1Department of Physics, University of California San Diego, La Jolla, CA 92093, USA
2Department of Physics, Harvard University, 17 Oxford Street, Cambridge, MA 02138, USA
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
Classical shadow tomography is a powerful randomized measurement protocol for predicting many properties of a quantum state with few measurements. Two classical shadow protocols have been extensively studied in the literature: the single-qubit (local) Pauli measurement, which is well suited for predicting local operators but inefficient for large operators; and the global Clifford measurement, which is efficient for low-rank operators but infeasible on near-term quantum devices due to the extensive gate overhead. In this work, we demonstrate a scalable classical shadow tomography approach for generic randomized measurements implemented with finite-depth local Clifford random unitary circuits, which interpolates between the limits of Pauli and Clifford measurements. The method combines the recently proposed locally-scrambled classical shadow tomography framework with tensor network techniques to achieve scalability for computing the classical shadow reconstruction map and evaluating various physical properties. The method enables classical shadow tomography to be performed on shallow quantum circuits with superior sample efficiency and minimal gate overhead and is friendly to noisy intermediate-scale quantum (NISQ) devices. We show that the shallow-circuit measurement protocol provides immediate, exponential advantages over the Pauli measurement protocol for predicting quasi-local operators. It also enables a more efficient fidelity estimation compared to the Pauli measurement.
Featured image: Classical shadow tomography in a system of n qubits by randomized measurement via a local, finite-depth random unitary circuit of L layers.
Popular summary
► BibTeX data
► References
[1] Scott Aaronson. Shadow Tomography of Quantum States. arXiv e-prints, art. arXiv:1711.01053, November 2017.
arXiv:1711.01053
[2] Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Phys. Rev. A, 70: 052328, Nov 2004. 10.1103/PhysRevA.70.052328. URL https://link.aps.org/doi/10.1103/PhysRevA.70.052328.
https://doi.org/10.1103/PhysRevA.70.052328
[3] Scott Aaronson and Guy N. Rothblum. Gentle Measurement of Quantum States and Differential Privacy. arXiv e-prints, art. arXiv:1904.08747, April 2019.
arXiv:1904.08747
[4] A. A. Akhtar and Yi-Zhuang You. Multiregion entanglement in locally scrambled quantum dynamics. Phys. Rev. B, 102 (13): 134203, October 2020. 10.1103/PhysRevB.102.134203.
https://doi.org/10.1103/PhysRevB.102.134203
[5] Mirko Arienzo, Markus Heinrich, Ingo Roth, and Martin Kliesch. Closed-form analytic expressions for shadow estimation with brickwork circuits, 2022. URL https://arxiv.org/abs/2211.09835.
arXiv:2211.09835
[6] Yimu Bao, Soonwon Choi, and Ehud Altman. Theory of the phase transition in random unitary circuits with measurements. Phys. Rev. B, 101 (10): 104301, March 2020. 10.1103/PhysRevB.101.104301.
https://doi.org/10.1103/PhysRevB.101.104301
[7] Christian Bertoni, Jonas Haferkamp, Marcel Hinsche, Marios Ioannou, Jens Eisert, and Hakop Pashayan. Shallow shadows: Expectation estimation using low-depth random Clifford circuits. arXiv e-prints, art. arXiv:2209.12924, September 2022.
arXiv:2209.12924
[8] Kaifeng Bu, Dax Enshan Koh, Roy J. Garcia, and Arthur Jaffe. Classical shadows with pauli-invariant unitary ensembles, 2022. URL https://arxiv.org/abs/2202.03272.
arXiv:2202.03272
[9] Carlton M. Caves, Christopher A. Fuchs, and Rüdiger Schack. Unknown quantum states: The quantum de Finetti representation. Journal of Mathematical Physics, 43 (9): 4537–4559, September 2002. 10.1063/1.1494475.
https://doi.org/10.1063/1.1494475
[10] Senrui Chen, Wenjun Yu, Pei Zeng, and Steven T. Flammia. Robust shadow estimation. PRX Quantum, 2: 030348, Sep 2021. 10.1103/PRXQuantum.2.030348. URL https://link.aps.org/doi/10.1103/PRXQuantum.2.030348.
https://doi.org/10.1103/PRXQuantum.2.030348
[11] Xiao Chen and Tianci Zhou. Quantum chaos dynamics in long-range power law interaction systems. Phys. Rev. B, 100 (6): 064305, August 2019. 10.1103/PhysRevB.100.064305.
