Overlapped grouping measurement: A unified framework for measuring quantum states

Bujiao Wu1,2, Jinzhao Sun3,1, Qi Huang4,1, and Xiao Yuan1,2

1Center on Frontiers of Computing Studies, Peking University, Beijing 100871, China
2School of Computer Science, Peking University, Beijing 100871, China
3Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
4School of Physics, Peking University, Beijing 100871, China

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

Abstract

Quantum algorithms designed for realistic quantum many-body systems, such as chemistry and materials, usually require a large number of measurements of the Hamiltonian. Exploiting different ideas, such as importance sampling, observable compatibility, or classical shadows of quantum states, different advanced measurement schemes have been proposed to greatly reduce the large measurement cost. Yet, the underline cost reduction mechanisms seem distinct from each other, and how to systematically find the optimal scheme remains a critical challenge. Here, we address this challenge by proposing a unified framework of quantum measurements, incorporating advanced measurement methods as special cases. Our framework allows us to introduce a general scheme – overlapped grouping measurement, which simultaneously exploits the advantages of most existing methods. An intuitive understanding of the scheme is to partition the measurements into overlapped groups with each one consisting of compatible measurements. We provide explicit grouping strategies and numerically verify its performance for different molecular Hamiltonians with up to 16 qubits. Our numerical result shows significant improvements over existing schemes. Our work paves the way for efficient quantum measurement and fast quantum processing with current and near-term quantum devices.

► BibTeX data

► References

[1] Scott Aaronson. Shadow tomography of quantum states. SIAM Journal on Computing, 49 (5): STOC18–368, 2019. 10.1145/​3188745.3188802. URL https:/​/​doi.org/​10.1145/​3188745.3188802.
https:/​/​doi.org/​10.1145/​3188745.3188802

[2] Atithi Acharya, Siddhartha Saha, and Anirvan M Sengupta. Informationally complete povm-based shadow tomography, 2021. URL https:/​/​doi.org/​10.48550/​arXiv.2105.05992.
https:/​/​doi.org/​10.48550/​arXiv.2105.05992

[3] Ryan Babbush, Nathan Wiebe, Jarrod McClean, James McClain, Hartmut Neven, and Garnet Kin-Lic Chan. Low-depth quantum simulation of materials. Phys. Rev. X, 8: 011044, Mar 2018. 10.1103/​PhysRevX.8.011044. URL https:/​/​doi.org/​10.1103/​PhysRevX.8.011044.
https:/​/​doi.org/​10.1103/​PhysRevX.8.011044

[4] 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 (nisq) algorithms, 2021. URL https:/​/​doi.org/​10.1103/​RevModPhys.94.015004.
https:/​/​doi.org/​10.1103/​RevModPhys.94.015004

[5] Carlos Bravo-Prieto, Ryan LaRose, M. Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick J. Coles. Variational quantum linear solver, 2019. URL https:/​/​doi.org/​10.48550/​arXiv.1909.05820.
https:/​/​doi.org/​10.48550/​arXiv.1909.05820

[6] Sergey Bravyi, Sarah Sheldon, Abhinav Kandala, David C. Mckay, and Jay M. Gambetta. Mitigating measurement errors in multiqubit experiments. Phys. Rev. A, 103: 042605, Apr 2021. 10.1103/​PhysRevA.103.042605. URL https:/​/​doi.org/​10.1103/​PhysRevA.103.042605.
https:/​/​doi.org/​10.1103/​PhysRevA.103.042605

[7] Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, Mária Kieferová, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P. D. Sawaya, Sukin Sim, Libor Veis, and Alán Aspuru-Guzik. Quantum chemistry in the age of quantum computing. Chemical Reviews, 119 (19): 10856–10915, 2019. 10.1021/​acs.chemrev.8b00803. URL https:/​/​doi.org/​10.1021/​acs.chemrev.8b00803. PMID: 31469277.
https:/​/​doi.org/​10.1021/​acs.chemrev.8b00803

[8] Juan Carrasquilla, Giacomo Torlai, Roger G Melko, and Leandro Aolita. Reconstructing quantum states with generative models. Nature Machine Intelligence, 1 (3): 155–161, 2019. 10.1038/​s42256-019-0028-1. URL https:/​/​doi.org/​10.1038/​s42256-019-0028-1.
https:/​/​doi.org/​10.1038/​s42256-019-0028-1

[9] Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, et al. Variational quantum algorithms. Nature Reviews Physics, 3 (9): 625–644, 2021. 10.1038/​s42254-021-00348-9. URL https:/​/​doi.org/​10.1038/​s42254-021-00348-9.
https:/​/​doi.org/​10.1038/​s42254-021-00348-9

[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:/​/​doi.org/​10.1103/​PRXQuantum.2.030348.
https:/​/​doi.org/​10.1103/​PRXQuantum.2.030348

