Analytic Theory for the Dynamics of Wide Quantum Neural Networks

Liu, Junyu; Najafi, Khadijeh; Sharma, Kunal; Tacchino, Francesco; Jiang, Liang; Mezzacapo, Antonio

doi:10.1103/physrevlett.130.150601

Analytic Theory for the Dynamics of Wide Quantum Neural Networks

Journal Article · Mon Apr 10 00:00:00 EDT 2023 · Physical Review Letters

DOI:https://doi.org/10.1103/physrevlett.130.150601· OSTI ID:2423784

Liu, Junyu ^[1]; Najafi, Khadijeh ^[2]; Sharma, Kunal ^[3]; ^[4]; Jiang, Liang ^[1]; Mezzacapo, Antonio ^[2]

The University of Chicago, IL (United States); Chicago Quantum Exchange, IL (United States)
IBM T. J. Watson Research Center, Yorktown Heights, NY (United States)
IBM T. J. Watson Research Center, Yorktown Heights, NY (United States); University of Maryland, College Park, MD (United States)
IBM Quantum, Zurich, Rüschlikon (Switzerland)

Here, parametrized quantum circuits can be used as quantum neural networks and have the potential to outperform their classical counterparts when trained for addressing learning problems. To date, much of the results on their performance on practical problems are heuristic in nature. In particular, the convergence rate for the training of quantum neural networks is not fully understood. Here, we analyze the dynamics of gradient descent for the training error of a class of variational quantum machine learning models. We define wide quantum neural networks as parametrized quantum circuits in the limit of a large number of qubits and variational parameters. Then, we find a simple analytic formula that captures the average behavior of their loss function and discuss the consequences of our findings. For example, for random quantum circuits, we predict and characterize an exponential decay of the residual training error as a function of the parameters of the system. Finally, we validate our analytic results with numerical experiments.

View Accepted Manuscript (DOE)

Research Organization:: National Quantum Information Science (QIS) Research Centers (United States). Next Generation Quantum Science and Engineering (Q-NEXT)

Sponsoring Organization:: AFOSR MURI; AFRL; ARO; ARO MURI; Packard Foundation; USDOE; USDOE Office of Science (SC), Basic Energy Sciences (BES). Scientific User Facilities (SUF)

Grant/Contract Number:: AC02-06CH11357; AC05-00OR22725

OSTI ID:: 2423784

Alternate ID(s):: OSTI ID: 1970414

Journal Information:: Physical Review Letters, Journal Name: Physical Review Letters Journal Issue: 15 Vol. 130; ISSN 0031-9007

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

References (33)