https://doi.org/10.1103/PhysRevB.100.064305
[12] Soonwon Choi, Yimu Bao, Xiao-Liang Qi, and Ehud Altman. Quantum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition. Phys. Rev. B, 125 (3): 030505, July 2020. 10.1103/PhysRevLett.125.030505.
https://doi.org/10.1103/PhysRevLett.125.030505
[13] Ze-Pei Cian, Hossein Dehghani, Andreas Elben, Benoı̂t Vermersch, Guanyu Zhu, Maissam Barkeshli, Peter Zoller, and Mohammad Hafezi. Many-body chern number from statistical correlations of randomized measurements. Phys. Rev. Lett., 126: 050501, Feb 2021. 10.1103/PhysRevLett.126.050501. URL https://link.aps.org/doi/10.1103/PhysRevLett.126.050501.
https://doi.org/10.1103/PhysRevLett.126.050501
[14] J. Ignacio Cirac, David Pé rez-García, Norbert Schuch, and Frank Verstraete. Matrix product states and projected entangled pair states: Concepts, symmetries, theorems. Reviews of Modern Physics, 93 (4), dec 2021. 10.1103/revmodphys.93.045003. URL https://doi.org/10.1103.
https://doi.org/10.1103/revmodphys.93.045003
[15] G. M. D’Ariano and P. Perinotti. Optimal Data Processing for Quantum Measurements. Phys. Rev. B, 98 (2): 020403, January 2007. 10.1103/PhysRevLett.98.020403.
https://doi.org/10.1103/PhysRevLett.98.020403
[16] Andreas Elben, Steven T. Flammia, Hsin-Yuan Huang, Richard Kueng, John Preskill, Benoı̂t Vermersch, and Peter Zoller. The randomized measurement toolbox. arXiv e-prints, art. arXiv:2203.11374, March 2022. 10.1038/s42254-022-00535-2.
https://doi.org/10.1038/s42254-022-00535-2
arXiv:2203.11374
[17] Ruihua Fan, Sagar Vijay, Ashvin Vishwanath, and Yi-Zhuang You. Self-organized error correction in random unitary circuits with measurement. Phys. Rev. B, 103 (17): 174309, May 2021. 10.1103/PhysRevB.103.174309.
https://doi.org/10.1103/PhysRevB.103.174309
[18] 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, September 2012. 10.1088/1367-2630/14/9/095022.
https://doi.org/10.1088/1367-2630/14/9/095022
[19] Chenhua Geng, Hong-Ye Hu, and Yijian Zou. Differentiable programming of isometric tensor networks. Machine Learning: Science and Technology, 3 (1): 015020, jan 2022. 10.1088/2632-2153/ac48a2. URL https://doi.org/10.1088/2632-2153/ac48a2.
https://doi.org/10.1088/2632-2153/ac48a2
[20] Hrant Gharibyan, Masanori Hanada, Stephen H. Shenker, and Masaki Tezuka. Onset of random matrix behavior in scrambling systems. Journal of High Energy Physics, 2018 (7): 124, July 2018. 10.1007/JHEP07(2018)124.
https://doi.org/10.1007/JHEP07(2018)124
[21] Daniel Gottesman. The heisenberg representation of quantum computers. 1998. 10.48550/ARXIV.QUANT-PH/9807006. URL https://arxiv.org/abs/quant-ph/9807006.
https://doi.org/10.48550/ARXIV.QUANT-PH/9807006
arXiv:quant-ph/9807006
[22] Tarun Grover and Matthew P. A. Fisher. Entanglement and the sign structure of quantum states. Physical Review A, 92 (4), oct 2015. 10.1103/physreva.92.042308. URL https://doi.org/10.1103.
https://doi.org/10.1103/physreva.92.042308
[23] Madalin Guta, Jonas Kahn, Richard Kueng, and Joel A. Tropp. Fast state tomography with optimal error bounds. arXiv e-prints, art. arXiv:1809.11162, September 2018.
arXiv:1809.11162
[24] Jeongwan Haah, Aram W. Harrow, Zhengfeng Ji, Xiaodi Wu, and Nengkun Yu. Sample-optimal tomography of quantum states. arXiv e-prints, art. arXiv:1508.01797, August 2015. 10.1109/TIT.2017.2719044.