[11] Kenny Choo, Antonio Mezzacapo, and Giuseppe Carleo. Fermionic neural-network states for ab-initio electronic structure. Nature communications, 11 (1): 1–7, 2020. 10.1038/​s41467-020-15724-9. URL https:/​/​doi.org/​10.1038/​s41467-020-15724-9.
https:/​/​doi.org/​10.1038/​s41467-020-15724-9

[12] Cristina Cirstoiu, Zoe Holmes, Joseph Iosue, Lukasz Cincio, Patrick J Coles, and Andrew Sornborger. Variational fast forwarding for quantum simulation beyond the coherence time. npj Quantum Information, 6 (1): 1–10, 2020. URL https:/​/​doi.org/​10.1038/​s41534-020-00302-0.
https:/​/​doi.org/​10.1038/​s41534-020-00302-0

[13] J. I. Colless, V. V. Ramasesh, D. Dahlen, M. S. Blok, M. E. Kimchi-Schwartz, J. R. McClean, J. Carter, W. A. de Jong, and I. Siddiqi. Computation of molecular spectra on a quantum processor with an error-resilient algorithm. Phys. Rev. X, 8: 011021, Feb 2018. 10.1103/​PhysRevX.8.011021. URL https:/​/​doi.org/​10.1103/​PhysRevX.8.011021.
https:/​/​doi.org/​10.1103/​PhysRevX.8.011021

[14] Benjamin Commeau, M. Cerezo, Zoë Holmes, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger. Variational hamiltonian diagonalization for dynamical quantum simulation, 2020. URL https:/​/​doi.org/​10.48550/​arXiv.2009.02559.
https:/​/​doi.org/​10.48550/​arXiv.2009.02559

[15] Jordan Cotler and Frank Wilczek. Quantum overlapping tomography. Phys. Rev. Lett., 124: 100401, Mar 2020. 10.1103/​PhysRevLett.124.100401. URL https:/​/​doi.org/​10.1103/​PhysRevLett.124.100401.
https:/​/​doi.org/​10.1103/​PhysRevLett.124.100401

[16] Ophelia Crawford, Barnaby van Straaten, Daochen Wang, Thomas Parks, Earl Campbell, and Stephen Brierley. Efficient quantum measurement of pauli operators in the presence of finite sampling error. Quantum, 5: 385, 2021. 10.22331/​q-2021-01-20-385. URL https:/​/​doi.org/​10.22331%2Fq-2021-01-20-385.
https:/​/​doi.org/​10.22331/​q-2021-01-20-385

[17] E. F. Dumitrescu, A. J. McCaskey, G. Hagen, G. R. Jansen, T. D. Morris, T. Papenbrock, R. C. Pooser, D. J. Dean, and P. Lougovski. Cloud quantum computing of an atomic nucleus. Phys. Rev. Lett., 120: 210501, May 2018. 10.1103/​PhysRevLett.120.210501. URL https:/​/​doi.org/​10.1103/​PhysRevLett.120.210501.
https:/​/​doi.org/​10.1103/​PhysRevLett.120.210501

[18] Suguru Endo, Simon C. Benjamin, and Ying Li. Practical quantum error mitigation for near-future applications. Phys. Rev. X, 8: 031027, Jul 2018. 10.1103/​PhysRevX.8.031027. URL https:/​/​doi.org/​10.1103/​PhysRevX.8.031027.
https:/​/​doi.org/​10.1103/​PhysRevX.8.031027

[19] Suguru Endo, Jinzhao Sun, Ying Li, Simon C. Benjamin, and Xiao Yuan. Variational quantum simulation of general processes. Phys. Rev. Lett., 125: 010501, Jun 2020. 10.1103/​PhysRevLett.125.010501. URL https:/​/​doi.org/​10.1103/​PhysRevLett.125.010501.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.010501

[20] Suguru Endo, Zhenyu Cai, Simon C. Benjamin, and Xiao Yuan. Hybrid quantum-classical algorithms and quantum error mitigation. Journal of the Physical Society of Japan, 90 (3): 032001, 2021. 10.7566/​JPSJ.90.032001. URL https:/​/​doi.org/​10.7566/​JPSJ.90.032001.
https:/​/​doi.org/​10.7566/​JPSJ.90.032001

[21] Keisuke Fujii, Kaoru Mizuta, Hiroshi Ueda, Kosuke Mitarai, Wataru Mizukami, and Yuya O. Nakagawa. Deep variational quantum eigensolver: A divide-and-conquer method for solving a larger problem with smaller size quantum computers. PRX Quantum, 3: 010346, Mar 2022. 10.1103/​PRXQuantum.3.010346. URL https:/​/​doi.org/​10.1103/​PRXQuantum.3.010346.
https:/​/​doi.org/​10.1103/​PRXQuantum.3.010346

