Adaptive estimation of quantum observables

Ariel Shlosberg1,2, Andrew J. Jena3,4, Priyanka Mukhopadhyay3,4, Jan F. Haase3,5,6, Felix Leditzky3,4,7,8, and Luca Dellantonio3,5,9

1JILA, University of Colorado and National Institute of Standards and Technology, Boulder, CO 80309, USA
2Department of Physics, University of Colorado, Boulder, CO 80309, USA
3Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada
4Department of Combinatorics & Optimization, University of Waterloo, Waterloo, ON N2L 3G1, Canada
5Department of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada
6Institute of Theoretical Physics and IQST, Universität Ulm, D-89069 Ulm, Germany
7Department of Mathematics and IQUIST, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA
8Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
9Department of Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom

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


The accurate estimation of quantum observables is a critical task in science. With progress on the hardware, measuring a quantum system will become increasingly demanding, particularly for variational protocols that require extensive sampling. Here, we introduce a measurement scheme that adaptively modifies the estimator based on previously obtained data. Our algorithm, which we call AEQuO, continuously monitors both the estimated average and the associated error of the considered observable, and determines the next measurement step based on this information. We allow both for overlap and non-bitwise commutation relations in the subsets of Pauli operators that are simultaneously probed, thereby maximizing the amount of gathered information. AEQuO comes in two variants: a greedy bucket-filling algorithm with good performance for small problem instances, and a machine learning-based algorithm with more favorable scaling for larger instances. The measurement configuration determined by these subroutines is further post-processed in order to lower the error on the estimator. We test our protocol on chemistry Hamiltonians, for which AEQuO provides error estimates that improve on all state-of-the-art methods based on various grouping techniques or randomized measurements, thus greatly lowering the toll of measurements in current and future quantum applications.

Quantum systems, as opposed to classical ones, are irreversibly destroyed every time they are measured. This has deep implications when one wants to extract information from a quantum system. For instance, when one must estimate the average value of an observable, it is often required to repeat the whole experiment several times. Depending on the measurement strategy employed, the requirements to achieve the same precision vary considerably. In this work, we propose a new approach that considerably lowers the resources on the hardware. Our strategy is adaptive, in the sense that learns and improves the measurement allocation while data is being acquired. Furthermore, it allows for estimating both the average and the error affecting the desired observable at the same time. Compared with other state-of-the-art approaches, we demonstrate consistent and considerable improvement in the accuracy of estimation when our protocol is employed.

► BibTeX data

► References

[1] P. W. Shor ``Algorithms for quantum computation: discrete logarithms and factoring'' Proceedings 35th Annual Symposium on Foundations of Computer Science 124-134 (1994).

[2] Michael A. Nielsenand Issaac L. Chuang ``Quantum Computation and Quantum Information'' Cambridge University Press (2010).

[3] Antonio Acín, Immanuel Bloch, Harry Buhrman, Tommaso Calarco, Christopher Eichler, Jens Eisert, Daniel Esteve, Nicolas Gisin, Steffen J Glaser, Fedor Jelezko, Stefan Kuhr, Maciej Lewenstein, Max F Riedel, Piet O Schmidt, Rob Thew, Andreas Wallraff, Ian Walmsley, and Frank K Wilhelm, ``The quantum technologies roadmap: a European community view'' New Journal of Physics 20, 080201 (2018).

[4] John Preskill ``Quantum Computing in the NISQ era and beyond'' Quantum 2, 79 (2018).

[5] I. M. Georgescu, S. Ashhab, and Franco Nori, ``Quantum simulation'' Reviews of Modern Physics 86, 153–185 (2014).

[6] Mari Carmen Banuls, Rainer Blatt, Jacopo Catani, Alessio Celi, Juan Ignacio Cirac, Marcello Dalmonte, Leonardo Fallani, Karl Jansen, Maciej Lewenstein, and Simone Montangero, ``Simulating lattice gauge theories within quantum technologies'' The European Physical Journal D 74, 1–42 (2020).

[7] Jan F. Haase, Luca Dellantonio, Alessio Celi, Danny Paulson, Angus Kan, Karl Jansen, and Christine A Muschik, ``A resource efficient approach for quantum and classical simulations of gauge theories in particle physics'' Quantum 5, 393 (2021).

[8] Danny Paulson, Luca Dellantonio, Jan F. Haase, Alessio Celi, Angus Kan, Andrew Jena, Christian Kokail, Rick van Bijnen, Karl Jansen, Peter Zoller, and Christine A. Muschik, ``Simulating 2D Effects in Lattice Gauge Theories on a Quantum Computer'' PRX Quantum 2, 030334 (2021).

