050–1057 (2020).

[2] Andreas Elben, Steven T. Flammia, Hsin-Yuan Huang, Richard Kueng, John Preskill, Benoit Vermersch, and Peter Zoller. ``The randomized measurement toolbox''. Nature Reviews Physics 5, 9–24 (2023).

[3] Charles Hadfield, Sergey Bravyi, Rudy Raymond, and Antonio Mezzacapo. ``Measurements of Quantum Hamiltonians with Locally-Biased Classical Shadows'' (2020). arXiv:2006.15788.

[4] Senrui Chen, Wenjun Yu, Pei Zeng, and Steven T. Flammia. ``Robust Shadow Estimation''. PRX Quantum 2, 030348 (2021).

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

[6] 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 (2021).

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

[8] Jonathan Kunjummen, Minh C. Tran, Daniel Carney, and Jacob M. Taylor. ``Shadow process tomography of quantum channels''. Physical Review A 107, 042403 (2023).

[9] Hsin-Yuan Huang. ``Learning quantum states from their classical shadows''. Nature Reviews Physics 4, 81–81 (2022).

[10] Kianna Wan, William J. Huggins, Joonho Lee, and Ryan Babbush. ``Matchgate Shadows for Fermionic Quantum Simulation''. Communications in Mathematical Physics 404, 629–700 (2023).

[11] Kaifeng Bu, Dax Enshan Koh, Roy J. Garcia, and Arthur Jaffe. ``Classical shadows with Pauli-invariant unitary ensembles''. npj Quantum Information 10, 1–7 (2024).

[12] H. Chau Nguyen, Jan Lennart Bonsel, Jonathan Steinberg, and Otfried Guhne. ``Optimizing Shadow Tomography with Generalized Measurements''. Physical Review Letters 129, 220502 (2022).

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

[14] Daniel Grier, Hakop Pashayan, and Luke Schaeffer. ``Sample-optimal classical shadows for pure states'' (2022). arXiv:2211.11810.

[15] Simon Becker, Nilanjana Datta, Ludovico Lami, and Cambyse Rouze. ``Classical shadow tomography for continuous variables quantum systems'' (2022). arXiv:2211.07578.

[16] 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''. PRX Quantum 4, 010303 (2023).

[17] Katherine Van Kirk, Jordan Cotler, Hsin-Yuan Huang, and Mikhail D. Lukin. ``Hardware-efficient learning of quantum many-body states'' (2022). arXiv:2212.06084.

[18] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, et al. ``Quantum supremacy using a programmable superconducting processor''. Nature 574, 505–510 (2019).

[19] Ehud Altman, Kenneth R. Brown, Giuseppe Carleo, Lincoln D. Carr, Eugene Demler, Cheng Chin, Brian DeMarco, Sophia E. Economou, et al. ``Quantum Simulators: Architectures and Opportunities''. PRX Quantum 2, 017003 (2021).

[20] Sepehr Ebadi, Tout T. Wang, Harry Levine, Alexander Keesling, Giulia Semeghini, Ahmed Omran, Dolev Bluvstein, Rhine Samajdar, et al. ``Quantum phases of matter on a 256-atom programmable quantum simulator''. Nature 595, 227–232 (2021).

[21] Xiao Mi, Pedram Roushan, Chris Quintana, Salvatore Mandra, Jeffrey Marshall, Charles Neill, Frank Arute, Kunal Arya, et al. ``Information scrambling in quantum circuits''. Science 374, 1479–1483 (2021).

[22] Tiff Brydges, Andreas Elben, Petar Jurcevic, Benoit Vermersch, Christine Maier, Ben P. Lanyon, Peter Zoller, Rainer Blatt, et al. ``Probing Renyi entanglement entropy via randomized measurements''. Science 364, 260–263 (2019).

[23] A. Elben, B. Vermersch, C. F. Roos, and P. Zoller. ``Statistical correlations between locally randomized measurements: A toolbox for probing entanglement in many-body quantum states''. Phys. Rev. A 99, 052323 (2019).

