Tight and Efficient Gradient Bounds for Parameterized Quantum Circuits

Alistair Letcher; Stefan Woerner; Christa Zoufal

doi:10.22331/q-2024-09-25-1484

Tight and Efficient Gradient Bounds for Parameterized Quantum Circuits

Alistair Letcher, Stefan Woerner, and Christa Zoufal

IBM Quantum, IBM Research Europe – Zurich

Published:	2024-09-25, volume 8, page 1484
Eprint:	arXiv:2309.12681v3
Doi:	https://doi.org/10.22331/q-2024-09-25-1484
Citation:	Quantum 8, 1484 (2024).

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Abstract

The training of a parameterized model largely depends on the landscape of the underlying loss function. In particular, vanishing gradients are a central bottleneck in the scalability of variational quantum algorithms (VQAs), and are known to arise in various ways. However, a caveat of most existing gradient bound results is the requirement of t-design circuit assumptions that are typically not satisfied in practice. In this work, we loosen these assumptions altogether and derive tight upper and lower bounds on loss and gradient concentration for a large class of parameterized quantum circuits and arbitrary observables, which are significantly stronger than prior work. Moreover, we show that these bounds, as well as the variance of the loss itself, can be estimated efficiently and classically-providing practical tools to study the loss landscapes of VQA models, including verifying whether or not a circuit/observable induces barren plateaus. In particular, our results can readily be leveraged to rule out barren plateaus for a realistic class of ansätze and mixed observables, namely, observables containing a non-vanishing local term. This insight has direct implications for hybrid Quantum Generative Adversarial Networks (qGANs). We prove that designing the discriminator appropriately leads to 1-local weights that stay constant in the number of qubits, regardless of discriminator depth. This implies that qGANs with appropriately chosen generators do not suffer from barren plateaus even at scale-making them a promising candidate for applications in generative quantum machine learning. We demonstrate this result by training a qGAN to learn a 2D mixture of Gaussian distributions with up to 16 qubits, and provide numerical evidence that global contributions to the gradient, while initially exponentially small, may kick in substantially over the course of training.

Featured image: Variance of the loss function and corresponding bounds compared with existing work for an EfficientSU2 ansatz of depth $d = \frac{n}{2}$ and a 1-local observable. Variance and bounds are estimated with 10000 samples of $\mathbf{\theta}\sim \left\{0, \frac{\pi}{2}\right\}^m$.

Popular summary

Variational quantum algorithms have been identified as tools for tackling multiple problems including time evolution, searching for ground states, and preparing thermal states. However, these systems often suffer from gradients that vanish exponentially in the system size and can, therefore, be difficult to train. Our research provides upper and lower bounds for the gradients that are not only tighter than existing ones and compatible with realistic circuit assumptions, but can also be efficiently evaluated with classical resources. This improves our understanding of whether it is worth approaching a given problem with a quantum computer. Furthermore, these results enable us to prove that certain variational quantum algorithms do not suffer from vanishing gradients, including an emerging class of algorithms called Quantum Generative Adversarial Networks – making them a promising candidate for applications in machine learning. The respective insights are supported by numerical experiments.

► BibTeX data

@article{Letcher2024tightefficient,
  doi = {10.22331/q-2024-09-25-1484},
  url = {https://doi.org/10.22331/q-2024-09-25-1484},
  title = {Tight and {E}fficient {G}radient {B}ounds for {P}arameterized {Q}uantum {C}ircuits},
  author = {Letcher, Alistair and Woerner, Stefan and Zoufal, Christa},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1484},
  month = sep,
  year = {2024}
}

► References

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

[2] M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles. ``Cost function dependent barren plateaus in shallow parametrized quantum circuits''. Nature Communications 12, 1791 (2021).
https://doi.org/10.1038/s41467-021-21728-w

[3] M Cerezo and Patrick J Coles. ``Higher order derivatives of quantum neural networks with barren plateaus''. Quantum Science and Technology 6, 035006 (2021).
https://doi.org/10.1088/2058-9565/abf51a

[4] Zoë Holmes, Kunal Sharma, M. Cerezo, and Patrick J. Coles. ``Connecting ansatz expressibility to gradient magnitudes and barren plateaus''. PRX Quantum 3, 345 (2022).
https://doi.org/10.1103/PRXQuantum.3.010313

