Fisher Information in Noisy Intermediate-Scale Quantum Applications

Johannes Jakob Meyer

Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
QMATH, Department of Mathematical Sciences, Københavns Universitet, 2100 København Ø, Denmark

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

Abstract

The recent advent of noisy intermediate-scale quantum devices, especially near-term quantum computers, has sparked extensive research efforts concerned with their possible applications. At the forefront of the considered approaches are variational methods that use parametrized quantum circuits. The classical and quantum Fisher information are firmly rooted in the field of quantum sensing and have proven to be versatile tools to study such parametrized quantum systems. Their utility in the study of other applications of noisy intermediate-scale quantum devices, however, has only been discovered recently. Hoping to stimulate more such applications, this article aims to further popularize classical and quantum Fisher information as useful tools for near-term applications beyond quantum sensing. We start with a tutorial that builds an intuitive understanding of classical and quantum Fisher information and outlines how both quantities can be calculated on near-term devices. We also elucidate their relationship and how they are influenced by noise processes. Next, we give an overview of the core results of the quantum sensing literature and proceed to a comprehensive review of recent applications in variational quantum algorithms and quantum machine learning.

► BibTeX data

► References

[1] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando G. S. L. Brandao, David A. Buell, Brian Burkett, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Andrew Dunsworth, Edward Farhi, Brooks Foxen, Austin Fowler, Craig Gidney, Marissa Giustina, Rob Graff, Keith Guerin, Steve Habegger, Matthew P. Harrigan, Michael J. Hartmann, Alan Ho, Markus Hoffmann, Trent Huang, Travis S. Humble, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Paul V. Klimov, Sergey Knysh, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Mike Lindmark, Erik Lucero, Dmitry Lyakh, Salvatore Mandrà , Jarrod R. McClean, Matthew McEwen, Anthony Megrant, Xiao Mi, Kristel Michielsen, Masoud Mohseni, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Murphy Yuezhen Niu, Eric Ostby, Andre Petukhov, John C. Platt, Chris Quintana, Eleanor G. Rieffel, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Kevin J. Sung, Matthew D. Trevithick, Amit Vainsencher, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, and John M. Martinis, ``Quantum supremacy using a programmable superconducting processor'' Nature 574, 505-510 (2019) Number: 7779 Publisher: Nature Publishing Group.
https:/​/​doi.org/​10.1038/​s41586-019-1666-5
https:/​/​www.nature.com/​articles/​s41586-019-1666-5

[2] John Preskill ``Quantum Computing in the NISQ era and beyond'' Quantum 2, 79 (2018) Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften.
https:/​/​doi.org/​10.22331/​q-2018-08-06-79
https:/​/​quantum-journal.org/​papers/​q-2018-08-06-79/​

[3] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles, ``Variational Quantum Algorithms'' arXiv:2012.09265 (2020).
arXiv:2012.09265

[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'' arXiv:2101.08448 (2021).
arXiv:2101.08448

[5] 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) Publisher: IOP Publishing.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​2/​023023

[6] Marcello Benedetti, Erika Lloyd, Stefan Sack, and Mattia Fiorentini, ``Parameterized quantum circuits as machine learning models'' Quantum Science and Technology 4, 043001 (2019) Publisher: IOP Publishing.
https:/​/​doi.org/​10.1088/​2058-9565/​ab4eb5

[7] Jing Liu, Haidong Yuan, Xiao-Ming Lu, and Xiaoguang Wang, ``Quantum Fisher information matrix and multiparameter estimation'' Journal of Physics A: Mathematical and Theoretical 53, 023001 (2020).
https:/​/​doi.org/​10.1088/​1751-8121/​ab5d4d

[8] Jasminder S. Sidhu and Pieter Kok ``Geometric perspective on quantum parameter estimation'' AVS Quantum Science 2, 014701 (2020) Publisher: American Vacuum Society.
https:/​/​doi.org/​10.1116/​1.5119961

[9] Vishal Katariya and Mark M. Wilde ``Geometric distinguishability measures limit quantum channel estimation and discrimination'' Quantum Information Processing 20, 78 (2021).
https:/​/​doi.org/​10.1007/​s11128-021-02992-7

[10] Michael A. Nielsen and Isaac L. Chuang ``Quantum computation and quantum information'' Cambridge University Press (2010).

[11] E. L Lehmann and George Casella ``Theory of Point Estimation'' Springer London (1998) OCLC: 1229220122.