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

[25] Christian Bertoni, Jonas Haferkamp, Marcel Hinsche, Marios Ioannou, Jens Eisert, and Hakop Pashayan. ``Shallow shadows: Expectation estimation using low-depth random Clifford circuits'' (2022). arXiv:2209.12924.

[26] Mirko Arienzo, Markus Heinrich, Ingo Roth, and Martin Kliesch. ``Closed-form analytic expressions for shadow estimation with brickwork circuits''. Quantum Information and Computation 23, 961 (2023).

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

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

[29] Jutho Haegeman, David Perez-Garcia, Ignacio Cirac, and Norbert Schuch. ``Order Parameter for Symmetry-Protected Phases in One Dimension''. Physical Review Letters 109, 050402 (2012).

[30] H. Bombin. ``An Introduction to Topological Quantum Codes'' (2013). arXiv:1311.0277.

[31] D. J. Thouless. ``Exchange in solid 3He and the Heisenberg Hamiltonian''. Proceedings of the Physical Society 86, 893 (1965).

[32] Alexander Altland and Ben D. Simons. ``Condensed Matter Field Theory''. Cambridge University Press. Cambridge (2010). 2nd edition.

[33] Debanjan Chowdhury, Suvrat Raju, Subir Sachdev, Ajay Singh, and Philipp Strack. ``Multipoint correlators of conformal field theories: Implications for quantum critical transport''. Physical Review B 87, 085138 (2013).

[34] I. Kukuljan, S. Sotiriadis, and G. Takacs. ``Correlation functions of the quantum sine-gordon model in and out of equilibrium''. Phys. Rev. Lett. 121, 110402 (2018).

[35] Fabian B. Kugler, Seung-Sup B. Lee, and Jan von Delft. ``Multipoint correlation functions: Spectral representation and numerical evaluation''. Phys. Rev. X 11, 041006 (2021).

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

[37] Yi-Zhuang You and Yingfei Gu. ``Entanglement features of random Hamiltonian dynamics''. Physical Review B 98, 014309 (2018).

[38] Wei-Ting Kuo, A. A. Akhtar, Daniel P. Arovas, and Yi-Zhuang You. ``Markovian Entanglement Dynamics under Locally Scrambled Quantum Evolution''. Physical Review B 101, 224202 (2020).

[39] Matteo Ippoliti and Vedika Khemani. ``Learnability transitions in monitored quantum dynamics via eavesdropper's classical shadows'' (2023). arXiv:2307.15011.

[40] Peter Shor and Raymond Laflamme. ``Quantum Analog of the MacWilliams Identities for Classical Coding Theory''. Physical Review Letters 78, 1600–1602 (1997).

[41] ChunJun Cao, Michael J. Gullans, Brad Lackey, and Zitao Wang. ``Quantum Lego Expansion Pack: Enumerators from Tensor Networks'' (2023). arXiv:2308.05152.

[42] Daniel Miller, Daniel Loss, Ivano Tavernelli, Hermann Kampermann, Dagmar Bruss, and Nikolai Wyderka. ``Shor-Laflamme distributions of graph states and noise robustness of entanglement''. Journal of Physics A: Mathematical and Theoretical 56, 335303 (2023).

[43] Ikko Hamamura and Takashi Imamichi. ``Efficient evaluation of quantum observables using entangled measurements''. npj Quantum Information 6, 1–8 (2020).

[44] Ruho Kondo, Yuki Sato, Satoshi Koide, Seiji Kajita, and Hideki Takamatsu. ``Computationally Efficient Quantum Expectation with Extended Bell Measurements''. Quantum 6, 688 (2022).

[45] Francisco Escudero, David Fernandez-Fernandez, Gabriel Jauma, Guillermo F. Penas, and Luciano Pereira. ``Hardware-Efficient Entangled Measurements for Variational Quantum Algorithms''. Physical Review Applied 20, 034044 (2023).

[46] Zhang Jiang, Amir Kalev, Wojciech Mruczkiewicz, and Hartmut Neven. ``Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning''. Quantum 4, 276 (2020).