Expressibility and Entangling Capability of Parameterized Quantum Circuits for Hybrid Quantum‐Classical Algorithms Sim, Sukin; Johnson, Peter D.; Aspuru‐Guzik, Alán Advanced Quantum Technologies, Vol. 2, Issue 12 https://doi.org/10.1002/qute.201900070	journal	October 2019
The Principles of Deep Learning Theory Roberts, Daniel A.; Yaida, Sho; Hanin, Boris https://doi.org/10.1017/9781009023405	book	May 2022
Deep learning LeCun, Yann; Bengio, Yoshua; Hinton, Geoffrey Nature, Vol. 521, Issue 7553 https://doi.org/10.1038/nature14539	journal	May 2015
Quantum machine learning Biamonte, Jacob; Wittek, Peter; Pancotti, Nicola Nature, Vol. 549, Issue 7671 https://doi.org/10.1038/nature23474	journal	September 2017
Structural absorption by barbule microstructures of super black bird of paradise feathers McCoy, Dakota E.; Feo, Teresa; Harvey, Todd Alan Nature Communications, Vol. 9, Issue 1 https://doi.org/10.1038/s41467-017-02088-w	journal	January 2018
U1 snRNP regulates cancer cell migration and invasion in vitro Oh, Jung-Min; Venters, Christopher C.; Di, Chao Nature Communications, Vol. 11, Issue 1 https://doi.org/10.1038/s41467-019-13993-7	journal	January 2020
Anomalous collapses of Nares Strait ice arches leads to enhanced export of Arctic sea ice Moore, G. W. K.; Howell, S. E. L.; Brady, M. Nature Communications, Vol. 12, Issue 1 https://doi.org/10.1038/s41467-020-20314-w	journal	January 2021
Power of data in quantum machine learning Huang, Hsin-Yuan; Broughton, Michael; Mohseni, Masoud Nature Communications, Vol. 12, Issue 1 https://doi.org/10.1038/s41467-021-22539-9	journal	May 2021
Quantum algorithmic measurement Aharonov, Dorit; Cotler, Jordan; Qi, Xiao-Liang Nature Communications, Vol. 13, Issue 1 https://doi.org/10.1038/s41467-021-27922-0	journal	February 2022
Generalization in quantum machine learning from few training data Caro, Matthias C.; Huang, Hsin-Yuan; Cerezo, M. Nature Communications, Vol. 13, Issue 1 https://doi.org/10.1038/s41467-022-32550-3	journal	August 2022
Quantum convolutional neural networks Cong, Iris; Choi, Soonwon; Lukin, Mikhail D. Nature Physics, Vol. 15, Issue 12 https://doi.org/10.1038/s41567-019-0648-8	journal	August 2019
A rigorous and robust quantum speed-up in supervised machine learning Liu, Yunchao; Arunachalam, Srinivasan; Temme, Kristan Nature Physics, Vol. 17, Issue 9 https://doi.org/10.1038/s41567-021-01287-z	journal	July 2021
The power of quantum neural networks Abbas, Amira; Sutter, David; Zoufal, Christa Nature Computational Science, Vol. 1, Issue 6 https://doi.org/10.1038/s43588-021-00084-1	journal	June 2021
An introduction to quantum machine learning Schuld, Maria; Sinayskiy, Ilya; Petruccione, Francesco Contemporary Physics, Vol. 56, Issue 2 https://doi.org/10.1080/00107514.2014.964942	journal	October 2014
Machine learning & artificial intelligence in the quantum domain: a review of recent progress Dunjko, Vedran; Briegel, Hans J. Reports on Progress in Physics, Vol. 81, Issue 7 https://doi.org/10.1088/1361-6633/aab406	journal	June 2018
Exploring Entanglement and Optimization within the Hamiltonian Variational Ansatz Wiersema, Roeland; Zhou, Cunlu; de Sereville, Yvette PRX Quantum, Vol. 1, Issue 2 https://doi.org/10.1103/PRXQuantum.1.020319	journal	December 2020
Generalization in Quantum Machine Learning: A Quantum Information Standpoint Banchi, Leonardo; Pereira, Jason; Pirandola, Stefano PRX Quantum, Vol. 2, Issue 4 https://doi.org/10.1103/PRXQuantum.2.040321	journal	November 2021
Connecting Ansatz Expressibility to Gradient Magnitudes and Barren Plateaus Holmes, Zoë; Sharma, Kunal; Cerezo, M. PRX Quantum, Vol. 3, Issue 1 https://doi.org/10.1103/PRXQuantum.3.010313	journal	January 2022