[5] 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

[6] Samson Wang, Enrico Fontana, M. Cerezo, et al. ``Noise-induced barren plateaus in variational quantum algorithms''. Nature Communications 12, 6961 (2021).
https://doi.org/10.1038/s41467-021-27045-6

[7] John Napp. ``Quantifying the barren plateau phenomenon for a model of unstructured variational ansätze'' (2022). arXiv:2203.06174.
arXiv:2203.06174

[8] A V Uvarov and J D Biamonte. ``On barren plateaus and cost function locality in variational quantum algorithms''. Journal of Physics A: Mathematical and Theoretical 54, 245301 (2021).
https://doi.org/10.1088/1751-8121/abfac7

[9] Arthur Pesah, M. Cerezo, Samson Wang, et al. ``Absence of barren plateaus in quantum convolutional neural networks''. Phys. Rev. X 11, 041011 (2021).
https://doi.org/10.1103/PhysRevX.11.041011

[10] Kunal Sharma, M. Cerezo, Lukasz Cincio, and Patrick J. Coles. ``Trainability of dissipative perceptron-based quantum neural networks''. Phys. Rev. Lett. 128, 180505 (2022).
https://doi.org/10.1103/PhysRevLett.128.180505

[11] 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/10.22331/q-2019-12-09-214

[12] Manuel S. Rudolph, Jacob Miller, Danial Motlagh, et al. ``Synergistic pretraining of parametrized quantum circuits via tensor networks''. Nature Communications 14, 8367 (2023).
https://doi.org/10.1038/s41467-023-43908-6

[13] Chen Zhao and Xiao-Shan Gao. ``Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus''. Quantum 5, 466 (2021).
https://doi.org/10.22331/q-2021-06-04-466

[14] Yabo Wang, Bo Qi, Chris Ferrie, and Daoyi Dong. ``Trainability enhancement of parameterized quantum circuits via reduced-domain parameter initialization'' (2023). arXiv:2302.06858.
arXiv:2302.06858

[15] Kaining Zhang, Liu Liu, Min-Hsiu Hsieh, and Dacheng Tao. ``Escaping from the barren plateau via gaussian initializations in deep variational quantum circuits''. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho, editors, Advances in Neural Information Processing Systems. Volume 35. (2022). url: https://doi.org/10.48550/arXiv.2203.09376.
https://doi.org/10.48550/arXiv.2203.09376

[16] Chiara Leadbeater, Louis Sharrock, Brian Coyle, and Marcello Benedetti. ``F-Divergences and Cost Function Locality in Generative Modelling with Quantum Circuits''. Entropy 23 (2021).
https://doi.org/10.3390/e23101281

[17] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. ``A quantum approximate optimization algorithm applied to a bounded occurrence constraint problem'' (2015). arXiv:1412.6062.
arXiv:1412.6062

[18] Amira Abbas, Andris Ambainis, Brandon Augustino, et al. ``Quantum optimization: Potential, challenges, and the path forward'' (2023). arXiv:2312.02279.
arXiv:2312.02279

[19] Vojtěch Havlíček, Antonio D. Córcoles, Kristan Temme, et al. ``Supervised learning with quantum-enhanced feature spaces''. Nature 567, 209–212 (2019).
https://doi.org/10.1038/s41586-019-0980-2

[20] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, et al. ``A variational eigenvalue solver on a photonic quantum processor''. Nature Communications 5, 4213 (2014).
https://doi.org/10.1038/ncomms5213

[21] Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. ``Exact and approximate unitary 2-designs and their application to fidelity estimation''. Phys. Rev. A 80, 012304 (2009).
https://doi.org/10.1103/PhysRevA.80.012304

[22] Manuel S. Rudolph, Sacha Lerch, Supanut Thanasilp, et al. ``Trainability barriers and opportunities in quantum generative modeling'' (2023). arXiv:2305.02881.
arXiv:2305.02881

[23] Carlos Bravo-Prieto, Ryan LaRose, M. Cerezo, et al. ``Variational Quantum Linear Solver''. Quantum 7, 1188 (2023).
https://doi.org/10.22331/q-2023-11-22-1188