[22] Joe Gibbs, Kaitlin Gili, Zoë Holmes, Benjamin Commeau, Andrew Arrasmith, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger. Long-time simulations with high fidelity on quantum hardware, 2021. URL https:/​/​arxiv.org/​abs/​2102.04313.
arXiv:2102.04313

[23] Tudor Giurgica-Tiron, Yousef Hindy, Ryan LaRose, Andrea Mari, and William J. Zeng. Digital zero noise extrapolation for quantum error mitigation. In 2020 IEEE International Conference on Quantum Computing and Engineering (QCE), pages 306–316, 2020. 10.1109/​QCE49297.2020.00045. URL https:/​/​doi.org/​10.1109/​QCE49297.2020.00045.
https:/​/​doi.org/​10.1109/​QCE49297.2020.00045

[24] Pranav Gokhale, Olivia Angiuli, Yongshan Ding, Kaiwen Gui, Teague Tomesh, Martin Suchara, Margaret Martonosi, and Frederic T Chong. Minimizing state preparations in variational quantum eigensolver by partitioning into commuting families. URL https:/​/​doi.org/​10.48550/​arXiv.1907.13623.
https:/​/​doi.org/​10.48550/​arXiv.1907.13623

[25] Harper R Grimsley, Sophia E Economou, Edwin Barnes, and Nicholas J Mayhall. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nature comm., 10 (1): 1–9, 2019. 10.1038/​s41467-018-07090-4. URL https:/​/​doi.org/​10.1038/​s41467-019-10988-2.
https:/​/​doi.org/​10.1038/​s41467-018-07090-4

[26] Charles Hadfield. Adaptive pauli shadows for energy estimation, 2021. URL https:/​/​doi.org/​10.48550/​arXiv.2105.12207.
https:/​/​doi.org/​10.48550/​arXiv.2105.12207

[27] Charles Hadfield, Sergey Bravyi, Rudy Raymond, and Antonio Mezzacapo. Measurements of quantum hamiltonians with locally-biased classical shadows. Communications in Mathematical Physics, 391 (3): 951–967, 2022. 10.1007/​s00220-022-04343-8. URL https:/​/​doi.org/​10.1007/​s00220-022-04343-8.
https:/​/​doi.org/​10.1007/​s00220-022-04343-8

[28] Cornelius Hempel, Christine Maier, Jonathan Romero, Jarrod McClean, Thomas Monz, Heng Shen, Petar Jurcevic, Ben P. Lanyon, Peter Love, Ryan Babbush, Alán Aspuru-Guzik, Rainer Blatt, and Christian F. Roos. Quantum chemistry calculations on a trapped-ion quantum simulator. Phys. Rev. X, 8: 031022, Jul 2018. 10.1103/​PhysRevX.8.031022. URL https:/​/​doi.org/​10.1103/​PhysRevX.8.031022.
https:/​/​doi.org/​10.1103/​PhysRevX.8.031022

[29] Oscar Higgott, Daochen Wang, and Stephen Brierley. Variational Quantum Computation of Excited States. Quantum, 3: 156, July 2019. ISSN 2521-327X. 10.22331/​q-2019-07-01-156. URL https:/​/​doi.org/​10.22331/​q-2019-07-01-156.
https:/​/​doi.org/​10.22331/​q-2019-07-01-156

[30] 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. 10.1109/​QCE52317.2021.00018. URL https:/​/​doi.org/​10.1109/​QCE52317.2021.00018.
https:/​/​doi.org/​10.1109/​QCE52317.2021.00018

[31] 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, 2020. 10.1038/​s41567-020-0932-7. URL https:/​/​doi.org/​10.1038/​s41567-020-0932-7.
https:/​/​doi.org/​10.1038/​s41567-020-0932-7

[32] Hsin-Yuan Huang, Kishor Bharti, and Patrick Rebentrost. Near-term quantum algorithms for linear systems of equations with regression loss functions. New Journal of Physics, 23 (11): 113021, nov 2021a. 10.1088/​1367-2630/​ac325f. URL https:/​/​doi.org/​10.1088/​1367-2630/​ac325f.
https:/​/​doi.org/​10.1088/​1367-2630/​ac325f

[33] Hsin-Yuan Huang, Richard Kueng, and John Preskill. Efficient estimation of pauli observables by derandomization. Phys. Rev. Lett., 127: 030503, Jul 2021b. 10.1103/​PhysRevLett.127.030503. URL https:/​/​doi.org/​10.1103/​PhysRevLett.127.030503.
https:/​/​doi.org/​10.1103/​PhysRevLett.127.030503

