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Layerwise learning for quantum neural networks
Open accessAbstract
Abstract With the increased focus on quantum circuit learning for near-term applications on quantum devices, in conjunction with unique challenges presented by cost function landscapes of parametrized quantum circuits, strategies for effective training are becoming increasingly important. In order to ameliorate some of these challenges, we investigate a layerwise learning strategy for parametrized quantum circuits. The circuit depth is incrementally grown during optimization, and only subsets of parameters are updated in each training step. We show that when considering sampling noise, this strategy can help avoid the problem of barren plateaus of the error surface due to the low depth of circuits, low number of parameters trained in one step, and larger magnitude of gradients compared to training the full circuit. These properties make our algorithm preferable for execution on noisy intermediate-scale quantum devices. We demonstrate our approach on an image-classification task on handwritten digits, and show that layerwise learning attains an 8% lower generalization error on average in comparison to standard learning schemes for training quantum circuits of the same size. Additionally, the percentage of runs that reach lower test errors is up to 40% larger compared to training the full circuit, which is susceptible to creeping onto a plateau during training.
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“Additionally, in Refs. [29, 205] an initialization strategy was developed specifically to address BPs in deep circuits. Here, one selects at random a subset of parameters in θ , and chooses the value of the remaining ones so that the circuit is a sequence of shallow unitaries that evaluates to the identity.”
From Variational quantum algorithms (2021) · cites this paper as [205]
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“Regardless of the conditions of their advantages, these approaches need careful consideration of subtle issues (see, e.g., Refs.[112-115]) that do not appear in classical algorithms and are, in particular, strongly intertwined with the underlying hardware.”
From Neutral atom quantum computing hardware: performance and end-user perspective (2023) · cites this paper as [113]
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“It was found that the entries of the Hessian vanish exponentially with the size of the system in models that are in a barren plateau.”
From The power of quantum neural networks (2021) · cites this paper as [32]
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“In the quantum realm, the negative effect of barren plateaus on training quantum neural networks has been linked to the Hessian matrix [32].”
From The power of quantum neural networks (2021) · cites this paper as [32]
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Cites
Adam: A Method for Stochastic Optimization
Kingma, D. P., & Ba, J. (2014) · arXiv · doi:10.48550/arxiv.1412.6980 · Cited by 98,946
Learning representations by back-propagating errors
Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986) · Nature · doi:10.1038/323533a0 · Cited by 31,892
A fast learning algorithm for deep belief nets
Hinton, G. E., Osindero, S., & Teh, Y.-W. (2006) · Neural Computation · doi:10.1162/neco.2006.18.7.1527 · Cited by 16,336
Quantum supremacy using a programmable superconducting processor
Arute, F., Arya, K., Babbush, R., Bacon, D., Bardin, J. C., Barends, R., Biswas, R., Boixo, S., Brandao, F. G. S. L., Buell, D. A., et al. (2019) · Nature · doi:10.1038/s41586-019-1666-5 · Cited by 7,720
Self-citationGreedy layer-wise training of deep networks
