Entanglement Entropy Production In Quantum Neural Networks

افلاطون کے ذریعہ دوبارہ شائع کیا گیا۔

فالونگ: 0

Marco Ballarin^1,2,3, Stefano Mangini^1,4,5, Simone Montangero^2,3,6, Chiara Macchiavello^4,5,7, and Riccardo Mengoni⁸

¹یہ مصنفین نے اس کام کو برابر طور پر بھیجا
²Dipartimento di Fisica e Astronomia "G. Galilei", via Marzolo 8, I-35131, Padova, Italy
³INFN, Sezione di Padova, via Marzolo 8, I-35131, Padova, Italy
⁴Dipartimento di Fisica, Università di Pavia, Via Bassi 6, I-27100, Pavia, Italy
⁵INFN Sezione di Pavia, Via Bassi 6, I-27100, Pavia, Italy
⁶Padua Quantum Technologies Research Center, Università degli Studi di Padova
⁷CNR-INO - Largo E. Fermi 6, I-50125, Firenze, Italy
⁸CINECA Quantum Computing Lab,Via Magnanelli, 6/3, 40033 Casalecchio di Reno, Bologna, Italy

اس کاغذ کو دلچسپ لگتا ہے یا اس پر بات کرنا چاہتے ہیں؟ SciRate پر تبصرہ کریں یا چھوڑیں۔.

خلاصہ

Quantum Neural Networks (QNN) are considered a candidate for achieving quantum advantage in the Noisy Intermediate Scale Quantum computer (NISQ) era. Several QNN architectures have been proposed and successfully tested on benchmark datasets for machine learning. However, quantitative studies of the QNN-generated entanglement have been investigated only for up to few qubits. Tensor network methods allow to emulate quantum circuits with a large number of qubits in a wide variety of scenarios. Here, we employ matrix product states to characterize recently studied QNN architectures with random parameters up to fifty qubits showing that their entanglement, measured in terms of entanglement entropy between qubits, tends to that of Haar distributed random states as the depth of the QNN is increased. We certify the randomness of the quantum states also by measuring the expressibility of the circuits, as well as using tools from random matrix theory. We show a universal behavior for the rate at which entanglement is created in any given QNN architecture, and consequently introduce a new measure to characterize the entanglement production in QNNs: the entangling speed. Our results characterise the entanglement properties of quantum neural networks, and provides new evidence of the rate at which these approximate random unitaries.

Featured image: Graphical representation of QNN and MPS. (a) QNN structure with alternating feature map F and variational ansatz V . Note that the ansatz parameters are different in each layer, while the feature map parameters are the same throughout the whole circuit. (b) MPS diagram. Each sphere is a tensor, representing a qubit qj . The entanglement entropy between bi-partitions A and B is computed by "cutting" the connecting edge ej . (c) Circuits analyzed in the manuscript, depicted with a linear entanglement topology, i.e. entangling gates are only applied between nearest neighbors on a line. (d) Different entanglement topologies: circular, with the first and last qubit of the line connected, and full, where the entangling gates are applied between each pair of qubits.

► BibTeX ڈیٹا

@article{Ballarin2023entanglemententropy, doi = {10.22331/q-2023-05-31-1023}, url = {https://doi.org/10.22331/q-2023-05-31-1023}, title = {Entanglement entropy production in {Q}uantum {N}eural {N}etworks}, author = {Ballarin, Marco and Mangini, Stefano and Montangero, Simone and Macchiavello, Chiara and Mengoni, Riccardo}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {1023}, month = may, year = {2023}
}

► حوالہ جات

ہے [1] Michael A. Nielsen and Isaac L. Chuang. ``Quantum computation and quantum information''. Cambridge University Press. Cambridge, UK (2010). 10th anniversary ed. (2010) edition.
https://doi.org/10.1017/CBO9780511976667

ہے [2] Ian Goodfellow, Yoshua Bengio, and Aaron Courville. ``Deep learning''. MIT Press. (2016). url: http://www.deeplearningbook.org.
http://www.deeplearningbook.org