[9] 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, 10856–10915 (2019).

[10] John Preskill ``Quantum computing 40 years later'' arXiv preprint (2021).

[11] Heinz-Peter Breuerand Francesco Petruccione ``The theory of open quantum systems'' Oxford University Press on Demand (2002).

[12] Y. Cao, J. Romero, and A. Aspuru-Guzik, ``Potential of quantum computing for drug discovery'' IBM Journal of Research and Development 62, 6:1–6:20 (2018).

[13] W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland, ``Quantum projection noise: Population fluctuations in two-level systems'' Physical Review A 47, 3554–3570 (1993).

[14] Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, and Lukasz Cincio, ``Variational quantum algorithms'' Nature Reviews Physics 3, 625–644 (2021).

[15] R. R. Ferguson, L. Dellantonio, A. Al Balushi, K. Jansen, W. Dür, and C. A. Muschik, ``Measurement-Based Variational Quantum Eigensolver'' Physical Review Letters 126, 220501 (2021).

[16] Andrew Jena, Scott Genin, and Michele Mosca, ``Pauli Partitioning with Respect to Gate Sets'' arXiv preprint (2019).

[17] Jarrod R. McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik, ``The theory of variational hybrid quantum-classical algorithms'' New Journal of Physics 18, 023023 (2016).

[18] 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, 124114 (2020).

[19] Andrew Arrasmith, Lukasz Cincio, Rolando D. Somma, and Patrick J. Coles, ``Operator Sampling for Shot-frugal Optimization in Variational Algorithms'' arXiv preprint (2020).

[20] 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).

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

[22] Giacomo Torlai, Guglielmo Mazzola, Giuseppe Carleo, and Antonio Mezzacapo, ``Precise measurement of quantum observables with neural-network estimators'' Physical Review Research 2, 022060 (2020).

[23] Stefan Hillmich, Charles Hadfield, Rudy Raymond, Antonio Mezzacapo, and Robert Wille, ``Decision Diagrams for Quantum Measurements with Shallow Circuits'' 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) 24–34 (2021).

[24] 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).

[25] Charles Hadfield, Sergey Bravyi, Rudy Raymond, and Antonio Mezzacapo, ``Measurements of Quantum Hamiltonians with Locally-Biased Classical Shadows'' Communications in Mathematical Physics 391, 951–967 (2022).

[26] Charles Hadfield ``Adaptive Pauli Shadows for Energy Estimation'' arXiv preprint (2021).

[27] Bujiao Wu, Jinzhao Sun, Qi Huang, and Xiao Yuan, ``Overlapped grouping measurement: A unified framework for measuring quantum states'' arXiv preprint (2021).

[28] Masaya Kohda, Ryosuke Imai, Keita Kanno, Kosuke Mitarai, Wataru Mizukami, and Yuya O. Nakagawa, ``Quantum expectation-value estimation by computational basis sampling'' Phys. Rev. Res. 4, 033173 (2022).

[29] 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'' arXiv preprint (2019).

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

[31] 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, 2400–2409 (2020).

[32] 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, 190–195 (2020).

[33] Cambyse Rouzéand Daniel Stilck França ``Learning quantum many-body systems from a few copies'' arXiv preprint (2021).

[34] Andrew J. Jenaand Ariel Shlosberg ``VQE measurement optimization (GitHub repository)'' https:/​/​​AndrewJena/​VQE_measurement_optimization (2021).

[35] Scott Aaronsonand Daniel Gottesman ``Improved simulation of stabilizer circuits'' Physical Review A 70, 052328 (2004).

[36] Coen Bronand Joep Kerbosch ``Algorithm 457: finding all cliques of an undirected graph'' Communications of the ACM 16, 575–577 (1973).

[37] Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein, ``Introduction to algorithms'' MIT press (2009).

[38] Stephan Hoyer, Jascha Sohl-Dickstein, and Sam Greydanus, ``Neural reparameterization improves structural optimization'' NeurIPS 2019 Deep Inverse Workshop (2019).

[39] Herbert Robbinsand Sutton Monro ``A stochastic approximation method'' The Annals of Mathematical Statistics 400–407 (1951).

[40] Diederik P. Kingmaand Jimmy Ba ``Adam: A Method for Stochastic Optimization'' 3rd International Conference on Learning Representations (2015).

[41] Stephen Wrightand Jorge Nocedal ``Numerical Optimization'' Springer Science 35, 7 (1999).