Bengio, Y., Lamblin, P., Popovici, D., & Larochelle, H. (2007) · Advances in Neural Information Processing Systems 19 · doi:10.7551/mitpress/7503.003.0024 · Cited by 4,841
A variational eigenvalue solver on a photonic quantum processor
Peruzzo, A., McClean, J., Shadbolt, P., Yung, M.-H., Zhou, X.-Q., Love, P. J., Aspuru-Guzik, A., & O’Brien, J. L. (2014) · Nature Communications · doi:10.1038/ncomms5213 · Cited by 4,552
Self-citationHardware-efficient variational quantum eigensolver for small molecules and quantum magnets
Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J. M., & Gambetta, J. M. (2017) · Nature · doi:10.1038/nature23879 · Cited by 2,982
Supervised learning with quantum-enhanced feature spaces
Havlíček, V., Córcoles, A. D., Temme, K., Harrow, A. W., Kandala, A., Chow, J. M., & Gambetta, J. M. (2019) · Nature · doi:10.1038/s41586-019-0980-2 · Cited by 2,572
The Cascade-Correlation learning architecture
Fahlman, S. E., & Lebiere, C. J. (2018) · doi:10.1184/r1/6610403 · Cited by 2,391
Barren plateaus in quantum neural network training landscapes
McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R., & Neven, H. (2018) · Nature Communications · doi:10.1038/s41467-018-07090-4 · Cited by 2,329
Self-citationThe theory of variational hybrid quantum-classical algorithms
McClean, J. R., Romero, J., Babbush, R., & Aspuru-Guzik, A. (2016) · New Journal of Physics · doi:10.1088/1367-2630/18/2/023023 · Cited by 2,227
Self-citationA Quantum Approximate Optimization Algorithm
Farhi, E., Goldstone, J., & Gutmann, S. (2014) · arXiv · doi:10.48550/arxiv.1411.4028 · Cited by 1,789
Mitarai, K., Negoro, M., Kitagawa, M., & Fujii, K. (2018) · Physical Review A · doi:10.1103/physreva.98.032309 · Cited by 1,639
Ghaffari, S., Chandorkar, S. A., Wang, S., Ng, E. J., Ahn, C. H., Hong, V. A., Yang, Y., & Kenny, T. W. (2014) · Scientific Reports · doi:10.1038/srep04331 · Cited by 1,521
Quantum convolutional neural networks
Cong, I., Choi, S., & Lukin, M. D. (2019) · Nature Physics · doi:10.1038/s41567-019-0648-8 · Cited by 1,517
Classification with Quantum Neural Networks on Near Term Processors
Farhi, E., & Neven, H. (2020) · doi:10.37686/qrl.v1i2.80 · Cited by 1,176
Characterizing quantum supremacy in near-term devices
Boixo, S., Isakov, S. V., Smelyanskiy, V. N., Babbush, R., Ding, N., Jiang, Z., Bremner, M. J., Martinis, J. M., & Neven, H. (2018) · Nature Physics · doi:10.1038/s41567-018-0124-x · Cited by 1,175
Evaluating analytic gradients on quantum hardware
Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., & Killoran, N. (2019) · Physical Review A · doi:10.1103/physreva.99.032331 · Cited by 1,072
Scalable Quantum Simulation of Molecular Energies
O’Malley, P. J. J., Babbush, R., Kivlichan, I. D., Romero, J., McClean, J. R., Barends, R., Kelly, J., Roushan, P., Tranter, A., Ding, N., et al. (2016) · Physical Review X · doi:10.1103/physrevx.6.031007 · Cited by 1,059
Self-citationAn adaptive variational algorithm for exact molecular simulations on a quantum computer
Grimsley, H. R., Economou, S. E., Barnes, E., & Mayhall, N. J. (2019) · Nature Communications · doi:10.1038/s41467-019-10988-2 · Cited by 1,033
Zhou, L., Wang, S.-T., Choi, S., Pichler, H., & Lukin, M. D. (2020) · Physical Review X · doi:10.1103/physrevx.10.021067 · Cited by 913
From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz
Hadfield, S., Wang, Z., O’Gorman, B., Rieffel, E. G., Venturelli, D., & Biswas, R. (2019) · Algorithms · doi:10.3390/a12020034 · Cited by 772
Progress towards practical quantum variational algorithms
Wecker, D., Hastings, M. B., & Troyer, M. (2015) · Physical Review A · doi:10.1103/physreva.92.042303 · Cited by 716
Quantum Chemistry Calculations on a Trapped-Ion Quantum Simulator