ہے [3] Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. ``Deep learning''. Nature 521, 436–444 (2015).
https://doi.org/10.1038/nature14539

ہے [4] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E. Hinton. ``Imagenet classification with deep convolutional neural networks''. In Proceedings of the 25th International Conference on Neural Information Processing Systems - Volume 1. Page 1097–1105. NIPS'12Red Hook, NY, USA (2012). Curran Associates Inc.
https://doi.org/10.1145/3065386

ہے [5] David Silver, Aja Huang, Chris J. Maddison, Arthur Guez, Laurent Sifre, George van den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, Sander Dieleman, Dominik Grewe, John Nham, Nal Kalchbrenner, Ilya Sutskever, Timothy Lillicrap, Madeleine Leach, Koray Kavukcuoglu, Thore Graepel, and Demis Hassabis. ``Mastering the game of Go with deep neural networks and tree search''. Nature 529, 484–489 (2016).
https://doi.org/10.1038/nature16961

ہے [6] Jonas Degrave, Federico Felici, Jonas Buchli, Michael Neunert, Brendan Tracey, Francesco Carpanese, Timo Ewalds, Roland Hafner, Abbas Abdolmaleki, Diego de las Casas, Craig Donner, Leslie Fritz, Cristian Galperti, Andrea Huber, James Keeling, Maria Tsimpoukelli, Jackie Kay, Antoine Merle, Jean-Marc Moret, Seb Noury, Federico Pesamosca, David Pfau, Olivier Sauter, Cristian Sommariva, Stefano Coda, Basil Duval, Ambrogio Fasoli, Pushmeet Kohli, Koray Kavukcuoglu, Demis Hassabis, and Martin Riedmiller. ``Magnetic control of tokamak plasmas through deep reinforcement learning''. Nature 602, 414–419 (2022).
https://doi.org/10.1038/s41586-021-04301-9

ہے [7] جیکب بیامونٹے، پیٹر وٹیک، نکولا پینکوٹی، پیٹرک ریبینٹروسٹ، ناتھن ویبی، اور سیٹھ لائیڈ۔ ''کوانٹم مشین لرننگ''۔ فطرت 549، 195–202 (2017)۔
https://doi.org/10.1038/nature23474

ہے [8] Vedran Dunjko and Peter Wittek. ``A non-review of quantum machine learning: trends and explorations''. Quantum 4, 32 (2020).
https://doi.org/10.22331/qv-2020-03-17-32

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

ہے [10] S. Mangini, F. Tacchino, D. Gerace, D. Bajoni, and C. Macchiavello. ``Quantum computing models for artificial neural networks''. Europhysics Letters 134, 10002 (2021).
https://doi.org/10.1209/0295-5075/134/10002

ہے [11] 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 algorithms''. Rev. Mod. Phys. 94, 015004 (2022).
https:///doi.org/10.1103/RevModPhys.94.015004

ہے [12] John Preskill. ``Quantum Computing in the NISQ era and beyond''. Quantum 2, 79 (2018).
https://doi.org/10.22331/q-2018-08-06-79

ہے [13] 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''. Nat. Commun. 5 (2014).
https://doi.org/10.1038/ncomms5213

ہے [14] 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

ہے [15] Hsin-Yuan Huang, Richard Kueng, and John Preskill. ``Information-theoretic bounds on quantum advantage in machine learning''. Phys. Rev. Lett. 126, 190505 (2021).
https:///doi.org/10.1103/PhysRevLett.126.190505

ہے [16] Hsin-Yuan Huang, Michael Broughton, Masoud Mohseni, Ryan Babbush, Sergio Boixo, Hartmut Neven, and Jarrod R. McClean. ``Power of data in quantum machine learning''. Nature Communications 12, 2631 (2021).
https://doi.org/10.1038/s41467-021-22539-9