[24] 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

[25] Sumeet Khatri, Ryan LaRose, Alexander Poremba, et al. ``Quantum-assisted quantum compiling''. Quantum 3, 140 (2019).
https://doi.org/10.22331/q-2019-05-13-140

[26] Cristina Cı̂rstoiu, Zoë Holmes, Joseph Iosue, et al. ``Variational fast forwarding for quantum simulation beyond the coherence time''. npj Quantum Information 6, 82 (2020).
https://doi.org/10.1038/s41534-020-00302-0

[27] Eric R. Anschuetz and Bobak T. Kiani. ``Quantum variational algorithms are swamped with traps''. Nature Communications 13, 7760 (2022).
https://doi.org/10.1038/s41467-022-35364-5

[28] M. Cerezo, Kunal Sharma, Andrew Arrasmith, and Patrick J. Coles. ``Variational quantum state eigensolver''. npj Quantum Information 8, 113 (2022).
https://doi.org/10.1038/s41534-022-00611-6

[29] Christa Zoufal, Aurélien Lucchi, and Stefan Woerner. ``Quantum generative adversarial networks for learning and loading random distributions''. npj Quantum Information 5 (2019).
https://doi.org/10.1038/s41534-019-0223-2

[30] Haozhen Situ, Zhimin He, Yuyi Wang, Lvzhou Li, and Shenggen Zheng. ``Quantum generative adversarial network for generating discrete distribution''. Information Sciences 538, 193–208 (2020).
https://doi.org/10.1016/j.ins.2020.05.127

[31] Pierre-Luc Dallaire-Demers and Nathan Killoran. ``Quantum generative adversarial networks''. Phys. Rev. A 98, 012324 (2018).
https://doi.org/10.1103/PhysRevA.98.012324

[32] Jonathan Romero and Alán Aspuru-Guzik. ``Variational quantum generators: Generative adversarial quantum machine learning for continuous distributions''. Advanced Quantum Technologies 4, 2000003 (2021).
https://doi.org/10.1002/qute.202000003

[33] Jinfeng Zeng, Yufeng Wu, Jin-Guo Liu, Lei Wang, and Jiangping Hu. ``Learning and inference on generative adversarial quantum circuits''. Phys. Rev. A 99, 052306 (2019).
https://doi.org/10.1103/PhysRevA.99.052306

[34] Alistair Letcher, Jakob Foerster, David Balduzzi, et al. ``Stable opponent shaping in differentiable games''. International Conference on Learning Representations (2019). url: https://doi.org/10.48550/arXiv.1811.08469.
https://doi.org/10.48550/arXiv.1811.08469

[35] M. Cerezo, Andrew Arrasmith, Ryan Babbush, et al. ``Variational quantum algorithms''. Nature Reviews Physics 3, 625–644 (2021).
https://doi.org/10.1038/s42254-021-00348-9

[36] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, et al. ``Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets''. Nature 549, 242–246 (2017).
https://doi.org/10.1038/nature23879

[37] Christa Zoufal, Ryan V. Mishmash, Nitin Sharma, et al. ``Variational quantum algorithm for unconstrained black box binary optimization: Application to feature selection''. Quantum 7, 909 (2023).
https://doi.org/10.22331/q-2023-01-26-909

[38] Tadayoshi Matsumori, Masato Taki, and Tadashi Kadowaki. ``Application of qubo solver using black-box optimization to structural design for resonance avoidance''. Scientific Reports 12, 12143 (2022).
https://doi.org/10.1038/s41598-022-16149-8

[39] Syun Izawa, Koki Kitai, Shu Tanaka, Ryo Tamura, and Koji Tsuda. ``Continuous black-box optimization with an ising machine and random subspace coding''. Phys. Rev. Res. 4, 023062 (2022).
https://doi.org/10.1103/PhysRevResearch.4.023062

[40] Javier Alcazar, Mohammad Ghazi Vakili, Can B. Kalayci, and Alejandro Perdomo-Ortiz. ``Enhancing combinatorial optimization with classical and quantum generative models''. Nature Communications 15, 2761 (2024).
https://doi.org/10.1038/s41467-024-46959-5