Hempel, C., Maier, C., Romero, J., McClean, J., Monz, T., Shen, H., Jurcevic, P., Lanyon, B. P., Love, P., Babbush, R., et al. (2018) · Physical Review X · doi:10.1103/physrevx.8.031022 · Cited by 629
Self-citationQuantum autoencoders for efficient compression of quantum data
Romero, J., Olson, J. P., & Aspuru-Guzik, A. (2017) · Quantum Science and Technology · doi:10.1088/2058-9565/aa8072 · Cited by 559
Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states
McClean, J. R., Kimchi-Schwartz, M. E., Carter, J., & de Jong, W. A. (2017) · Physical Review A · doi:10.1103/physreva.95.042308 · Cited by 516
Self-citationComputation of Molecular Spectra on a Quantum Processor with an Error-Resilient Algorithm
Colless, J. I., Ramasesh, V. V., Dahlen, D., Blok, M. S., Kimchi-Schwartz, M. E., McClean, J. R., Carter, J., de Jong, W. A., & Siddiqi, I. (2018) · Physical Review X · doi:10.1103/physrevx.8.011021 · Cited by 460
Self-citationAn initialization strategy for addressing barren plateaus in parametrized quantum circuits
Grant, E., Wossnig, L., Ostaszewski, M., & Benedetti, M. (2019) · Quantum · doi:10.22331/q-2019-12-09-214 · Cited by 406
Hierarchical quantum classifiers
Grant, E., Benedetti, M., Cao, S., Hallam, A., Lockhart, J., Stojevic, V., Green, A. G., & Simone Severini, B. (2018) · npj Quantum Information · doi:10.1038/s41534-018-0116-9 · Cited by 387
A generative modeling approach for benchmarking and training shallow quantum circuits
Benedetti, M., Garcia-Pintos, D., Perdomo, O., Leyton-Ortega, V., Nam, Y., & Perdomo-Ortiz, A. (2019) · npj Quantum Information · doi:10.1038/s41534-019-0157-8 · Cited by 379
Quantum approximate optimization algorithm for MaxCut: A fermionic view
Wang, Z., Hadfield, S., Jiang, Z., & Rieffel, E. G. (2018) · Physical Review A · doi:10.1103/physreva.97.022304 · Cited by 353
An initialization strategy for addressing barren plateaus in parametrized quantum circuits
Grant, E., Wossnig, L., Ostaszewski, M., & Benedetti, M. (2019) · UCL Discovery (University College London) · Cited by 338
Random Quantum Circuits are Approximate 2-designs
Harrow, A. W., & Low, R. A. (2009) · Communications in Mathematical Physics · doi:10.1007/s00220-009-0873-6 · Cited by 324
TensorFlow Quantum: A Software Framework for Quantum Machine Learning
Broughton, M., Verdon, G., McCourt, T., Martinez, A. J., Yoo, J. H., Isakov, S. V., Massey, P., Halavati, R., Niu, M. Y., Zlokapa, A., et al. (2020) · arXiv · doi:10.48550/arxiv.2003.02989 · Cited by 308
Self-citationHierarchical quantum classifiers
Grant, E., Benedetti, M., Cao, S., Hallam, A., Lockhart, J., Stojevic, V, Green, A. G., & Simone Severini, B. (2018) · UCL Discovery (University College London) · Cited by 296
Differentiable learning of quantum circuit Born machines
Liu, J.-G., & Wang, L. (2018) · Physical Review A · doi:10.1103/physreva.98.062324 · Cited by 279
Quantum implementation of the unitary coupled cluster for simulating molecular electronic structure
Shen, Y., Zhang, X., Zhang, S., Zhang, J.-N., Yung, M.-H., & Kim, K. (2015) · Physical Review A · doi:10.1103/physreva.95.020501 · Cited by 262
Conservation prioritization can resolve the flagship species conundrum
McGowan, J., Beaumont, L. J., Smith, R. J., Chauvenet, A. L. M., Harcourt, R. G., Atkinson, S., Mittermeier, J. C., Esperón‐Rodríguez, M., Baumgartner, J. B., Beattie, A. J., et al. (2020) · Nature Communications · doi:10.1038/s41467-020-14554-z · Cited by 254
Learning to learn with quantum neural networks via classical neural networksPreprint
Verdon, G., Broughton, M., McClean, J. R., Sung, K. J., Babbush, R., Zhang, J., Neven, H., & Mohseni, M. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1907.05415 · Cited by 250