[12] E. A. Morozova and N. N. Chentsov ``Markov invariant geometry on manifolds of states'' Journal of Soviet Mathematics 56, 2648–2669 (1991).
https:/​/​doi.org/​10.1007/​BF01095975

[13] Clément L. Canonne ``A short note on learning discrete distributions'' arXiv:2002.11457 (2020).
arXiv:2002.11457

[14] Giacomo Torlaiand Roger G. Melko ``Machine-Learning Quantum States in the NISQ Era'' Annual Review of Condensed Matter Physics 11, 325–344 (2020) _eprint: https:/​/​doi.org/​10.1146/​annurev-conmatphys-031119-050651.
https:/​/​doi.org/​10.1146/​annurev-conmatphys-031119-050651

[15] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, ``Quantum circuit learning'' Physical Review A 98, 032309 (2018) Publisher: American Physical Society.
https:/​/​doi.org/​10.1103/​PhysRevA.98.032309

[16] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran, ``Evaluating analytic gradients on quantum hardware'' Physical Review A 99, 032331 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.032331
arXiv:1811.11184

[17] Luigi Seveso, Francesco Albarelli, Marco G. Genoni, and Matteo G. A. Paris, ``On the discontinuity of the quantum Fisher information for quantum statistical models with parameter dependent rank'' Journal of Physics A: Mathematical and Theoretical 53, 02LT01 (2020).
https:/​/​doi.org/​10.1088/​1751-8121/​ab599b
arXiv:1906.06185

[18] Mark M. Wilde ``From Classical to Quantum Shannon Theory'' arXiv:1106.1445 (2017).
https:/​/​doi.org/​10.1017/​9781316809976.001

[19] Michael M. Wolf ``Quantum Channels & Operations Guided Tour'' (2014).

[20] Dénes Petz ``Monotone metrics on matrix spaces'' Linear Algebra and its Applications 244, 81–96 (1996).
https:/​/​doi.org/​10.1016/​0024-3795(94)00211-8
https:/​/​linkinghub.elsevier.com/​retrieve/​pii/​0024379594002118

[21] C.W. Helstrom ``Minimum mean-squared error of estimates in quantum statistics'' Physics Letters A 25, 101–102 (1967).
https:/​/​doi.org/​10.1016/​0375-9601(67)90366-0
https:/​/​linkinghub.elsevier.com/​retrieve/​pii/​0375960167903660

[22] Ran Cheng ``Quantum Geometric Tensor (Fubini-Study Metric) in Simple Quantum System: A pedagogical Introduction'' arXiv:1012.1337 (2013).
arXiv:1012.1337

[23] James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo, ``Quantum Natural Gradient'' Quantum 4, 269 (2020).
https:/​/​doi.org/​10.22331/​q-2020-05-25-269
https:/​/​quantum-journal.org/​papers/​q-2020-05-25-269/​

[24] Andrea Mari, Thomas R. Bromley, and Nathan Killoran, ``Estimating the gradient and higher-order derivatives on quantum hardware'' Physical Review A 103, 012405 (2021).
https:/​/​doi.org/​10.1103/​PhysRevA.103.012405

[25] Vojtěch Havlíček,, Antonio D. Córcoles, Kristan Temme, Aram W. Harrow, Abhinav Kandala, Jerry M. Chow, and Jay M. Gambetta, ``Supervised learning with quantum-enhanced feature spaces'' Nature 567, 209–212 (2019).
https:/​/​doi.org/​10.1038/​s41586-019-0980-2
https:/​/​www.nature.com/​articles/​s41586-019-0980-2

[26] Julien Gacon, Christa Zoufal, Giuseppe Carleo, and Stefan Woerner, ``Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information'' arXiv:2103.09232 (2021).
arXiv:2103.09232

[27] Samuel L. Braunsteinand Carlton M. Caves ``Statistical distance and the geometry of quantum states'' Physical Review Letters 72, 3439–3443 (1994).
https:/​/​doi.org/​10.1103/​PhysRevLett.72.3439

[28] O. E. Barndorff-Nielsenand R. D. Gill ``Fisher information in quantum statistics'' Journal of Physics A: Mathematical and General 33, 4481 (2000) Publisher: IOP Publishing.
https:/​/​doi.org/​10.1088/​0305-4470/​33/​24/​306