ہے [17] Franz J. Schreiber, Jens Eisert, and Johannes Jakob Meyer. ``Classical surrogates for quantum learning models'' (2022) arXiv:2206.11740.
آر ایکس سی: 2206.11740

ہے [18] Thomas Hubregtsen, Josef Pichlmeier, Patrick Stecher, and Koen Bertels. ``Evaluation of parameterized quantum circuits: On the relation between classification accuracy, expressibility, and entangling capability''. Quantum Machine Intelligence 3, 9 (2021).
https://doi.org/10.1007/s42484-021-00038-w

ہے [19] M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles. ``Cost function dependent barren plateaus in shallow parametrized quantum circuits''. Nat. Commun. 12 (2021).
https://doi.org/10.1038/s41467-021-21728-w

ہے [20] Iris Cong, Soonwon Choi, and Mikhail D. Lukin. ``Quantum convolutional neural networks''. Nature Physics 15, 1273–1278 (2019).
https://doi.org/10.1038/s41567-019-0648-8

ہے [21] Johannes Jakob Meyer, Marian Mularski, Elies Gil-Fuster, Antonio Anna Mele, Francesco Arzani, Alissa Wilms, and Jens Eisert. ``Exploiting symmetry in variational quantum machine learning''. PRX Quantum 4, 010328 (2023).
https:///doi.org/10.1103/PRXQuantum.4.010328

ہے [22] Andrea Skolik, Michele Cattelan, Sheir Yarkoni, Thomas Bäck, and Vedran Dunjko. ``Equivariant quantum circuits for learning on weighted graphs''. npj Quantum Information 9, 47 (2023).
https://doi.org/10.1038/s41534-023-00710-y

ہے [23] Sukin Sim, Peter D. Johnson, and Alán Aspuru-Guzik. ``Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms''. Adv. Quantum Technol. 2, 1900070 (2019).
https://doi.org/10.1002/qute.201900070

ہے [24] Adrián Pérez-Salinas, Alba Cervera-Lierta, Elies Gil-Fuster, and José I. Latorre. ``Data re-uploading for a universal quantum classifier''. Quantum 4, 226 (2020).
https://doi.org/10.22331/q-2020-02-06-226

ہے [25] Maria Schuld, Ryan Sweke, and Johannes Jakob Meyer. ``Effect of data encoding on the expressive power of variational quantum-machine-learning models''. Phys. Rev. A 103, 032430 (2021).
https:///doi.org/10.1103/PhysRevA.103.032430

ہے [26] Francesco Tacchino, Stefano Mangini, Panagiotis Kl. Barkoutsos, Chiara Macchiavello, Dario Gerace, Ivano Tavernelli, and Daniele Bajoni. ``Variational learning for quantum artificial neural networks''. IEEE Transactions on Quantum Engineering 2, 1–10 (2021).
https://doi.org/10.1109/TQE.2021.3062494

ہے [27] B Jaderberg, L W Anderson, W Xie, S Albanie, M Kiffner, and D Jaksch. ``Quantum self-supervised learning''. Quantum Science and Technology 7, 035005 (2022).
https://doi.org/10.1088/2058-9565/ac6825

ہے [28] David A. Meyer and Nolan R. Wallach. ``Global entanglement in multiparticle systems''. Journal of Mathematical Physics 43, 4273–4278 (2002).
https://doi.org/10.1063/1.1497700

ہے [29] Pietro Silvi, Ferdinand Tschirsich, Matthias Gerster, Johannes Jünemann, Daniel Jaschke, Matteo Rizzi, and Simone Montangero. ``The tensor networks anthology: Simulation techniques for many-body quantum lattice systems''. SciPost Physics Lecture Notes (2019).
https:///doi.org/10.21468/scipostphyslectnotes.8

ہے [30] S. Montangero. ``Introduction to tensor network methods''. Springer International Publishing. Cham, CH (2018).
https://doi.org/10.1007/978-3-030-01409-4