[47] Ruben Verresen. ``Everything is a quantum Ising model'' (2023). arXiv:2301.11917.

[48] Charles Hadfield. ``Adaptive Pauli Shadows for Energy Estimation'' (2021). arXiv:2105.12207.

[49] 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. (2021).

[50] Tzu-Ching Yen, Aadithya Ganeshram, and Artur F. Izmaylov. ``Deterministic improvements of quantum measurements with grouping of compatible operators, non-local transformations, and covariance estimates''. npj Quantum Information 9, 1–7 (2023).

[51] Bujiao Wu, Jinzhao Sun, Qi Huang, and Xiao Yuan. ``Overlapped grouping measurement: A unified framework for measuring quantum states''. Quantum 7, 896 (2023).

[52] Minh C. Tran, Daniel K. Mark, Wen Wei Ho, and Soonwon Choi. ``Measuring Arbitrary Physical Properties in Analog Quantum Simulation''. Physical Review X 13, 011049 (2023).

[53] Max McGinley and Michele Fava. ``Shadow Tomography from Emergent State Designs in Analog Quantum Simulators''. Physical Review Letters 131, 160601 (2023).

[54] Joonhee Choi, Adam L. Shaw, Ivaylo S. Madjarov, Xin Xie, Ran Finkelstein, Jacob P. Covey, Jordan S. Cotler, Daniel K. Mark, et al. ``Preparing random states and benchmarking with many-body quantum chaos''. Nature 613, 468–473 (2023).

[55] Jordan S. Cotler, Daniel K. Mark, Hsin-Yuan Huang, Felipe Hernandez, Joonhee Choi, Adam L. Shaw, Manuel Endres, and Soonwon Choi. ``Emergent Quantum State Designs from Individual Many-Body Wave Functions''. PRX Quantum 4, 010311 (2023).

[56] Wen Wei Ho and Soonwon Choi. ``Exact Emergent Quantum State Designs from Quantum Chaotic Dynamics''. Physical Review Letters 128, 060601 (2022).

[57] Pieter W. Claeys and Austen Lamacraft. ``Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics''. Quantum 6, 738 (2022).

[58] Matteo Ippoliti and Wen Wei Ho. ``Dynamical Purification and the Emergence of Quantum State Designs from the Projected Ensemble''. PRX Quantum 4, 030322 (2023).

[59] Matteo Ippoliti and Wen Wei Ho. ``Solvable model of deep thermalization with distinct design times''. Quantum 6, 886 (2022).

[60] Pieter W. Claeys. ``Universality in quantum snapshots''. Quantum Views 7, 71 (2023).

Cited by

[1] Bujiao Wu and Dax Enshan Koh, "Error-mitigated fermionic classical shadows on noisy quantum devices", arXiv:2310.12726, (2023).

[2] Benoît Vermersch, Marko Ljubotina, J. Ignacio Cirac, Peter Zoller, Maksym Serbyn, and Lorenzo Piroli, "Many-body entropies and entanglement from polynomially-many local measurements", arXiv:2311.08108, (2023).

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

[4] Dominik Šafránek and Dario Rosa, "Measuring energy by measuring any other observable", Physical Review A 108 2, 022208 (2023).

[5] Tianren Gu, Xiao Yuan, and Bujiao Wu, "Efficient measurement schemes for bosonic systems", Quantum Science and Technology 8 4, 045008 (2023).

[6] Arkopal Dutt, William Kirby, Rudy Raymond, Charles Hadfield, Sarah Sheldon, Isaac L. Chuang, and Antonio Mezzacapo, "Practical Benchmarking of Randomized Measurement Methods for Quantum Chemistry Hamiltonians", arXiv:2312.07497, (2023).

[7] Yuxuan Du, Yibo Yang, Tongliang Liu, Zhouchen Lin, Bernard Ghanem, and Dacheng Tao, "ShadowNet for Data-Centric Quantum System Learning", arXiv:2308.11290, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2024-04-12 11:30:58). 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 2024-04-12 11:30:57).