[29] Luca Pezzè, Mario A. Ciampini, Nicolò Spagnolo, Peter C. Humphreys, Animesh Datta, Ian A. Walmsley, Marco Barbieri, Fabio Sciarrino, and Augusto Smerzi, ``Optimal Measurements for Simultaneous Quantum Estimation of Multiple Phases'' Physical Review Letters 119 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.130504

[30] Jing Yang, Shengshi Pang, Yiyu Zhou, and Andrew N. Jordan, ``Optimal measurements for quantum multiparameter estimation with general states'' Physical Review A 100, 032104 (2019) Publisher: American Physical Society.
https:/​/​doi.org/​10.1103/​PhysRevA.100.032104

[31] Sammy Ragy, Marcin Jarzyna, and Rafał Demkowicz-Dobrzański, ``Compatibility in multiparameter quantum metrology'' Physical Review A 94, 052108 (2016) Publisher: American Physical Society.
https:/​/​doi.org/​10.1103/​PhysRevA.94.052108

[32] Xiao-Ming Luand Xiaoguang Wang ``Incorporating Heisenberg's Uncertainty Principle into Quantum Multiparameter Estimation'' Physical Review Letters 126, 120503 (2021) Publisher: American Physical Society.
https:/​/​doi.org/​10.1103/​PhysRevLett.126.120503

[33] Federico Belliardoand Vittorio Giovannetti ``Incompatibility in quantum parameter estimation'' New Journal of Physics 23, 063055 (2021) Publisher: IOP Publishing.
https:/​/​doi.org/​10.1088/​1367-2630/​ac04ca

[34] Jing Liu, Heng-Na Xiong, Fei Song, and Xiaoguang Wang, ``Fidelity susceptibility and quantum Fisher information for density operators with arbitrary ranks'' Physica A: Statistical Mechanics and its Applications 410, 167–173 (2014).
https:/​/​doi.org/​10.1016/​j.physa.2014.05.028
http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0378437114003926

[35] Dominik Šafránek ``Discontinuities of the quantum Fisher information and the Bures metric'' Physical Review A 95, 052320 (2017) Publisher: American Physical Society.
https:/​/​doi.org/​10.1103/​PhysRevA.95.052320

[36] T. Baumgratz, A. Nüßeler, M. Cramer, and M. B. Plenio, ``A scalable maximum likelihood method for quantum state tomography'' New Journal of Physics 15, 125004 (2013) Publisher: IOP Publishing.
https:/​/​doi.org/​10.1088/​1367-2630/​15/​12/​125004

[37] Geza Toth ``Lower bounds on the quantum Fisher information based on the variance and various types of entropies'' arXiv:1701.07461 (2018).
arXiv:1701.07461

[38] Bálint Koczor and Simon C. Benjamin ``Quantum natural gradient generalised to non-unitary circuits'' arXiv:1912.08660 (2020).
arXiv:1912.08660

[39] Yuxuan Duand Dacheng Tao ``On exploring practical potentials of quantum auto-encoder with advantages'' arXiv:2106.15432 (2021).
arXiv:2106.15432

[40] Jonathan Romero, Jonathan P. Olson, and Alan Aspuru-Guzik, ``Quantum autoencoders for efficient compression of quantum data'' Quantum Science and Technology 2, 045001 (2017) Publisher: IOP Publishing.
https:/​/​doi.org/​10.1088/​2058-9565/​aa8072

[41] Ranyiliu Chen, Zhixin Song, Xuanqiang Zhao, and Xin Wang, ``Variational Quantum Algorithms for Trace Distance and Fidelity Estimation'' arXiv:2012.05768 (2020).
arXiv:2012.05768

[42] Kok Chuan Tan and Tyler Volkoff ``Variational quantum algorithms to estimate rank, quantum entropies, fidelity and Fisher information via purity minimization'' arXiv:2103.15956 (2021).
arXiv:2103.15956

[43] Akira Sone, M. Cerezo, Jacob L. Beckey, and Patrick J. Coles, ``A Generalized Measure of Quantum Fisher Information'' arXiv:2010.02904 (2021).
arXiv:2010.02904

[44] Jacob L. Beckey, M. Cerezo, Akira Sone, and Patrick J. Coles, ``Variational Quantum Algorithm for Estimating the Quantum Fisher Information'' arXiv:2010.10488 (2020).
arXiv:2010.10488