[41] Ling Hu, Shu-Hao Wu, Weizhou Cai, et al. ``Quantum generative adversarial learning in a superconducting quantum circuit''. Science Advances 5, eaav2761 (2019).
https://doi.org/10.1126/sciadv.aav2761

[42] Seth Lloyd and Christian Weedbrook. ``Quantum generative adversarial learning''. Phys. Rev. Lett. 121, 040502 (2018).
https://doi.org/10.1103/PhysRevLett.121.040502

[43] Igor O. Sokolov, Panagiotis Kl. Barkoutsos, Pauline J. Ollitrault, et al. ``Quantum orbital-optimized unitary coupled cluster methods in the strongly correlated regime: Can quantum algorithms outperform their classical equivalents?''. The Journal of Chemical Physics 152, 123 (2020). url: https://doi.org/10.1063/1.5141835.
https://doi.org/10.1063/1.5141835

[44] Guillermo García-Pérez, Matteo A.C. Rossi, Boris Sokolov, et al. ``Learning to measure: Adaptive informationally complete generalized measurements for quantum algorithms''. PRX Quantum 2, 040342 (2021).
https://doi.org/10.1103/PRXQuantum.2.040342

[45] Johannes Hachmann, Wim Cardoen, and Garnet Kin-Lic Chan. ``Multireference correlation in long molecules with the quadratic scaling density matrix renormalization group''. The Journal of Chemical Physics 125, 144101 (2006).
https://doi.org/10.1063/1.2345196

[46] Peter A. Limacher, Paul W. Ayers, Paul A. Johnson, et al. ``A new mean-field method suitable for strongly correlated electrons: Computationally facile antisymmetric products of nonorthogonal geminals''. Journal of Chemical Theory and Computation 9, 1394–1401 (2013). url: http://dx.doi.org/10.1021/ct300902c.
https://doi.org/10.1021/ct300902c

[47] Mario Motta et al. ``Towards the solution of the many-electron problem in real materials: Equation of state of the hydrogen chain with state-of-the-art many-body methods''. Phys. Rev. X 7, 031059 (2017).
https://doi.org/10.1103/PhysRevX.7.031059

[48] Pauline J. Ollitrault, Alberto Baiardi, Markus Reiher, and Ivano Tavernelli. ``Hardware efficient quantum algorithms for vibrational structure calculations''. Chem. Sci. 11, 6842–6855 (2020).
https://doi.org/10.1039/D0SC01908A

[49] Sam McArdle, Alexander Mayorov, Xiao Shan, Simon Benjamin, and Xiao Yuan. ``Digital quantum simulation of molecular vibrations''. Chem. Sci. 10, 5725–5735 (2019).
https://doi.org/10.1039/C9SC01313J

[50] Nicolas P. D. Sawaya, Francesco Paesani, and Daniel P. Tabor. ``Near- and long-term quantum algorithmic approaches for vibrational spectroscopy''. Phys. Rev. A 104, 062419 (2021).
https://doi.org/10.1103/PhysRevA.104.062419

[51] Karol Kowalski. ``Dimensionality reduction of the many-body problem using coupled-cluster subsystem flow equations: Classical and quantum computing perspective''. Phys. Rev. A 104, 032804 (2021).
https://doi.org/10.1103/PhysRevA.104.032804

[52] Renke Huang, Chenyang Li, and Francesco A. Evangelista. ``Leveraging small-scale quantum computers with unitarily downfolded hamiltonians''. PRX Quantum 4, 020313 (2023).
https://doi.org/10.1103/PRXQuantum.4.020313

[53] Nicholas P. Bauman, Bo Peng, and Karol Kowalski. ``Chapter two - coupled-cluster downfolding techniques: A review of existing applications in classical and quantum computing for chemical systems''. In Monika Musiał and Ireneusz Grabowski, editors, Polish Quantum Chemistry from Kołos to Now. Volume 87 of Advances in Quantum Chemistry, pages 141–166. Academic Press (2023).
https://doi.org/10.1016/bs.aiq.2023.03.006

[54] Andrew Lucas. ``Ising formulations of many np problems''. Frontiers in Physics 2 (2014).
https://doi.org/10.3389/fphy.2014.00005