[34] William J Huggins, Jarrod R McClean, Nicholas C Rubin, Zhang Jiang, Nathan Wiebe, K Birgitta Whaley, and Ryan Babbush. Efficient and noise resilient measurements for quantum chemistry on near-term quantum computers. npj Quantum Information, 7 (1): 1–9, 2021. 10.1038/​s41534-020-00341-7. URL https:/​/​doi.org/​10.1038/​s41534-020-00341-7.
https:/​/​doi.org/​10.1038/​s41534-020-00341-7

[35] Artur F Izmaylov, Tzu-Ching Yen, Robert A Lang, and Vladyslav Verteletskyi. Unitary partitioning approach to the measurement problem in the variational quantum eigensolver method. Journal of chemical theory and computation, 16 (1): 190–195, 2019a. 10.1021/​acs.jctc.9b00791. URL https:/​/​doi.org/​10.1021/​acs.jctc.9b00791.
https:/​/​doi.org/​10.1021/​acs.jctc.9b00791

[36] Artur F Izmaylov, Tzu-Ching Yen, and Ilya G Ryabinkin. Revising the measurement process in the variational quantum eigensolver: is it possible to reduce the number of separately measured operators? Chemical science, 10 (13): 3746–3755, 2019b. 10.1039/​C8SC05592K. URL https:/​/​doi.org/​10.1039/​C8SC05592K.
https:/​/​doi.org/​10.1039/​C8SC05592K

[37] Andrew Jena, Scott Genin, and Michele Mosca. Pauli partitioning with respect to gate sets, 2019. URL https:/​/​doi.org/​10.48550/​arXiv.1907.07859.
https:/​/​doi.org/​10.48550/​arXiv.1907.07859

[38] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M Chow, and Jay M Gambetta. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature, 549 (7671): 242–246, 2017. 10.1038/​nature23879. URL https:/​/​doi.org/​10.1038/​nature23879.
https:/​/​doi.org/​10.1038/​nature23879

[39] Ying Li and Simon C. Benjamin. Efficient variational quantum simulator incorporating active error minimization. Phys. Rev. X, 7: 021050, Jun 2017. 10.1103/​PhysRevX.7.021050. URL https:/​/​doi.org/​10.1103/​PhysRevX.7.021050.
https:/​/​doi.org/​10.1103/​PhysRevX.7.021050

[40] Jin-Guo Liu, Yi-Hong Zhang, Yuan Wan, and Lei Wang. Variational quantum eigensolver with fewer qubits. Phys. Rev. Research, 1: 023025, Sep 2019. 10.1103/​PhysRevResearch.1.023025. URL https:/​/​doi.org/​10.1103/​PhysRevResearch.1.023025.
https:/​/​doi.org/​10.1103/​PhysRevResearch.1.023025

[41] He Ma, Marco Govoni, and Giulia Galli. Quantum simulations of materials on near-term quantum computers. npj Computational Materials, 6 (1): 1–8, 2020. 10.1038/​s41524-020-00353-z. URL https:/​/​doi.org/​10.1038/​s41524-020-00353-z.
https:/​/​doi.org/​10.1038/​s41524-020-00353-z

[42] Sam McArdle, Tyson Jones, Suguru Endo, Ying Li, Simon C Benjamin, and Xiao Yuan. Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Information, 5 (1): 1–6, 2019. 10.1038/​s41534-019-0187-2. URL https:/​/​doi.org/​10.1038/​s41534-019-0187-2.
https:/​/​doi.org/​10.1038/​s41534-019-0187-2

[43] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan. Quantum computational chemistry. Rev. Mod. Phys., 92: 015003, Mar 2020. 10.1103/​RevModPhys.92.015003. URL https:/​/​doi.org/​10.1103/​RevModPhys.92.015003.
https:/​/​doi.org/​10.1103/​RevModPhys.92.015003

[44] Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18 (2): 023023, feb 2016. 10.1088/​1367-2630/​18/​2/​023023. URL https:/​/​doi.org/​10.1088/​1367-2630/​18/​2/​023023.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​2/​023023

[45] Jarrod R McClean, Mollie E Kimchi-Schwartz, Jonathan Carter, and Wibe A de Jong. Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states. Physical Review A, 95 (4): 042308, 2017. URL https:/​/​doi.org/​10.1103/​PhysRevA.95.042308.
https:/​/​doi.org/​10.1103/​PhysRevA.95.042308

[46] Jarrod R McClean, Zhang Jiang, Nicholas C Rubin, Ryan Babbush, and Hartmut Neven. Decoding quantum errors with subspace expansions. Nature Communications, 11 (1): 1–9, 2020. 10.1038/​s41467-020-14341-w. URL https:/​/​doi.org/​10.1038/​s41467-020-14341-w.
https:/​/​doi.org/​10.1038/​s41467-020-14341-w