ہے [31] J. Eisert. ``Entanglement and tensor network states'' (2013). arXiv:1308.3318.
آر ایکس سی: 1308.3318

ہے [32] Sebastian Paeckel, Thomas Köhler, Andreas Swoboda, Salvatore R. Manmana, Ulrich Schollwöck, and Claudius Hubig. ``Time-evolution methods for matrix-product states''. Annals of Physics 411, 167998 (2019).
https:///doi.org/10.1016/j.aop.2019.167998

ہے [33] Patrick Hayden, Debbie W. Leung, and Andreas Winter. ``Aspects of Generic Entanglement''. Communications in Mathematical Physics 265, 95–117 (2006).
https://doi.org/10.1007/s00220-006-1535-6

ہے [34] Elizabeth S. Meckes. ``The Random Matrix Theory of the Classical Compact Groups''. Cambridge Tracts in Mathematics. Cambridge University Press. Cambridge (2019).
https://doi.org/10.1017/9781108303453

ہے [35] Alan Edelman and N. Raj Rao. ``Random matrix theory''. Acta Numerica 14, 233–297 (2005).
https:///doi.org/10.1017/S0962492904000236

ہے [36] Don N. Page. ``Average entropy of a subsystem''. Phys. Rev. Lett. 71, 1291–1294 (1993).
https:///doi.org/10.1103/PhysRevLett.71.1291

ہے [37] Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. ``The theory of variational hybrid quantum-classical algorithms''. New J. Phys. 18, 023023 (2016).
https://doi.org/10.1088/1367-2630/18/2/023023

ہے [38] Francisco Javier Gil Vidal and Dirk Oliver Theis. ``Input redundancy for parameterized quantum circuits''. Front. Phys. 8, 297 (2020).
https://doi.org/10.3389/fphy.2020.00297

ہے [39] E. Torrontegui and J. J. Garcia-Ripoll. ``Unitary quantum perceptron as efficient universal approximator''. EPL 125, 30004 (2019).
https://doi.org/10.1209/0295-5075/125/30004

ہے [40] Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven. ``Barren plateaus in quantum neural network training landscapes''. Nat. Commun. 9, 4812 (2018).
https://doi.org/10.1038/s41467-018-07090-4

ہے [41] ماریا شولڈ، ویل برگھولم، کرسچن گوگولن، جوش آئیزاک، اور ناتھن کلوران۔ ``کوانٹم ہارڈ ویئر پر تجزیاتی میلان کا اندازہ لگانا''۔ طبیعیات Rev. A 99, 032331 (2019)۔
https:///doi.org/10.1103/PhysRevA.99.032331

ہے [42] Andrew Arrasmith, M. Cerezo, Piotr Czarnik, Lukasz Cincio, and Patrick J. Coles. ``Effect of barren plateaus on gradient-free optimization''. Quantum 5, 558 (2021).
https://doi.org/10.22331/q-2021-10-05-558

ہے [43] Zoë Holmes, Kunal Sharma, M. Cerezo, and Patrick J. Coles. ``Connecting ansatz expressibility to gradient magnitudes and barren plateaus''. PRX Quantum 3, 010313 (2022).
https:///doi.org/10.1103/PRXQuantum.3.010313

ہے [44] Carlos Ortiz Marrero, Mária Kieferová, and Nathan Wiebe. ``Entanglement-induced barren plateaus''. PRX Quantum 2, 040316 (2021).
https:///doi.org/10.1103/PRXQuantum.2.040316

ہے [45] Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J. Coles. ``Noise-induced barren plateaus in variational quantum algorithms''. Nature Communications 12, 6961 (2021).
https://doi.org/10.1038/s41467-021-27045-6

ہے [46] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. ``Exact and approximate unitary 2-designs and their application to fidelity estimation''. Physical Review A 80 (2009).
https:///doi.org/10.1103/physreva.80.012304