[55] F Barahona. ``On the computational complexity of ising spin glass models''. Journal of Physics A: Mathematical and General 15, 3241 (1982).
https://doi.org/10.1088/0305-4470/15/10/028

[56] Stasja Stanisic et al. ``Observing ground-state properties of the fermi-hubbard model using a scalable algorithm on a quantum computer''. Nature communications 13, 5743 (2022).
https://doi.org/10.1038/s41467-022-33335-4

[57] Chris Cade, Lana Mineh, Ashley Montanaro, and Stasja Stanisic. ``Strategies for solving the fermi-hubbard model on near-term quantum computers''. Phys. Rev. B 102, 235122 (2020).
https://doi.org/10.1103/PhysRevB.102.235122

[58] Zhang Jiang, Kevin J. Sung, Kostyantyn Kechedzhi, Vadim N. Smelyanskiy, and Sergio Boixo. ``Quantum algorithms to simulate many-body physics of correlated fermions''. Phys. Rev. Appl. 9, 044036 (2018).
https://doi.org/10.1103/PhysRevApplied.9.044036

[59] Pauline J Ollitrault, Sven Jandura, Alexander Miessen, et al. ``Quantum algorithms for grid-based variational time evolution''. Quantum 7, 1139 (2023).
https://doi.org/10.22331/q-2023-10-12-1139

[60] P. Jordan and E. Wigner. ``Über das Paulische äquivalenzverbot''. Zeitschrift für Physik 47, 631–651 (1928). url: https://doi.org/10.1007/BF01331938.
https://doi.org/10.1007/BF01331938

[61] Jacob T. Seeley, Martin J. Richard, and Peter J. Love. ``The Bravyi-Kitaev transformation for quantum computation of electronic structure''. The Journal of Chemical Physics 137, 224109 (2012).
https://doi.org/10.1063/1.4768229

[62] Sergey B. Bravyi and Alexei Yu. Kitaev. ``Fermionic quantum computation''. Annals of Physics 298, 210–226 (2002). url: https://doi.org/10.1006/aphy.2002.6254.
https://doi.org/10.1006/aphy.2002.6254

[63] Andrew Arrasmith, Zoë Holmes, M 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

[64] Lorenzo Leone, Salvatore F.E. Oliviero, Lukasz Cincio, and M. Cerezo. ``On the practical usefulness of the Hardware Efficient Ansatz''. Quantum 8, 1395 (2024).
https://doi.org/10.22331/q-2024-07-03-1395

[65] Martin Larocca, Piotr Czarnik, Kunal Sharma, et al. ``Diagnosing Barren Plateaus with Tools from Quantum Optimal Control''. Quantum 6, 824 (2022).
https://doi.org/10.22331/q-2022-09-29-824

[66] Enrico Fontana, Dylan Herman, Shouvanik Chakrabarti, et al. ``Characterizing barren plateaus in quantum ansätze with the adjoint representation''. Nature Communications 15, 7171 (2024).
https://doi.org/10.1038/s41467-024-49910-w

[67] Michael Ragone, Bojko N. Bakalov, Frédéric Sauvage, Alexander F. Kemper, Carlos Ortiz Marrero, Martín Larocca, and M. Cerezo. ``A lie algebraic theory of barren plateaus for deep parameterized quantum circuits''. Nature Communications 15, 7172 (2024).
https://doi.org/10.1038/s41467-024-49909-3

[68] https://qiskit.org/documentation/stubs/qiskit.circuit.library.EfficientSU2.html (2023).
https://qiskit.org/documentation/stubs/qiskit.circuit.library.EfficientSU2.html

[69] https://qiskit.org/documentation/stubs/qiskit.circuit.library.RealAmplitudes.html (2023).
https://qiskit.org/documentation/stubs/qiskit.circuit.library.RealAmplitudes.html

[70] Louis Schatzki, Martín Larocca, Quynh T. Nguyen, Frédéric Sauvage, and M. Cerezo. ``Theoretical guarantees for permutation-equivariant quantum neural networks''. npj Quantum Information 10, 12 (2024).
https://doi.org/10.1038/s41534-024-00804-1