[47] Nikolaj Moll, Panagiotis Barkoutsos, Lev S Bishop, Jerry M Chow, Andrew Cross, Daniel J Egger, Stefan Filipp, Andreas Fuhrer, Jay M Gambetta, Marc Ganzhorn, et al. Quantum optimization using variational algorithms on near-term quantum devices. Quantum Science and Technology, 3 (3): 030503, 2018. 10.1088/​2058-9565/​aab822. URL https:/​/​doi.org/​10.1088/​2058-9565/​aab822.
https:/​/​doi.org/​10.1088/​2058-9565/​aab822

[48] Ken M Nakanishi, Kosuke Mitarai, and Keisuke Fujii. Subspace-search variational quantum eigensolver for excited states. Physical Review Research, 1 (3): 033062, 2019. 10.1103/​PhysRevResearch.1.033062. URL https:/​/​doi.org/​10.1103/​PhysRevResearch.1.033062.
https:/​/​doi.org/​10.1103/​PhysRevResearch.1.033062

[49] Bryan O'Gorman, William J Huggins, Eleanor G Rieffel, and K Birgitta Whaley. Generalized swap networks for near-term quantum computing, 2019. URL https:/​/​doi.org/​10.48550/​arXiv.1905.05118.
https:/​/​doi.org/​10.48550/​arXiv.1905.05118

[50] P. J. J. O'Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Jeffrey, E. Lucero, A. Megrant, J. Y. Mutus, M. Neeley, C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, P. V. Coveney, P. J. Love, H. Neven, A. Aspuru-Guzik, and J. M. Martinis. Scalable quantum simulation of molecular energies. Phys. Rev. X, 6: 031007, Jul 2016. 10.1103/​PhysRevX.6.031007. URL https:/​/​doi.org/​10.1103/​PhysRevX.6.031007.
https:/​/​doi.org/​10.1103/​PhysRevX.6.031007

[51] Matthew Otten and Stephen K Gray. Accounting for errors in quantum algorithms via individual error reduction. Npj Quantum Inf., 5 (1): 11, 2019. 10.1038/​s41534-019-0125-3. URL https:/​/​doi.org/​10.1038/​s41534-019-0125-3.
https:/​/​doi.org/​10.1038/​s41534-019-0125-3

[52] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J Love, Alán Aspuru-Guzik, and Jeremy L O’brien. A variational eigenvalue solver on a photonic quantum processor. Nature comm., 5: 4213, 2014. 10.1038/​ncomms5213. URL https:/​/​doi.org/​10.1038/​ncomms5213.
https:/​/​doi.org/​10.1038/​ncomms5213

[53] John Preskill. Quantum computing in the nisq era and beyond. Quantum, 2: 79, 2018. 10.22331/​q-2018-08-06-79. URL https:/​/​doi.org/​10.22331/​q-2018-08-06-79.
https:/​/​doi.org/​10.22331/​q-2018-08-06-79

[54] Google AI Quantum, Collaborators*†, Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B Buckley, et al. Hartree-fock on a superconducting qubit quantum computer. Science, 369 (6507): 1084–1089, 2020. 10.1126/​science.abb9811. URL https:/​/​doi.org/​10.1126/​science.abb9811.
https:/​/​doi.org/​10.1126/​science.abb9811

[55] Nicholas C Rubin, Ryan Babbush, and Jarrod McClean. Application of fermionic marginal constraints to hybrid quantum algorithms. New Journal of Physics, 20 (5): 053020, may 2018. 10.1088/​1367-2630/​aab919. URL https:/​/​doi.org/​10.1088/​1367-2630/​aab919.
https:/​/​doi.org/​10.1088/​1367-2630/​aab919

[56] Ariel Shlosberg, Andrew J. Jena, Priyanka Mukhopadhyay, Jan F. Haase, Felix Leditzky, and Luca Dellantonio. Adaptive estimation of quantum observables, 2021. URL https:/​/​doi.org/​10.48550/​arXiv.2110.15339.
https:/​/​doi.org/​10.48550/​arXiv.2110.15339

[57] Armands Strikis, Dayue Qin, Yanzhu Chen, Simon C. Benjamin, and Ying Li. Learning-based quantum error mitigation. PRX Quantum, 2: 040330, Nov 2021. 10.1103/​PRXQuantum.2.040330. URL https:/​/​doi.org/​10.1103/​PRXQuantum.2.040330.
https:/​/​doi.org/​10.1103/​PRXQuantum.2.040330

[58] 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, Jan 2021. 10.1103/​PRXQuantum.2.010307. URL https:/​/​doi.org/​10.1103/​PRXQuantum.2.010307.
https:/​/​doi.org/​10.1103/​PRXQuantum.2.010307