ہے [47] Andrew Arrasmith, Zoë Holmes, Marco Cerezo, and Patrick J Coles. ``Equivalence of quantum barren plateaus to cost concentration and narrow gorges''. Quantum Science and Technology 7, 045015 (2022).
https://doi.org/10.1088/2058-9565/ac7d06

ہے [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 (2022).
https:///doi.org/10.1103/PRXQuantum.3.020365

ہے [49] Taylor L. Patti, Khadijeh Najafi, Xun Gao, and Susanne F. Yelin. ``Entanglement devised barren plateau mitigation''. Phys. Rev. Research 3, 033090 (2021).
https:///doi.org/10.1103/PhysRevResearch.3.033090

ہے [50] Zi-Wen Liu, Seth Lloyd, Elton Zhu, and Huangjun Zhu. ``Entanglement, quantum randomness, and complexity beyond scrambling''. Journal of High Energy Physics 2018, 41 (2018).
https://doi.org/10.1007/JHEP07(2018)041

ہے [51] Edward Grant, Leonard Wossnig, Mateusz Ostaszewski, and Marcello Benedetti. ``An initialization strategy for addressing barren plateaus in parametrized quantum circuits''. Quantum 3, 214 (2019).
https://doi.org/10.48550/arXiv.1903.05076

ہے [52] Tyler Volkoff and Patrick J Coles. ``Large gradients via correlation in random parameterized quantum circuits''. Quantum Science and Technology 6, 025008 (2021).
https://doi.org/10.1088/2058-9565/abd891

ہے [53] Andrea Skolik, Jarrod R McClean, Masoud Mohseni, Patrick van der Smagt, and Martin Leib. ``Layerwise learning for quantum neural networks''. Quantum Machine Intelligence 3, 1–11 (2021).
https://doi.org/10.1007/s42484-020-00036-4

ہے [54] Joonho Kim and Yaron Oz. ``Entanglement diagnostics for efficient vqa optimization''. Journal of Statistical Mechanics: Theory and Experiment 2022, 073101 (2022).
https://doi.org/10.1088/1742-5468/ac7791

ہے [55] 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

ہے [56] Aram W. Harrow and Richard A. Low. ``Random Quantum Circuits are Approximate 2-designs''. Communications in Mathematical Physics 291, 257–302 (2009).
https://doi.org/10.1007/s00220-009-0873-6

ہے [57] Jonas Haferkamp and Nicholas Hunter-Jones. ``Improved spectral gaps for random quantum circuits: Large local dimensions and all-to-all interactions''. Phys. Rev. A 104, 022417 (2021).
https:///doi.org/10.1103/PhysRevA.104.022417

ہے [58] Maria Schuld. ``Supervised quantum machine learning models are kernel methods'' (2021) arXiv:2101.11020.
آر ایکس سی: 2101.11020

ہے [59] Sofiene Jerbi, Lukas J Fiderer, Hendrik Poulsen Nautrup, Jonas M Kübler, Hans J Briegel, and Vedran Dunjko. ``Quantum machine learning beyond kernel methods''. Nature Communications 14, 517 (2023).
https://doi.org/10.1038/s41467-023-36159-y

ہے [60] Seth Lloyd. ``Quantum approximate optimization is computationally universal'' (2018) arXiv:1812.11075.
آر ایکس سی: 1812.11075

ہے [61] M. E. S. Morales, J. D. Biamonte, and Z. Zimborás. ``On the universality of the quantum approximate optimization algorithm''. Quantum Information Processing 19, 291 (2020).
https://doi.org/10.1007/s11128-020-02748-9

ہے [62] Fernando G. S. L. Brandão, Aram W. Harrow, and Michał Horodecki. ``Local Random Quantum Circuits are Approximate Polynomial-Designs''. Communications in Mathematical Physics 346, 397–434 (2016).
https://doi.org/10.1007/s00220-016-2706-8

ہے [63] Aram W Harrow and Saeed Mehraban. ``Approximate unitary t-designs by short random quantum circuits using nearest-neighbor and long-range gates''. Communications in Mathematical PhysicsPages 1–96 (2023).
https://doi.org/10.1007/s00220-023-04675-z