[45] Aniket Rath, Cyril Branciard, Anna Minguzzi, and Benoı̂t Vermersch, ``Quantum Fisher information from randomized measurements'' arXiv:2105.13164 (2021).
arXiv:2105.13164

[46] 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) Number: 10 Publisher: Nature Publishing Group.
https:/​/​doi.org/​10.1038/​s41567-020-0932-7
https:/​/​www.nature.com/​articles/​s41567-020-0932-7

[47] Chao Zhang, Benjamin Yadin, Zhi-Bo Hou, Huan Cao, Bi-Heng Liu, Yun-Feng Huang, Reevu Maity, Vlatko Vedral, Chuan-Feng Li, Guang-Can Guo, and Davide Girolami, ``Detecting metrologically useful asymmetry and entanglement by a few local measurements'' Physical Review A 96, 042327 (2017) Publisher: American Physical Society.
https:/​/​doi.org/​10.1103/​PhysRevA.96.042327

[48] Davide Girolami and Benjamin Yadin ``Witnessing Multipartite Entanglement by Detecting Asymmetry'' Entropy 19, 124 (2017) Number: 3 Publisher: Multidisciplinary Digital Publishing Institute.
https:/​/​doi.org/​10.3390/​e19030124
https:/​/​www.mdpi.com/​1099-4300/​19/​3/​124

[49] Carl W. Helstrom ``Quantum detection and estimation theory'' Journal of Statistical Physics 1, 231–252 (1969).
https:/​/​doi.org/​10.1007/​BF01007479

[50] A. S. Holevo ``Probabilistic and Statistical Aspects of Quantum Theory'' (2011) OCLC: 863661368.
https:/​/​doi.org/​10.1007/​978-88-7642-378-9

[51] Rafał Demkowicz-Dobrzański, Wojciech Górecki, and Mădălin Guţă, ``Multi-parameter estimation beyond quantum Fisher information'' Journal of Physics A: Mathematical and Theoretical 53, 363001 (2020) Publisher: IOP Publishing.
https:/​/​doi.org/​10.1088/​1751-8121/​ab8ef3

[52] Mankei Tsang, Francesco Albarelli, and Animesh Datta, ``Quantum Semiparametric Estimation'' Physical Review X 10, 031023 (2020) Publisher: American Physical Society.
https:/​/​doi.org/​10.1103/​PhysRevX.10.031023

[53] F. Albarelli, M. Barbieri, M. G. Genoni, and I. Gianani, ``A perspective on multiparameter quantum metrology: From theoretical tools to applications in quantum imaging'' Physics Letters A 384, 126311 (2020).
https:/​/​doi.org/​10.1016/​j.physleta.2020.126311
https:/​/​www.sciencedirect.com/​science/​article/​pii/​S0375960120301109

[54] J. J . Bollinger, Wayne M. Itano, D. J. Wineland, and D. J. Heinzen, ``Optimal frequency measurements with maximally correlated states'' Physical Review A 54, R4649–R4652 (1996) Publisher: American Physical Society.
https:/​/​doi.org/​10.1103/​PhysRevA.54.R4649

[55] S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, ``Improvement of Frequency Standards with Quantum Entanglement'' Physical Review Letters 79, 3865–3868 (1997).
https:/​/​doi.org/​10.1103/​PhysRevLett.79.3865

[56] Rafał Demkowicz-Dobrzański, Jan Kołodyński, and Mădălin Guţă, ``The elusive Heisenberg limit in quantum-enhanced metrology'' Nature Communications 3, 1063 (2012) Number: 1 Publisher: Nature Publishing Group.
https:/​/​doi.org/​10.1038/​ncomms2067
https:/​/​www.nature.com/​articles/​ncomms2067

[57] Sisi Zhou, Mengzhen Zhang, John Preskill, and Liang Jiang, ``Achieving the Heisenberg limit in quantum metrology using quantum error correction'' Nature Communications 9, 78 (2018) Number: 1 Publisher: Nature Publishing Group.
https:/​/​doi.org/​10.1038/​s41467-017-02510-3
https:/​/​www.nature.com/​articles/​s41467-017-02510-3

[58] Wojciech Górecki, Sisi Zhou, Liang Jiang, and Rafał Demkowicz-Dobrzański, ``Optimal probes and error-correction schemes in multi-parameter quantum metrology'' Quantum 4, 288 (2020) Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften.
https:/​/​doi.org/​10.22331/​q-2020-07-02-288
https:/​/​quantum-journal.org/​papers/​q-2020-07-02-288/​