Self-citationFrom transistor to trapped-ion computers for quantum chemistry
Yung, M.-H., Casanova, J., Mezzacapo, A., McClean, J., Lamata, L., Aspuru-Guzik, A., & Solano, E. (2014) · Scientific Reports · doi:10.1038/srep03589 · Cited by 221
Self-citationSequential minimal optimization for quantum-classical hybrid algorithms
Nakanishi, K. M., Fujii, K., & Todo, S. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.043158 · Cited by 221
Witnessing eigenstates for quantum simulation of Hamiltonian spectra
Santagati, R., Wang, J., Gentile, A. A., Paesani, S., Wiebe, N., McClean, J. R., Morley-Short, S., Shadbolt, P. J., Bonneau, D., Silverstone, J. W., et al. (2018) · Science Advances · doi:10.1126/sciadv.aap9646 · Cited by 211
Self-citationOptimal quantum measurements of expectation values of observables
Knill, E. H., Ortíz, G., & Somma, R. D. (2007) · Physical Review A · doi:10.1103/physreva.75.012328 · Cited by 206
Takeshita, T. Y., Rubin, N. C., Zhang, J., Lee, E., Babbush, R., & McClean, J. R. (2020) · Physical Review X · doi:10.1103/physrevx.10.011004 · Cited by 177
Self-citationApplication of fermionic marginal constraints to hybrid quantum algorithms
Rubin, N. C., Babbush, R., & McClean, J. (2018) · New Journal of Physics · doi:10.1088/1367-2630/aab919 · Cited by 152
Self-citationDecoding quantum errors with subspace expansions
McClean, J. R., Zhang, J., Rubin, N. C., Babbush, R., & Neven, H. (2020) · Nature Communications · doi:10.1038/s41467-020-14341-w · Cited by 148
Self-citationLarge gradients via correlation in random parameterized quantum circuits
Volkoff, T., & Coles, P. J. (2021) · Quantum Science and Technology · doi:10.1088/2058-9565/abd891 · Cited by 143
Performance of hybrid quantum-classical variational heuristics for combinatorial optimization
Nannicini, G. (2019) · Physical Review E · doi:10.1103/physreve.99.013304 · Cited by 138 · cited 1×
“Indeed a number of different optimization strategies for PQCs have been explored, including deterministic and stochastic optimizers (Wecker et al. 2015; Romero et al. 2017; Nannicini 2019; Nakanishi et al. 2020; Zhou et al. 2018; Sung et al. 2020).”
BackgroundTraining the quantum approximate optimization algorithm without access to a quantum processing unit
Streif, M., & Leib, M. (2020) · Quantum Science and Technology · doi:10.1088/2058-9565/ab8c2b · Cited by 125
Self-citationDecoding quantum errors with subspace expansions
McClean, J. R., Zhang, J., Rubin, N. C., Babbush, R., & Neven, H. (2020) · RePEc: Research Papers in Economics · Cited by 105
Self-citationRobust determination of molecular spectra on a quantum processorPreprint
Colless, J., Ramasesh, V., Dahlen, D., et al. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1707.06408 · Cited by 95
Universal discriminative quantum neural networks
Chen, H., Wossnig, L., Severini, S., Neven, H., & Mohseni, M. (2021) · Quantum Machine Intelligence · doi:10.1007/s42484-020-00025-7 · Cited by 89
Self-citationAdversarial quantum circuit learning for pure state approximation
Benedetti, M., Grant, E., Wossnig, L., & Simone Severini, B. (2019) · New Journal of Physics · doi:10.1088/1367-2630/ab14b5 · Cited by 88
Variational quantum unsampling on a quantum photonic processor
Carolan, J., Mohseni, M., Olson, J. P., Prabhu, M., Chen, C., Bunandar, D., Niu, M. Y., Harris, N. C., Wong, F. N. C., Hochberg, M., et al. (2020) · Nature Physics · doi:10.1038/s41567-019-0747-6 · Cited by 88
Self-citationCost-Function-Dependent Barren Plateaus in Shallow Quantum Neural Networks.Preprint
Cerezo, M., Sone, A., Volkoff, T. J., Cincio, Ł., & Coles, P. J. (2020) · arXiv (Cornell University) · Cited by 83
Parrish, R. M., Iosue, J. T., Ozaeta, A., & McMahon, P. L. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1904.03206 · Cited by 67