ہے [64] Pasquale Calabrese and John Cardy. ``Evolution of entanglement entropy in one-dimensional systems''. Journal of Statistical Mechanics: Theory and Experiment 2005, P04010 (2005).
https://doi.org/10.1088/1742-5468/2005/04/p04010

ہے [65] Tianci Zhou and Adam Nahum. ``Emergent statistical mechanics of entanglement in random unitary circuits''. Phys. Rev. B 99, 174205 (2019).
https:///doi.org/10.1103/PhysRevB.99.174205

ہے [66] Adam Nahum, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. ``Quantum entanglement growth under random unitary dynamics''. Phys. Rev. X 7, 031016 (2017).
https:///doi.org/10.1103/PhysRevX.7.031016

ہے [67] M. Aeberhard, Stefan & Forina. ``Wine''. UCI Machine Learning Repository (1991). DOI: https://doi.org/10.24432/C5PC7J.
https://doi.org/10.24432/C5PC7J

ہے [68] Milan Zwitter, Matjaz & Soklic. ``Breast Cancer''. UCI Machine Learning Repository (1988). DOI: https://doi.org/10.24432/C51P4M.
https://doi.org/10.24432/C51P4M

ہے [69] Marko Žnidarič. ``Entanglement of random vectors''. Journal of Physics A: Mathematical and Theoretical 40, F105 (2006).
https://doi.org/10.1088/1751-8113/40/3/F04

ہے [70] Daniel Jaschke and Simone Montangero. ``Is quantum computing green? an estimate for an energy-efficiency quantum advantage''. Quantum Science and Technology (2022).
https://doi.org/10.1088/2058-9565/acae3e

ہے [71] V A Marčenko and L A Pastur. ``Distribution of eigenvalues for some sets of random matrices''. Mathematics of the USSR-Sbornik 1, 457 (1967).
https://doi.org/10.1070/SM1967v001n04ABEH001994

ہے [72] Zbigniew Puchała, Łukasz Pawela, and Karol Życzkowski. ``Distinguishability of generic quantum states''. Physical Review A 93, 062112 (2016).
https:///doi.org/10.1103/PhysRevA.93.062112

ہے [73] Maxime Dupont, Nicolas Didier, Mark J. Hodson, Joel E. Moore, and Matthew J. Reagor. ``Entanglement perspective on the quantum approximate optimization algorithm''. Phys. Rev. A 106, 022423 (2022).
https:///doi.org/10.1103/PhysRevA.106.022423

ہے [74] Andreas J. C. Woitzik, Panagiotis Kl. Barkoutsos, Filip Wudarski, Andreas Buchleitner, and Ivano Tavernelli. ``Entanglement production and convergence properties of the variational quantum eigensolver''. Phys. Rev. A 102, 042402 (2020).
https:///doi.org/10.1103/PhysRevA.102.042402

ہے [75] Michael Ragone, Paolo Braccia, Quynh T. Nguyen, Louis Schatzki, Patrick J. Coles, Frederic Sauvage, Martin Larocca, and M. Cerezo. ``Representation theory for geometric quantum machine learning'' (2022) arXiv:2210.07980.
آر ایکس سی: 2210.07980

ہے [76] Kunal Sharma, M. Cerezo, Zoë Holmes, Lukasz Cincio, Andrew Sornborger, and Patrick J. Coles. ``Reformulation of the no-free-lunch theorem for entangled datasets''. Phys. Rev. Lett. 128, 070501 (2022).
https:///doi.org/10.1103/PhysRevLett.128.070501

ہے [77] Martin Larocca, Nathan Ju, Diego García-Martín, Patrick J. Coles, and M. Cerezo. ``Theory of overparametrization in quantum neural networks'' (2021) arXiv:2109.11676.
آر ایکس سی: 2109.11676