https://doi.org/10.1109/TIT.2017.2719044
arXiv:1508.01797
[25] Charles Hadfield, Sergey Bravyi, Rudy Raymond, and Antonio Mezzacapo. Measurements of Quantum Hamiltonians with Locally-Biased Classical Shadows. arXiv e-prints, art. arXiv:2006.15788, June 2020.
arXiv:2006.15788
[26] Guang Hao Low. Classical shadows of fermions with particle number symmetry. arXiv e-prints, art. arXiv:2208.08964, August 2022.
arXiv:2208.08964
[27] Markus Hauru, Maarten Van Damme, and Jutho Haegeman. Riemannian optimization of isometric tensor networks. SciPost Phys., 10: 40, 2021. 10.21468/SciPostPhys.10.2.040. URL https://scipost.org/10.21468/SciPostPhys.10.2.040.
https://doi.org/10.21468/SciPostPhys.10.2.040
[28] Hong-Ye Hu and Yi-Zhuang You. Hamiltonian-driven shadow tomography of quantum states. Physical Review Research, 4 (1): 013054, January 2022. 10.1103/PhysRevResearch.4.013054.
https://doi.org/10.1103/PhysRevResearch.4.013054
[29] Hong-Ye Hu, Soonwon Choi, and Yi-Zhuang You. Classical Shadow Tomography with Locally Scrambled Quantum Dynamics. arXiv e-prints, art. arXiv:2107.04817, July 2021.
arXiv:2107.04817
[30] Hong-Ye Hu, Ryan LaRose, Yi-Zhuang You, Eleanor Rieffel, and Zhihui Wang. Logical shadow tomography: Efficient estimation of error-mitigated observables. arXiv e-prints, art. arXiv:2203.07263, March 2022.
arXiv:2203.07263
[31] Hongye Hu. Efficient representation and learning of quantum many-body states. PhD thesis, UC San Diego, 2022.
[32] 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, June 2020. 10.1038/s41567-020-0932-7.
https://doi.org/10.1038/s41567-020-0932-7
[33] Matteo Ippoliti, Yaodong Li, Tibor Rakovszky, and Vedika Khemani. Operator relaxation and the optimal depth of classical shadows, 2023.
[34] Daniel F. V. James, Paul G. Kwiat, William J. Munro, and Andrew G. White. Measurement of qubits. Physical Review A, 64 (5): 052312, November 2001. 10.1103/PhysRevA.64.052312.
https://doi.org/10.1103/PhysRevA.64.052312
[35] Dax Enshan Koh and Sabee Grewal. Classical Shadows With Noise. Quantum, 6: 776, August 2022. ISSN 2521-327X. 10.22331/q-2022-08-16-776. URL https://doi.org/10.22331/q-2022-08-16-776.
https://doi.org/10.22331/q-2022-08-16-776
[36] Wei-Ting Kuo, A. A. Akhtar, Daniel P. Arovas, and Yi-Zhuang You. Markovian entanglement dynamics under locally scrambled quantum evolution. Phys. Rev. B, 101 (22): 224202, June 2020. 10.1103/PhysRevB.101.224202.
https://doi.org/10.1103/PhysRevB.101.224202
[37] Nima Lashkari, Douglas Stanford, Matthew Hastings, Tobias Osborne, and Patrick Hayden. Towards the fast scrambling conjecture. Journal of High Energy Physics, 2013: 22, April 2013. 10.1007/JHEP04(2013)022.
https://doi.org/10.1007/JHEP04(2013)022
[38] Ryan Levy, Di Luo, and Bryan K. Clark. Classical Shadows for Quantum Process Tomography on Near-term Quantum Computers. arXiv e-prints, art. arXiv:2110.02965, October 2021.
arXiv:2110.02965
[39] Adam Nahum, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. Quantum Entanglement Growth under Random Unitary Dynamics. Physical Review X, 7 (3): 031016, July 2017. 10.1103/PhysRevX.7.031016.
https://doi.org/10.1103/PhysRevX.7.031016
[40] Adam Nahum, Sagar Vijay, and Jeongwan Haah. Operator Spreading in Random Unitary Circuits. Physical Review X, 8 (2): 021014, April 2018. 10.1103/PhysRevX.8.021014.