Zhou, L., Wang, S.-T., Choi, S., Pichler, H., & Lukin, M. D. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1812.01041 · Cited by 54
Learning Unitaries by Gradient DescentPreprint
Kiani, B. T., Lloyd, S., & Maity, R. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2001.11897 · Cited by 44
Forward Thinking: Building and Training Neural Networks One Layer at a TimePreprint
Hettinger, C., Christensen, T., Ehlert, B., Humpherys, J., Jarvis, T. J., & Wade, S. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1706.02480 · Cited by 28
Accelerated variational algorithms for digital quantum simulation of many-body ground states
Lyu, C., Montenegro, V., & Bayat, A. (2020) · Quantum · doi:10.22331/q-2020-09-16-324 · Cited by 27
Sung, K. J., Harrigan, M. P., Rubin, N. C., Zhang, J., Babbush, R., & McClean, J. R. (2020) · arXiv (Cornell University) · Cited by 4
Self-citation
Research works printed but not linked: 6
- Cerezo M, Sone A, Volkoff T, Cincio L, Coles PJ (2020) arXiv:2001.00550 · arXiv
- Chen H, Wossnig L, Severini S, Neven H, Mohseni M (2018) arXiv:1805.08654 · arXiv
- Fahlman S E, Lebiere C (1990) Adv Neural Inf Process 2:524–532. https://doi.org/10.1.1.125.6421
- Hettinger C, Christensen T, Ehlert B, Humpherys J, Jarvis T, Wade S (2017) In: 31st Conference on Neural Information Processing Systems. 1706.02480
- Volkoff T, Coles PJ (2020) arXiv:2005.12200 · arXiv
- Yung M-H, Casanova J, Mezzacapo A, McClean J, Lamata L, Aspuru-Guzik A, Solano E (2014) Sci Rep 4(3589):1–7. https://www.nature.com/articles/srep03589
Cited by
Sorted by citations
Variational quantum algorithms
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“[178] introduced a method where one employs layerby-layer training.”
MethodSelf-citationNoisy intermediate-scale quantum algorithms
Bharti, K., Cervera-Lierta, A., Kyaw, T. H., Haug, T., Alperin-Lea, S., Anand, A., Degroote, M., Heimonen, H., Kottmann, J. S., Menke, T., et al. (2022) · Reviews of Modern Physics · doi:10.1103/revmodphys.94.015004 · Cited by 2,231
The power of quantum neural networks
Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., & Woerner, S. (2021) · Nature Computational Science · doi:10.1038/s43588-021-00084-1 · Cited by 1,082
Cost function dependent barren plateaus in shallow parametrized quantum circuits
Cerezo, M., Sone, A., Volkoff, T., Cincio, L., & Coles, P. J. (2021) · Nature Communications · doi:10.1038/s41467-021-21728-w · Cited by 1,047
The Variational Quantum Eigensolver: A review of methods and best practices
Tilly, J., Chen, H., Cao, S., Picozzi, D., Setia, K., Li, Y., Grant, E., Wossnig, L., Rungger, I., Booth, G. H., et al. (2022) · Physics Reports · doi:10.1016/j.physrep.2022.08.003 · Cited by 1,031
Metadata
- Authors
- Skolik, A., McClean, J. R., Mohseni, M., van der Smagt, P., & Leib, M.
- Affiliations
- Leiden University (NL)
- Google (United States) (US)
- Corresponding author
- Skolik, Andrea
- Venue
- Quantum Machine Intelligence
- Published
- 2021-01-25
- Volume
- 3
- Issue
- 1
- ISSN
- 2524-4906 (ISSN-L)
- 2524-4914
- OpenAlex
- W3004252283
- Cited by
- 324
- References
- 63
- Citations by year
- 2026: 43 · 2025: 58 · 2024: 65 · 2023: 59 · 2022: 46 · 2021: 48
- FWCI
- 19.09
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Locations
- Quantum Machine Intelligence (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Bulletin of the American Physical Society
- Funders
- Horizon 2020 (828826)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Keywords
- quantum neural networks, variational quantum circuits, barren plateaus, NISQ devices, circuit depth, sampling noise, image classification, handwritten digits, generalization error
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