[42] Philip E. Gilland Walter Murray ``Quasi-Newton methods for unconstrained optimization'' IMA Journal of Applied Mathematics 9, 91–108 (1972).

[43] Chigozie Nwankpa, Winifred Ijomah, Anthony Gachagan, and Stephen Marshall, ``Activation Functions: Comparison of trends in Practice and Research for Deep Learning'' arXiv preprint (2018).

[44] Fabian H.L. Essler, Holger Frahm, Frank Göhmann, Andreas Klümper, and Vladimir E Korepin, ``The one-dimensional Hubbard model'' Cambridge University Press (2005).

[45] Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S. Yu, ``A Comprehensive Survey on Graph Neural Networks'' IEEE Transactions on Neural Networks and Learning Systems 32, 4–24 (2021).

[46] J. F. Haase, P. J. Vetter, T. Unden, A. Smirne, J. Rosskopf, B. Naydenov, A. Stacey, F. Jelezko, M. B. Plenio, and S. F. Huelga, ``Controllable Non-Markovianity for a Spin Qubit in Diamond'' Physical Review Letters 121, 060401 (2018).

[47] Nicholas C. Rubin, Ryan Babbush, and Jarrod McClean, ``Application of fermionic marginal constraints to hybrid quantum algorithms'' New Journal of Physics 20, 053020 (2018).

[48] John Kruschke ``Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan'' Academic Press (2014).

[49] Andrew Gelman, John B. Carlin, Hal S. Stern, and Donald B. Rubin, ``Bayesian data analysis'' Chapman Hall/​CRC (1995).

[50] Paolo Fornasini ``The uncertainty in physical measurements: an introduction to data analysis in the physics laboratory'' Springer (2008).

[51] Roger A. Hornand Charles R. Johnson ``Matrix analysis'' Cambridge University Press (2012).

[52] J. W. Moonand L. Moser ``On cliques in graphs'' Israel Journal of Mathematics 3, 23–28 (1965).

[53] Dong C. Liuand Jorge Nocedal ``On the limited memory BFGS method for large scale optimization'' Mathematical programming 45, 503–528 (1989).

Cited by

[1] David Peral-García, Juan Cruz-Benito, and Francisco José García-Peñalvo, "Comparing Natural Language Processing and Quantum Natural Processing approaches in text classification tasks", Expert Systems with Applications 254, 124427 (2024).

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

[3] 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).

[4] Lane G. Gunderman, Andrew Jena, and Luca Dellantonio, "Minimal qubit representations of Hamiltonians via conserved charges", Physical Review A 109 2, 022618 (2024).

[5] Laurin E. Fischer, Timothée Dao, Ivano Tavernelli, and Francesco Tacchino, "Dual-frame optimization for informationally complete quantum measurements", Physical Review A 109 6, 062415 (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] Ruidi Zhu, Ciara Pike-Burke, and Florian Mintert, "Active learning for quantum mechanical measurements", Physical Review A 109 6, 062404 (2024).