Is Quantum Advantage the Right Goal for Quantum Machine Learning? Schuld, Maria; Killoran, Nathan PRX Quantum, Vol. 3, Issue 3 https://doi.org/10.1103/PRXQuantum.3.030101	journal	July 2022
Representation Learning via Quantum Neural Tangent Kernels Liu, Junyu; Tacchino, Francesco; Glick, Jennifer R. PRX Quantum, Vol. 3, Issue 3 https://doi.org/10.1103/PRXQuantum.3.030323	journal	August 2022
Exact and approximate unitary 2-designs and their application to fidelity estimation Dankert, Christoph; Cleve, Richard; Emerson, Joseph Physical Review A, Vol. 80, Issue 1 https://doi.org/10.1103/PhysRevA.80.012304	journal	July 2009
Progress towards practical quantum variational algorithms Wecker, Dave; Hastings, Matthew B.; Troyer, Matthias Physical Review A, Vol. 92, Issue 4 https://doi.org/10.1103/PhysRevA.92.042303	journal	October 2015
Information-Theoretic Bounds on Quantum Advantage in Machine Learning Huang, Hsin-Yuan; Kueng, Richard; Preskill, John Physical Review Letters, Vol. 126, Issue 19 https://doi.org/10.1103/PhysRevLett.126.190505	journal	May 2021
Reformulation of the No-Free-Lunch Theorem for Entangled Datasets Sharma, Kunal; Cerezo, M.; Holmes, Zoë Physical Review Letters, Vol. 128, Issue 7 https://doi.org/10.1103/PhysRevLett.128.070501	journal	February 2022
Efficient Measure for the Expressivity of Variational Quantum Algorithms Du, Yuxuan; Tu, Zhuozhuo; Yuan, Xiao Physical Review Letters, Vol. 128, Issue 8 https://doi.org/10.1103/PhysRevLett.128.080506	journal	February 2022
Trainability of Dissipative Perceptron-Based Quantum Neural Networks Sharma, Kunal; Cerezo, M.; Cincio, Lukasz Physical Review Letters, Vol. 128, Issue 18 https://doi.org/10.1103/PhysRevLett.128.180505	journal	May 2022
Presence and Absence of Barren Plateaus in Tensor-Network Based Machine Learning Liu, Zidu; Yu, Li-Wei; Duan, L. -M. Physical Review Letters, Vol. 129, Issue 27 https://doi.org/10.1103/PhysRevLett.129.270501	journal	December 2022
Entanglement devised barren plateau mitigation Patti, Taylor L.; Najafi, Khadijeh; Gao, Xun Physical Review Research, Vol. 3, Issue 3 https://doi.org/10.1103/PhysRevResearch.3.033090	journal	July 2021
Absence of Barren Plateaus in Quantum Convolutional Neural Networks Pesah, Arthur; Cerezo, M.; Wang, Samson Physical Review X, Vol. 11, Issue 4 https://doi.org/10.1103/PhysRevX.11.041011	journal	October 2021
Quantum computing models for artificial neural networks Mangini, S.; Tacchino, F.; Gerace, D. Europhysics Letters, Vol. 134, Issue 1 https://doi.org/10.1209/0295-5075/134/10002	journal	April 2021
Expressibility of the alternating layered ansatz for quantum computation Nakaji, Kouhei; Yamamoto, Naoki Quantum, Vol. 5 https://doi.org/10.22331/q-2021-04-19-434	journal	April 2021
Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus Zhao, Chen; Gao, Xiao-Shan Quantum, Vol. 5 https://doi.org/10.22331/q-2021-06-04-466	journal	June 2021
Diagnosing Barren Plateaus with Tools from Quantum Optimal Control Larocca, Martin; Czarnik, Piotr; Sharma, Kunal Quantum, Vol. 6 https://doi.org/10.22331/q-2022-09-29-824	journal	September 2022

Similar Records

Representation Learning via Quantum Neural Tangent Kernels

Journal Article · Tue Aug 16 20:00:00 EDT 2022 · PRX Quantum · OSTI ID:1982853

Hybrid Quantum-Classical Neural Networks

Conference · Tue Nov 01 00:00:00 EDT 2022 · OSTI ID:1905393

Uncertainty quantification of graph convolution neural network models of evolving processes

Journal Article · Tue Jul 02 20:00:00 EDT 2024 · Computer Methods in Applied Mechanics and Engineering · OSTI ID:2440697

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
Artificial neural networks
Machine learning
Physics
Quantum algorithms
Quantum simulation

Analytic Theory for the Dynamics of Wide Quantum Neural Networks

Citation Formats

References (33)

Similar Records

Related Subjects