ہے [78] Bobak Toussi Kiani, Seth Lloyd, and Reevu Maity. ``Learning unitaries by gradient descent'' (2020) arXiv:2001.11897.
آر ایکس سی: 2001.11897

ہے [79] Eric R. Anschuetz and Bobak T. Kiani. ``Quantum variational algorithms are swamped with traps''. Nature Communications 13 (2022).
https://doi.org/10.1038/s41467-022-35364-5

ہے [80] Md Sajid Anis et al. ``Qiskit: An open-source framework for quantum computing''. Zenodo (2021).
https://doi.org/10.5281/zenodo.2562111

ہے [81] Marco Ballarin. ``Quantum computer simulation via tensor networks''. Università degli Studi di Padova, Master Thesis (2021). url: https://hdl.handle.net/20.500.12608/21799.
https:///hdl.handle.net/20.500.12608/21799

ہے [82] Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, M Sohaib Alam, Shahnawaz Ahmed, Juan Miguel Arrazola, Carsten Blank, Alain Delgado, Soran Jahangiri, et al. ``Pennylane: Automatic differentiation of hybrid quantum-classical computations'' (2018). arXiv:1811.04968.
آر ایکس سی: 1811.04968

ہے [83] Julian Havil. ``Gamma: exploring euler's constant''. The Australian Mathematical SocietyPage 250 (2003). url: https://ieeexplore.ieee.org/document/9452347.
https://ieeexplore.ieee.org/document/9452347

ہے [84] Juan Carlos Garcia-Escartin and Pedro Chamorro-Posada. ``Equivalent quantum circuits'' (2011). arXiv:1110.2998.
آر ایکس سی: 1110.2998

ہے [85] Karol Życzkowski and Hans-Jürgen Sommers. ``Average fidelity between random quantum states''. Phys. Rev. A 71, 032313 (2005).
https:///doi.org/10.1103/PhysRevA.71.032313

کی طرف سے حوالہ دیا گیا

[1] Yuchen Guo and Shuo Yang, "Noise effects on purity and quantum entanglement in terms of physical implementability", npj کوانٹم معلومات 9, 11 (2023).

[2] Dirk Heimann, Gunnar Schönhoff, and Frank Kirchner, "Learning capability of parametrized quantum circuits", آر ایکس سی: 2209.10345, (2022).

مذکورہ بالا اقتباسات سے ہیں۔ SAO/NASA ADS (آخری بار کامیابی کے ساتھ 2023-06-06 14:08:58)۔ فہرست نامکمل ہو سکتی ہے کیونکہ تمام ناشرین مناسب اور مکمل حوالہ ڈیٹا فراہم نہیں کرتے ہیں۔

On Crossref کی طرف سے پیش خدمت کاموں کے حوالے سے کوئی ڈیٹا نہیں ملا (آخری کوشش 2023-06-06 14:08:57)۔

یہ مقالہ کوانٹم میں کے تحت شائع کیا گیا ہے۔ Creative Commons انتساب 4.0 انٹرنیشنل (CC BY 4.0) لائسنس کاپی رائٹ اصل کاپی رائٹ ہولڈرز جیسے مصنفین یا ان کے اداروں کے پاس رہتا ہے۔

SEO سے چلنے والا مواد اور PR کی تقسیم۔ آج ہی بڑھا دیں۔
پلیٹوآئ اسٹریم۔ ویب 3 ڈیٹا انٹیلی جنس۔ علم میں اضافہ۔ یہاں تک رسائی حاصل کریں۔
ایڈریین ایشلے کے ساتھ مستقبل کا نقشہ بنانا۔ یہاں تک رسائی حاصل کریں۔
PREIPO® کے ساتھ PRE-IPO کمپنیوں میں حصص خریدیں اور بیچیں۔ یہاں تک رسائی حاصل کریں۔
ماخذ: https://quantum-journal.org/papers/q-2023-05-31-1023/

ٹائم اسٹیمپ: 31 فرمائے، 2023