[71] Nikitas Stamatopoulos, Daniel J. Egger, Yue Sun, et al. ``Option Pricing using Quantum Computers''. Quantum 4, 291 (2020).
https://doi.org/10.22331/q-2020-07-06-291

[72] Brian Coyle, Daniel Mills, Vincent Danos, and Elham Kashefi. ``The born supremacy: quantum advantage and training of an ising born machine''. npj Quantum Information 6, 60 (2020).
https://doi.org/10.1038/s41534-020-00288-9

[73] Marcello Benedetti, Delfina Garcia-Pintos, Oscar Perdomo, Vicente Leyton-Ortega, Yunseong Nam, and Alejandro Perdomo-Ortiz. ``A generative modeling approach for benchmarking and training shallow quantum circuits''. npj Quantum Information 5, 45 (2019).
https://doi.org/10.1038/s41534-019-0157-8

[74] Christa Zoufal, Aurélien Lucchi, and Stefan Woerner. ``Variational quantum Boltzmann machines''. Quantum Machine Intelligence 3, 7 (2021).
https://doi.org/10.1007/s42484-020-00033-7

[75] Mohammad H. Amin, Evgeny Andriyash, Jason Rolfe, Bohdan Kulchytskyy, and Roger Melko. ``Quantum boltzmann machine''. Phys. Rev. X 8, 021050 (2018).
https://doi.org/10.1103/PhysRevX.8.021050

[76] Mária Kieferová and Nathan Wiebe. ``Tomography and Generative Training with Quantum Boltzmann Machines''. Phys. Rev. A 96, 062327 (2017). url: https://doi.org/10.1103/PhysRevA.96.062327.
https://doi.org/10.1103/PhysRevA.96.062327

[77] S. Kullback and R. A. Leibler. ``On information and sufficiency''. Ann. Math. Statist. 22, 79–86 (1951). url: https://doi.org/10.1214/aoms/1177729694.
https://doi.org/10.1214/aoms/1177729694

[78] Supanut Thanasilp, Samson Wang, Nhat Anh Nghiem, Patrick Coles, and Marco Cerezo. ``Subtleties in the trainability of quantum machine learning models''. Quantum Machine Intelligence 5 (2023).
https://doi.org/10.1007/s42484-023-00103-6

[79] Andrew L Maas et al. ``Rectifier nonlinearities improve neural network acoustic models''. In ICML. Volume 30, page 3. Atlanta, GA (2013). url: https://ai.stanford.edu/ amaas/papers/relu_hybrid_icml2013_final.pdf.
https://ai.stanford.edu/~amaas/papers/relu_hybrid_icml2013_final.pdf

[80] Xavier Glorot and Yoshua Bengio. ``Understanding the difficulty of training deep feedforward neural networks''. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Pages 249–256. (2010). url: https://proceedings.mlr.press/v9/glorot10a.html.
https://proceedings.mlr.press/v9/glorot10a.html

[81] K. He, X. Zhang, S. Ren, and J. Sun. ``Delving deep into rectifiers: Surpassing human-level performance on imagenet classification''. In 2015 IEEE International Conference on Computer Vision (ICCV). Los Alamitos, CA, USA (2015). IEEE Computer Society.

[82] Alec Radford, Luke Metz, and Soumith Chintala. ``Unsupervised representation learning with deep convolutional generative adversarial networks'' (2016). arXiv:1511.06434.
arXiv:1511.06434

[83] Andrew Brock, Jeff Donahue, and Karen Simonyan. ``Large Scale GAN Training for High Fidelity Natural Image Synthesis'' (2019). arXiv:1809.11096.
arXiv:1809.11096

[84] Tero Karras, Samuli Laine, Miika Aittala, et al. ``Analyzing and improving the image quality of stylegan''. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. Pages 8110–8119. (2020). url: https://doi.org/10.48550/arXiv.1912.04958.
https://doi.org/10.48550/arXiv.1912.04958

[85] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, et al. ``Generative adversarial nets''. In Advances in Neural Information Processing Systems 27. Pages 2672–2680. Curran Associates, Inc. (2014). url: https://doi.org/10.48550/arXiv.1406.2661.
https://doi.org/10.48550/arXiv.1406.2661