[59] J. F. Haase, A. Smirne, S. F. Huelga, J. Kołodynski, and R. Demkowicz-Dobrzański, ``Precision Limits in Quantum Metrology with Open Quantum Systems'' Quantum Measurements and Quantum Metrology 5, 13–39 (2016) Publisher: De Gruyter Open Section: Quantum Measurements and Quantum Metrology.
https:/​/​doi.org/​10.1515/​qmetro-2018-0002

[60] Guoming Wang, Dax Enshan Koh, Peter D. Johnson, and Yudong Cao, ``Minimizing Estimation Runtime on Noisy Quantum Computers'' PRX Quantum 2, 010346 (2021) Publisher: American Physical Society.
https:/​/​doi.org/​10.1103/​PRXQuantum.2.010346

[61] Dax Enshan Koh, Guoming Wang, Peter D. Johnson, and Yudong Cao, ``A framework for engineering quantum likelihood functions for expectation estimation'' arXiv:2006.09349 (2020).
arXiv:2006.09349

[62] Raphael Kaubruegger, Pietro Silvi, Christian Kokail, Rick van Bijnen, Ana Maria Rey, Jun Ye, Adam M. Kaufman, and Peter Zoller, ``Variational Spin-Squeezing Algorithms on Programmable Quantum Sensors'' Physical Review Letters 123, 260505 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.260505

[63] Bálint Koczor, Suguru Endo, Tyson Jones, Yuichiro Matsuzaki, and Simon C Benjamin, ``Variational-state quantum metrology'' New Journal of Physics 22, 083038 (2020).
https:/​/​doi.org/​10.1088/​1367-2630/​ab965e

[64] Johannes Jakob Meyer, Johannes Borregaard, and Jens Eisert, ``A variational toolbox for quantum multi-parameter estimation'' npj Quantum Information 7, 1–5 (2021).
https:/​/​doi.org/​10.1038/​s41534-021-00425-y
https:/​/​www.nature.com/​articles/​s41534-021-00425-y

[65] Ziqi Ma, Pranav Gokhale, Tian-Xing Zheng, Sisi Zhou, Xiaofei Yu, Liang Jiang, Peter Maurer, and Frederic T. Chong, ``Adaptive Circuit Learning for Quantum Metrology'' arXiv:2010.08702 (2020).
arXiv:2010.08702

[66] Martin Gärttner, Philipp Hauke, and Ana Maria Rey, ``Relating Out-of-Time-Order Correlations to Entanglement via Multiple-Quantum Coherences'' Physical Review Letters 120, 040402 (2018) Publisher: American Physical Society.
https:/​/​doi.org/​10.1103/​PhysRevLett.120.040402

[67] M. Cerezo, Akira Sone, Jacob L. Beckey, and Patrick J. Coles, ``Sub-quantum Fisher information'' Quantum Science and Technology 6, 035008 (2021) Publisher: IOP Publishing.
https:/​/​doi.org/​10.1088/​2058-9565/​abfbef

[68] David Wierichs, Christian Gogolin, and Michael Kastoryano, ``Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer'' Physical Review Research 2, 043246 (2020).
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.043246

[69] Barnaby van Straaten and Bálint Koczor ``Measurement Cost of Metric-Aware Variational Quantum Algorithms'' PRX Quantum 2, 030324 (2021) Publisher: American Physical Society.
https:/​/​doi.org/​10.1103/​PRXQuantum.2.030324

[70] Maria Schuld, Ilya Sinayskiy, and Francesco Petruccione, ``An introduction to quantum machine learning'' Contemporary Physics 56, 172–185 (2015).
https:/​/​doi.org/​10.1080/​00107514.2014.964942

[71] Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd, ``Quantum Machine Learning'' Nature 549, 195–202 (2017).
https:/​/​doi.org/​10.1038/​nature23474
arXiv:1611.09347

[72] Tengyuan Liang, Tomaso Poggio, Alexander Rakhlin, and James Stokes, ``Fisher-Rao Metric, Geometry, and Complexity of Neural Networks'' The 22nd International Conference on Artificial Intelligence and Statistics 9 (2019).