[59] Jinzhao Sun, Xiao Yuan, Takahiro Tsunoda, Vlatko Vedral, Simon C. Benjamin, and Suguru Endo. Mitigating realistic noise in practical noisy intermediate-scale quantum devices. Phys. Rev. Applied, 15: 034026, Mar 2021. 10.1103/​PhysRevApplied.15.034026. URL https:/​/​doi.org/​10.1103/​PhysRevApplied.15.034026.
https:/​/​doi.org/​10.1103/​PhysRevApplied.15.034026

[60] Jinzhao Sun, Suguru Endo, Huiping Lin, Patrick Hayden, Vlatko Vedral, and Xiao Yuan. Perturbative quantum simulation, Sep 2022. URL https:/​/​doi.org/​10.1103/​PhysRevLett.129.120505.
https:/​/​doi.org/​10.1103/​PhysRevLett.129.120505

[61] Kristan Temme, Sergey Bravyi, and Jay M. Gambetta. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett., 119: 180509, Nov 2017. 10.1103/​PhysRevLett.119.180509. URL https:/​/​doi.org/​10.1103/​PhysRevLett.119.180509.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.180509

[62] Giacomo Torlai, Guglielmo Mazzola, Juan Carrasquilla, Matthias Troyer, Roger Melko, and Giuseppe Carleo. Neural-network quantum state tomography. Nature Physics, 14 (5): 447–450, 2018. 10.1038/​s41567-018-0048-5. URL https:/​/​doi.org/​10.1038/​s41567-018-0048-5.
https:/​/​doi.org/​10.1038/​s41567-018-0048-5

[63] Giacomo Torlai, Guglielmo Mazzola, Giuseppe Carleo, and Antonio Mezzacapo. Precise measurement of quantum observables with neural-network estimators. Phys. Rev. Res., 2: 022060, Jun 2020. 10.1103/​PhysRevResearch.2.022060. URL https:/​/​doi.org/​10.1103/​PhysRevResearch.2.022060.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.022060

[64] Harish J Vallury, Michael A Jones, Charles D Hill, and Lloyd CL Hollenberg. Quantum computed moments correction to variational estimates. Quantum, 4: 373, 2020. 10.22331/​q-2020-12-15-373. URL https:/​/​doi.org/​10.22331/​q-2020-12-15-373.
https:/​/​doi.org/​10.22331/​q-2020-12-15-373

[65] Vladyslav Verteletskyi, Tzu-Ching Yen, and Artur F Izmaylov. Measurement optimization in the variational quantum eigensolver using a minimum clique cover. The Journal of chemical physics, 152 (12): 124114, 2020. 10.1063/​1.5141458. URL https:/​/​doi.org/​10.1063/​1.5141458.
https:/​/​doi.org/​10.1063/​1.5141458

[66] Samson Wang, Enrico Fontana, Marco Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J Coles. Noise-induced barren plateaus in variational quantum algorithms. Nature communications, 12 (1): 1–11, 2021. 10.1038/​s41467-021-27045-6. URL https:/​/​doi.org/​10.1038/​s41467-021-27045-6.
https:/​/​doi.org/​10.1038/​s41467-021-27045-6

[67] Dave Wecker, Matthew B. Hastings, and Matthias Troyer. Progress towards practical quantum variational algorithms. Phys. Rev. A, 92: 042303, Oct 2015. 10.1103/​PhysRevA.92.042303. URL https:/​/​doi.org/​10.1103/​PhysRevA.92.042303.
https:/​/​doi.org/​10.1103/​PhysRevA.92.042303

[68] Xiaosi Xu, Jinzhao Sun, Suguru Endo, Ying Li, Simon C. Benjamin, and Xiao Yuan. Variational algorithms for linear algebra. Science Bulletin, 2021. ISSN 2095-9273. 10.1016/​j.scib.2021.06.023. URL https:/​/​doi.org/​10.1016/​j.scib.2021.06.023.
https:/​/​doi.org/​10.1016/​j.scib.2021.06.023

[69] Zhi-Cheng Yang, Armin Rahmani, Alireza Shabani, Hartmut Neven, and Claudio Chamon. Optimizing variational quantum algorithms using pontryagin's minimum principle. Phys. Rev. X, 7: 021027, May 2017. 10.1103/​PhysRevX.7.021027. URL https:/​/​doi.org/​10.1103/​PhysRevX.7.021027.
https:/​/​doi.org/​10.1103/​PhysRevX.7.021027

[70] Tzu-Ching Yen, Vladyslav Verteletskyi, and Artur F Izmaylov. Measuring all compatible operators in one series of single-qubit measurements using unitary transformations. Journal of chemical theory and computation, 16 (4): 2400–2409, 2020. 10.1021/​acs.jctc.0c00008. URL https:/​/​doi.org/​10.1021/​acs.jctc.0c00008.
https:/​/​doi.org/​10.1021/​acs.jctc.0c00008