[8] Albie Chan, Zheng Shi, Luca Dellantonio, Wolfgang Dür, and Christine A. Muschik, "Measurement-Based Infused Circuits for Variational Quantum Eigensolvers", Physical Review Letters 132 24, 240601 (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] Yuri Alexeev, Maximilian Amsler, Marco Antonio Barroca, Sanzio Bassini, Torey Battelle, Daan Camps, David Casanova, Young jai Choi, Frederic T. Chong, Charles Chung, Christopher Codella, Antonio D. Córcoles, James Cruise, Alberto Di Meglio, Ivan Duran, Thomas Eckl, Sophia Economou, Stephan Eidenbenz, Bruce Elmegreen, Clyde Fare, Ismael Faro, Cristina Sanz Fernández, Rodrigo Neumann Barros Ferreira, Keisuke Fuji, Bryce Fuller, Laura Gagliardi, Giulia Galli, Jennifer R. Glick, Isacco Gobbi, Pranav Gokhale, Salvador de la Puente Gonzalez, Johannes Greiner, Bill Gropp, Michele Grossi, Emmanuel Gull, Burns Healy, Matthew R. Hermes, Benchen Huang, Travis S. Humble, Nobuyasu Ito, Artur F. Izmaylov, Ali Javadi-Abhari, Douglas Jennewein, Shantenu Jha, Liang Jiang, Barbara Jones, Wibe Albert de Jong, Petar Jurcevic, William Kirby, Stefan Kister, Masahiro Kitagawa, Joel Klassen, Katherine Klymko, Kwangwon Koh, Masaaki Kondo, Dog̃a Murat Kürkçüog̃lu, Krzysztof Kurowski, Teodoro Laino, Ryan Landfield, Matt Leininger, Vicente Leyton-Ortega, Ang Li, Meifeng Lin, Junyu Liu, Nicolas Lorente, Andre Luckow, Simon Martiel, Francisco Martin-Fernandez, Margaret Martonosi, Claire Marvinney, Arcesio Castaneda Medina, Dirk Merten, Antonio Mezzacapo, Kristel Michielsen, Abhishek Mitra, Tushar Mittal, Kyungsun Moon, Joel Moore, Sarah Mostame, Mario Motta, Young-Hye Na, Yunseong Nam, Prineha Narang, Yu-ya Ohnishi, Daniele Ottaviani, Matthew Otten, Scott Pakin, Vincent R. Pascuzzi, Edwin Pednault, Tomasz Piontek, Jed Pitera, Patrick Rall, Gokul Subramanian Ravi, Niall Robertson, Matteo A.C. Rossi, Piotr Rydlichowski, Hoon Ryu, Georgy Samsonidze, Mitsuhisa Sato, Nishant Saurabh, Vidushi Sharma, Kunal Sharma, Soyoung Shin, George Slessman, Mathias Steiner, Iskandar Sitdikov, In-Saeng Suh, Eric D. Switzer, Wei Tang, Joel Thompson, Synge Todo, Minh C. Tran, Dimitar Trenev, Christian Trott, Huan-Hsin Tseng, Norm M. Tubman, Esin Tureci, David García Valiñas, Sofia Vallecorsa, Christopher Wever, Konrad Wojciechowski, Xiaodi Wu, Shinjae Yoo, Nobuyuki Yoshioka, Victor Wen-zhe Yu, Seiji Yunoki, Sergiy Zhuk, and Dmitry Zubarev, "Quantum-centric supercomputing for materials science: A perspective on challenges and future directions", Future Generation Computer Systems (2024).

[11] Lane G. Gunderman, "Transforming collections of Pauli operators into equivalent collections of Pauli operators over minimal registers", Physical Review A 107 6, 062416 (2023).

[12] Andreas Elben, Steven T. Flammia, Hsin-Yuan Huang, Richard Kueng, John Preskill, Benoît Vermersch, and Peter Zoller, "The randomized measurement toolbox", Nature Reviews Physics 5 1, 9 (2023).

[13] Valeria Cimini, Mauro Valeri, Emanuele Polino, Simone Piacentini, Francesco Ceccarelli, Giacomo Corrielli, Nicolò Spagnolo, Roberto Osellame, and Fabio Sciarrino, "Deep reinforcement learning for quantum multiparameter estimation", Advanced Photonics 5, 016005 (2023).

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

[15] 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).

[16] Bujiao Wu, Jinzhao Sun, Qi Huang, and Xiao Yuan, "Overlapped grouping measurement: A unified framework for measuring quantum states", arXiv:2105.13091, (2021).

[17] Bojia Duan and Chang-Yu Hsieh, "Hamiltonian-based data loading with shallow quantum circuits", Physical Review A 106 5, 052422 (2022).

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

[19] Zachary Pierce Bansingh, Tzu-Ching Yen, Peter D. Johnson, and Artur F. Izmaylov, "Fidelity overhead for non-local measurements in variational quantum algorithms", arXiv:2205.07113, (2022).

[20] Albie Chan, Zheng Shi, Luca Dellantonio, Wolfgang Dür, and Christine A. Muschik, "Measurement-based infused circuits for variational quantum eigensolvers", arXiv:2305.19200, (2023).

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

[22] 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).

[23] Francisco Escudero, David Fernández-Fernández, Gabriel Jaumà, Guillermo F. Peñas, and Luciano Pereira, "Hardware-Efficient Entangled Measurements for Variational Quantum Algorithms", Physical Review Applied 20 3, 034044 (2023).

[24] 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).

[25] Zachary Pierce Bansingh, Tzu-Ching Yen, Peter D. Johnson, and Artur F. Izmaylov, "Fidelity Overhead for Nonlocal Measurements in Variational Quantum Algorithms", Journal of Physical Chemistry A 126 39, 7007 (2022).

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

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-13 22:11:55) and SAO/NASA ADS (last updated successfully 2024-06-13 22:11:56). The list may be incomplete as not all publishers provide suitable and complete citation data.