https://doi.org/10.1103/PhysRevX.8.021014
[41] Simone Notarnicola, Andreas Elben, Thierry Lahaye, Antoine Browaeys, Simone Montangero, and Benoit Vermersch. A randomized measurement toolbox for rydberg quantum technologies, 2021. URL https://arxiv.org/abs/2112.11046.
arXiv:2112.11046
[42] Ryan O’Donnell and John Wright. Efficient quantum tomography. arXiv e-prints, art. arXiv:1508.01907, August 2015.
arXiv:1508.01907
[43] M. Ohliger, V. Nesme, and J. Eisert. Efficient and feasible state tomography of quantum many-body systems. New Journal of Physics, 15 (1): 015024, January 2013. 10.1088/1367-2630/15/1/015024.
https://doi.org/10.1088/1367-2630/15/1/015024
[44] Román Orús. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Annals of Physics, 349: 117–158, oct 2014. 10.1016/j.aop.2014.06.013. URL https://doi.org/10.1016.
https://doi.org/10.1016/j.aop.2014.06.013
[45] Marco Paini and Amir Kalev. An approximate description of quantum states. arXiv e-prints, art. arXiv:1910.10543, October 2019.
arXiv:1910.10543
[46] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Köpf, Edward Yang, Zach DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. PyTorch: An Imperative Style, High-Performance Deep Learning Library. Curran Associates Inc., Red Hook, NY, USA, 2019.
[47] Ruth Pordes, Don Petravick, Bill Kramer, Doug Olson, Miron Livny, Alain Roy, Paul Avery, Kent Blackburn, Torre Wenaus, Frank Würthwein, Ian Foster, Rob Gardner, Mike Wilde, Alan Blatecky, John McGee, and Rob Quick. The open science grid. In J. Phys. Conf. Ser., volume 78 of 78, page 012057, 2007. 10.1088/1742-6596/78/1/012057.
https://doi.org/10.1088/1742-6596/78/1/012057
[48] 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. 10.1103/PRXQuantum.3.020365. URL https://link.aps.org/doi/10.1103/PRXQuantum.3.020365.
https://doi.org/10.1103/PRXQuantum.3.020365
[49] 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 e-prints, art. arXiv:2203.07309, March 2022.
arXiv:2203.07309
[50] Igor Sfiligoi, Daniel C Bradley, Burt Holzman, Parag Mhashilkar, Sanjay Padhi, and Frank Wurthwein. The pilot way to grid resources using glideinwms. In 2009 WRI World Congress on Computer Science and Information Engineering, volume 2 of 2, pages 428–432, 2009. 10.1109/CSIE.2009.950.
https://doi.org/10.1109/CSIE.2009.950
[51] Shenglong Xu and Brian Swingle. Locality, Quantum Fluctuations, and Scrambling. Physical Review X, 9 (3): 031048, July 2019. 10.1103/PhysRevX.9.031048.
https://doi.org/10.1103/PhysRevX.9.031048
[52] Yi-Zhuang You and Yingfei Gu. Entanglement features of random Hamiltonian dynamics. Phys. Rev. B, 98 (1): 014309, July 2018. 10.1103/PhysRevB.98.014309.
https://doi.org/10.1103/PhysRevB.98.014309
[53] Yi-Zhuang You, Zhao Yang, and Xiao-Liang Qi. Machine learning spatial geometry from entanglement features. Phys. Rev. B, 97 (4): 045153, February 2018. 10.1103/PhysRevB.97.045153.
https://doi.org/10.1103/PhysRevB.97.045153
[54] Andrew Zhao, Nicholas C. Rubin, and Akimasa Miyake. Fermionic partial tomography via classical shadows. Phys. Rev. Lett., 127: 110504, Sep 2021. 10.1103/PhysRevLett.127.110504. URL https://link.aps.org/doi/10.1103/PhysRevLett.127.110504.
https://doi.org/10.1103/PhysRevLett.127.110504
[55] Tianci Zhou and Xiao Chen. Operator dynamics in a Brownian quantum circuit. Phys. Rev. B, 99 (5): 052212, May 2019. 10.1103/PhysRevE.99.052212.
https://doi.org/10.1103/PhysRevE.99.052212
[56] Tianci Zhou and Adam Nahum. Emergent statistical mechanics of entanglement in random unitary circuits. Phys. Rev. B, 99 (17): 174205, May 2019. 10.1103/PhysRevB.99.174205.
https://doi.org/10.1103/PhysRevB.99.174205
Cited by
[1] Hong-Ye Hu, Soonwon Choi, and Yi-Zhuang You, “Classical Shadow Tomography with Locally Scrambled Quantum Dynamics”, arXiv:2107.04817, (2021).