[71] 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, 2022. URL https:/​/​doi.org/​10.48550/​arXiv.2201.01471.
https:/​/​doi.org/​10.48550/​arXiv.2201.01471

[72] Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C Benjamin. Theory of variational quantum simulation. Quantum, 3: 191, 2019. 10.22331/​q-2019-10-07-191. URL https:/​/​doi.org/​10.22331/​q-2019-10-07-191.
https:/​/​doi.org/​10.22331/​q-2019-10-07-191

[73] Xiao Yuan, Jinzhao Sun, Junyu Liu, Qi Zhao, and You Zhou. Quantum simulation with hybrid tensor networks. Phys. Rev. Lett., 127: 040501, Jul 2021. 10.1103/​PhysRevLett.127.040501. URL https:/​/​doi.org/​10.1103/​PhysRevLett.127.040501.
https:/​/​doi.org/​10.1103/​PhysRevLett.127.040501

[74] Ting Zhang, Jinzhao Sun, Xiao-Xu Fang, Xiao-Ming Zhang, Xiao Yuan, and He Lu. Experimental quantum state measurement with classical shadows. Phys. Rev. Lett., 127: 200501, Nov 2021. 10.1103/​PhysRevLett.127.200501. URL https:/​/​doi.org/​10.1103/​PhysRevLett.127.200501.
https:/​/​doi.org/​10.1103/​PhysRevLett.127.200501

[75] Zi-Jian Zhang, Jinzhao Sun, Xiao Yuan, and Man-Hong Yung. Low-depth hamiltonian simulation by adaptive product formula, 2020. URL https:/​/​doi.org/​10.48550/​arXiv.2011.05283.
https:/​/​doi.org/​10.48550/​arXiv.2011.05283

[76] Andrew Zhao, Andrew Tranter, William M. Kirby, Shu Fay Ung, Akimasa Miyake, and Peter J. Love. Measurement reduction in variational quantum algorithms. Phys. Rev. A, 101: 062322, Jun 2020. 10.1103/​PhysRevA.101.062322. URL https:/​/​doi.org/​10.1103/​PhysRevA.101.062322.
https:/​/​doi.org/​10.1103/​PhysRevA.101.062322

[77] 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:/​/​doi.org/​10.1103/​PhysRevLett.127.110504.
https:/​/​doi.org/​10.1103/​PhysRevLett.127.110504

[78] Leo Zhou, Sheng-Tao Wang, Soonwon Choi, Hannes Pichler, and Mikhail D. Lukin. Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices. Phys. Rev. X, 10: 021067, Jun 2020. 10.1103/​PhysRevX.10.021067. URL https:/​/​doi.org/​10.1103/​PhysRevX.10.021067.
https:/​/​doi.org/​10.1103/​PhysRevX.10.021067

Cited by

[1] 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).

[2] Linqing Peng, Xing Zhang, and Garnet Kin-Lic Chan, "Fermionic Reduced Density Low-Rank Matrix Completion, Noise Filtering, and Measurement Reduction in Quantum Simulations", Journal of Chemical Theory and Computation 19 24, 9151 (2023).

[3] Yanqi Song, Yusen Wu, Shengyao Wu, Dandan Li, Qiaoyan Wen, Sujuan Qin, and Fei Gao, "A quantum federated learning framework for classical clients", Science China Physics, Mechanics & Astronomy 67 5, 250311 (2024).

[4] Kouhei Nakaji, Suguru Endo, Yuichiro Matsuzaki, and Hideaki Hakoshima, "Measurement optimization of variational quantum simulation by classical shadow and derandomization", Quantum 7, 995 (2023).

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

[6] Linghua Zhu, Senwei Liang, Chao Yang, and Xiaosong Li, "Optimizing Shot Assignment in Variational Quantum Eigensolver Measurement", Journal of Chemical Theory and Computation 20 6, 2390 (2024).

[7] 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).

[8] Yuma Nakamura, Yoshichika Yano, and Nobuyuki Yoshioka, "Adaptive measurement strategy for quantum subspace methods", New Journal of Physics 26 3, 033028 (2024).

[9] 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, 14 (2023).

[10] William Kirby, Mario Motta, and Antonio Mezzacapo, "Exact and efficient Lanczos method on a quantum computer", Quantum 7, 1018 (2023).

[11] Jiaqi Miao, Chang-Yu Hsieh, and Shi-Xin Zhang, "Neural-network-encoded variational quantum algorithms", Physical Review Applied 21 1, 014053 (2024).

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

[13] Antoine Michel, Sebastian Grijalva, Loïc Henriet, Christophe Domain, and Antoine Browaeys, "Blueprint for a digital-analog variational quantum eigensolver using Rydberg atom arrays", Physical Review A 107 4, 042602 (2023).