[73] Amira Abbas, David Sutter, Christa Zoufal, Aurelien Lucchi, Alessio Figalli, and Stefan Woerner, ``The power of quantum neural networks'' Nature Computational Science 1, 403–409 (2021).
https:/​/​doi.org/​10.1038/​s43588-021-00084-1
https:/​/​www.nature.com/​articles/​s43588-021-00084-1

[74] Tobias Haug, Kishor Bharti, and M. S. Kim, ``Capacity and quantum geometry of parametrized quantum circuits'' arXiv:2102.01659 (2021).
arXiv:2102.01659

[75] Yuxiang Yang, Renato Renner, and Giulio Chiribella, ``Optimal Universal Programming of Unitary Gates'' Physical Review Letters 125, 210501 (2020) Publisher: American Physical Society.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.210501

[76] Aleksander Kubica and Rafał Demkowicz-Dobrzański ``Using Quantum Metrological Bounds in Quantum Error Correction: A Simple Proof of the Approximate Eastin-Knill Theorem'' Physical Review Letters 126, 150503 (2021) Publisher: American Physical Society.
https:/​/​doi.org/​10.1103/​PhysRevLett.126.150503

[77] Kok Chuan Tan, Varun Narasimhachar, and Bartosz Regula, ``Fisher information universally identifies quantum resources'' arXiv:2104.01763 (2021).
arXiv:2104.01763

[78] Eric Chitambar and Gilad Gour ``Quantum resource theories'' Reviews of Modern Physics 91, 025001 (2019) Publisher: American Physical Society.
https:/​/​doi.org/​10.1103/​RevModPhys.91.025001

[79] Diego Paiva Pires, Marco Cianciaruso, Lucas C. Céleri, Gerardo Adesso, and Diogo O. Soares-Pinto, ``Generalized Geometric Quantum Speed Limits'' Physical Review X 6, 021031 (2016).
https:/​/​doi.org/​10.1103/​PhysRevX.6.021031

[80] D. Spehner, F. Illuminati, M. Orszag, and W. Roga, ``Geometric Measures of Quantum Correlations with Bures and Hellinger Distances'' Springer International Publishing (2017) Series Title: Quantum Science and Technology.
https:/​/​doi.org/​10.1007/​978-3-319-53412-1_6

Cited by

[1] Yong-Xin Yao, Niladri Gomes, Feng Zhang, Cai-Zhuang Wang, Kai-Ming Ho, Thomas Iadecola, and Peter P. Orth, "Adaptive Variational Quantum Dynamics Simulations", PRX Quantum 2 3, 030307 (2021).

[2] Tobias Haug, Kishor Bharti, and M. S. Kim, "Capacity and quantum geometry of parametrized quantum circuits", arXiv:2102.01659.

[3] Tobias Haug and M. S. Kim, "Optimal training of variational quantum algorithms without barren plateaus", arXiv:2104.14543.

[4] Francesco Albarelli and Rafal Demkowicz-Dobrzanski, "Probe incompatibility in multiparameter noisy quantum channel estimation", arXiv:2104.11264.

[5] David Wierichs, Josh Izaac, Cody Wang, and Cedric Yen-Yu Lin, "General parameter-shift rules for quantum gradients", arXiv:2107.12390.

[6] Kishor Bharti, Tobias Haug, Vlatko Vedral, and Leong-Chuan Kwek, "NISQ Algorithm for Semidefinite Programming", arXiv:2106.03891.

[7] Christa Zoufal, David Sutter, and Stefan Woerner, "Error Bounds for Variational Quantum Time Evolution", arXiv:2108.00022.

[8] Tobias Haug and M. S. Kim, "Natural parameterized quantum circuit", arXiv:2107.14063.

[9] Tomotaka Kuwahara and Keiji Saito, "Exponential clustering of bipartite quantum entanglement at arbitrary temperatures", arXiv:2108.12209.

[10] Joe H. Jenne and David R. M. Arvidsson-Shukur, "Quantum Learnability is Arbitrarily Distillable", arXiv:2104.09520.

[11] Sofia Qvarfort, Dennis Rätzel, and Stephen Stopyra, "Constraining modified gravity with quantum optomechanics", arXiv:2108.00742.

[12] Tobias Haug, Chris N. Self, and M. S. Kim, "Large-scale quantum machine learning", arXiv:2108.01039.

The above citations are from SAO/NASA ADS (last updated successfully 2021-09-23 06:29:52). 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 2021-09-23 06:29:51).