[2] Christian Bertoni, Jonas Haferkamp, Marcel Hinsche, Marios Ioannou, Jens Eisert, and Hakop Pashayan, “Shallow shadows: Expectation estimation using low-depth random Clifford circuits”, arXiv:2209.12924, (2022).
[3] Gregory Boyd and Bálint Koczor, “Training Variational Quantum Circuits with CoVaR: Covariance Root Finding with Classical Shadows”, Physical Review X 12 4, 041022 (2022).
[4] Minh C. Tran, Daniel K. Mark, Wen Wei Ho, and Soonwon Choi, “Measuring Arbitrary Physical Properties in Analog Quantum Simulation”, Physical Review X 13 1, 011049 (2023).
[5] Matteo Ippoliti, “Classical shadows based on locally-entangled measurements”, arXiv:2305.10723, (2023).
[6] Mirko Arienzo, Markus Heinrich, Ingo Roth, and Martin Kliesch, “Closed-form analytic expressions for shadow estimation with brickwork circuits”, arXiv:2211.09835, (2022).
[7] Arnaud Carignan-Dugas, Dar Dahlen, Ian Hincks, Egor Ospadov, Stefanie J. Beale, Samuele Ferracin, Joshua Skanes-Norman, Joseph Emerson, and Joel J. Wallman, “The Error Reconstruction and Compiled Calibration of Quantum Computing Cycles”, arXiv:2303.17714, (2023).
[8] Katherine Van Kirk, Jordan Cotler, Hsin-Yuan Huang, and Mikhail D. Lukin, “Hardware-efficient learning of quantum many-body states”, arXiv:2212.06084, (2022).
[9] Yusen Wu, Bujiao Wu, Yanqi Song, Xiao Yuan, and Jingbo B. Wang, “Complexity analysis of weakly noisy quantum states via quantum machine learning”, arXiv:2303.17813, (2023).
[10] Matthias C. Caro, “Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrix”, arXiv:2212.04471, (2022).
[11] Matteo Ippoliti, Yaodong Li, Tibor Rakovszky, and Vedika Khemani, “Operator relaxation and the optimal depth of classical shadows”, arXiv:2212.11963, (2022).
[12] Hans Hon Sang Chan, Richard Meister, Matthew L. Goh, and Bálint Koczor, “Algorithmic Shadow Spectroscopy”, arXiv:2212.11036, (2022).
[13] Markus Heinrich, Martin Kliesch, and Ingo Roth, “General guarantees for randomized benchmarking with random quantum circuits”, arXiv:2212.06181, (2022).
[14] Haoxiang Wang, Maurice Weber, Josh Izaac, and Cedric Yen-Yu Lin, “Predicting Properties of Quantum Systems with Conditional Generative Models”, arXiv:2211.16943, (2022).
[15] Hong-Ye Hu, Soonwon Choi, and Yi-Zhuang You, “Classical shadow tomography with locally scrambled quantum dynamics”, Physical Review Research 5 2, 023027 (2023).
[16] Zi-Jian Zhang, Kouhei Nakaji, Matthew Choi, and Alán Aspuru-Guzik, “A composite measurement scheme for efficient quantum observable estimation”, arXiv:2305.02439, (2023).
[17] Zheng An, Jiahui Wu, Muchun Yang, D. L. Zhou, and Bei Zeng, “Unified Quantum State Tomography and Hamiltonian Learning Using Transformer Models: A Language-Translation-Like Approach for Quantum Systems”, arXiv:2304.12010, (2023).
The above citations are from SAO/NASA ADS (last updated successfully 2023-06-01 22:48:29). 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 2023-06-01 22:48:27).
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.
- SEO Powered Content & PR Distribution. Get Amplified Today.
- PlatoAiStream. Web3 Data Intelligence. Knowledge Amplified. Access Here.
- Minting the Future w Adryenn Ashley. Access Here.
- Buy and Sell Shares in PRE-IPO Companies with PREIPO®. Access Here.
- Source: https://quantum-journal.org/papers/q-2023-06-01-1026/