[14] Andrew Zhao, Nicholas C. Rubin, and Akimasa Miyake, "Fermionic Partial Tomography via Classical Shadows", Physical Review Letters 127 11, 110504 (2021).

[15] Ting Zhang, Jinzhao Sun, Xiao-Xu Fang, Xiao-Ming Zhang, Xiao Yuan, and He Lu, "Experimental Quantum State Measurement with Classical Shadows", Physical Review Letters 127 20, 200501 (2021).

[16] Junyu Liu, Zimu Li, Han Zheng, Xiao Yuan, and Jinzhao Sun, "Towards a variational Jordan-Lee-Preskill quantum algorithm", Machine Learning: Science and Technology 3 4, 045030 (2022).

[17] Dax Enshan Koh and Sabee Grewal, "Classical Shadows With Noise", arXiv:2011.11580, (2020).

[18] Zi-Jian Zhang, Jinzhao Sun, Xiao Yuan, and Man-Hong Yung, "Low-Depth Hamiltonian Simulation by an Adaptive Product Formula", Physical Review Letters 130 4, 040601 (2023).

[19] Bryce Fuller, Charles Hadfield, Jennifer R. Glick, Takashi Imamichi, Toshinari Itoko, Richard J. Thompson, Yang Jiao, Marna M. Kagele, Adriana W. Blom-Schieber, Rudy Raymond, and Antonio Mezzacapo, "Approximate Solutions of Combinatorial Problems via Quantum Relaxations", arXiv:2111.03167, (2021).

[20] Masaya Kohda, Ryosuke Imai, Keita Kanno, Kosuke Mitarai, Wataru Mizukami, and Yuya O. Nakagawa, "Quantum expectation-value estimation by computational basis sampling", Physical Review Research 4 3, 033173 (2022).

[21] Ariel Shlosberg, Andrew J. Jena, Priyanka Mukhopadhyay, Jan F. Haase, Felix Leditzky, and Luca Dellantonio, "Adaptive estimation of quantum observables", Quantum 7, 906 (2023).

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

[23] Daniel Miller, Laurin E. Fischer, Igor O. Sokolov, Panagiotis Kl. Barkoutsos, and Ivano Tavernelli, "Hardware-Tailored Diagonalization Circuits", arXiv:2203.03646, (2022).

[24] Alexander Gresch and Martin Kliesch, "Guaranteed efficient energy estimation of quantum many-body Hamiltonians using ShadowGrouping", arXiv:2301.03385, (2023).

[25] Weitang Li, Zigeng Huang, Changsu Cao, Yifei Huang, Zhigang Shuai, Xiaoming Sun, Jinzhao Sun, Xiao Yuan, and Dingshun Lv, "Toward Practical Quantum Embedding Simulation of Realistic Chemical Systems on Near-term Quantum Computers", arXiv:2109.08062, (2021).

[26] Zhenhuan Liu, Pei Zeng, You Zhou, and Mile Gu, "Characterizing correlation within multipartite quantum systems via local randomized measurements", Physical Review A 105 2, 022407 (2022).

[27] Seonghoon Choi, Ignacio Loaiza, and Artur F. Izmaylov, "Fluid fermionic fragments for optimizing quantum measurements of electronic Hamiltonians in the variational quantum eigensolver", Quantum 7, 889 (2023).

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

[29] Andrew Jena, Scott N. Genin, and Michele Mosca, "Optimization of variational-quantum-eigensolver measurement by partitioning Pauli operators using multiqubit Clifford gates on noisy intermediate-scale quantum hardware", Physical Review A 106 4, 042443 (2022).

[30] Xiao-Ming Zhang, Zixuan Huo, Kecheng Liu, Ying Li, and Xiao Yuan, "Unbiased random circuit compiler for time-dependent Hamiltonian simulation", arXiv:2212.09445, (2022).

[31] Marco Majland, Rasmus Berg Jensen, Mads Greisen Højlund, Nikolaj Thomas Zinner, and Ove Christiansen, "Runtime optimization for vibrational structure on quantum computers: coordinates and measurement schemes", arXiv:2211.11615, (2022).

[32] Seonghoon Choi and Artur F. Izmaylov, "Measurement optimization techniques for excited electronic states in near-term quantum computing algorithms", arXiv:2302.11421, (2023).

[33] Yifei Chen, Zhan Yu, Chenghong Zhu, and Xin Wang, "Efficient information recovery from Pauli noise via classical shadow", arXiv:2305.04148, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-03-28 14:14:37) and SAO/NASA ADS (last updated successfully 2024-03-28 14:14:37). The list may be incomplete as not all publishers provide suitable and complete citation data.