[86] Martin Arjovsky, Soumith Chintala, and Léon Bottou. ``Wasserstein generative adversarial networks''. In Doina Precup and Yee Whye Teh, editors, Proceedings of the 34th International Conference on Machine Learning. Volume 70 of Proceedings of Machine Learning Research, pages 214–223. PMLR (2017). url: https://doi.org/10.48550/arXiv.1701.07875.
https://doi.org/10.48550/arXiv.1701.07875

[87] Xavier Glorot, Antoine Bordes, and Yoshua Bengio. ``Deep sparse rectifier neural networks''. In Geoffrey Gordon, David Dunson, and Miroslav Dudík, editors, Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics. Volume 15 of Proceedings of Machine Learning Research, pages 315–323. PMLR (2011). url: https://proceedings.mlr.press/v15/glorot11a/glorot11a.pdf.
https://proceedings.mlr.press/v15/glorot11a/glorot11a.pdf

[88] Qiskit contributors. ``Qiskit: An open-source framework for quantum computing'' (2023).

[89] Luke Metz, Ben Poole, David Pfau, and Jascha Sohl-Dickstein. ``Unrolled generative adversarial networks''. In International Conference on Learning Representations. (2016). url: https://doi.org/10.48550/arXiv.1611.02163.
https://doi.org/10.48550/arXiv.1611.02163

[90] Alistair Letcher, David Balduzzi, Sébastien Racanière, et al. ``Differentiable game mechanics''. Journal of Machine Learning Research 20 (2019). url: https://doi.org/10.48550/arXiv.1905.04926.
https://doi.org/10.48550/arXiv.1905.04926

[91] Adam Paszke, Sam Gross, Francisco Massa, et al. ``PyTorch: an imperative style, high-performance deep learning library''. Curran Associates Inc. Red Hook, NY, USA (2019). url: https://dl.acm.org/doi/10.5555/3454287.3455008.
https://dl.acm.org/doi/10.5555/3454287.3455008

[92] Diederik P. Kingma and Jimmy Ba. ``Adam: A Method for Stochastic Optimization''. In Yoshua Bengio and Yann LeCun, editors, 3rd International Conference on Learning Representations. (2015). url: https://doi.org/10.48550/arXiv.1412.6980.
https://doi.org/10.48550/arXiv.1412.6980

[93] Amira Abbas, David Sutter, Christa Zoufal, et al. ``The power of quantum neural networks''. Nature Computational Science 1, 12 (2021).
https://doi.org/10.1038/s43588-021-00084-1

[94] M. Cerezo, Martin Larocca, Diego García-Martín, et al. ``Does provable absence of barren plateaus imply classical simulability? or, why we need to rethink variational quantum computing'' (2024). arXiv:2312.09121.
arXiv:2312.09121

[95] Sunitha Basodi, Chunyan Ji, Haiping Zhang, and Yi Pan. ``An efficient way to train deep neural networks''. Big Data Mining and Analytics 3, 196–207 (2020). url: https://doi.org/10.48550/arXiv.2006.10560.
https://doi.org/10.48550/arXiv.2006.10560

[96] M.G. Bulmer. ``Principles of statistics''. Dover Books on Mathematics Series. Dover Publications. (1979). url: https://books.google.com/books?id=dh24EaSrmBkC.
https://books.google.com/books?id=dh24EaSrmBkC

[97] Gary Kochenberger, Jin-Kao Hao, Fred Glover, et al. ``The unconstrained binary quadratic programming problem: a survey''. Journal of combinatorial optimization 28, 58–81 (2014). url: https://doi.org/10.1007/s10878-014-9734-0.
https://doi.org/10.1007/s10878-014-9734-0

[98] James C Spall. ``Multivariate stochastic approximation using a simultaneous perturbation gradient approximation''. IEEE transactions on automatic control 37, 332–341 (1992). url: https://doi.org/10.1109/9.119632.
https://doi.org/10.1109/9.119632

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[5] Sabri Meyer, Francesco Scala, Francesco Tacchino, and Aurelien Lucchi, "Gradient scalability and Taylor surrogation of quantum cost landscapes", Physical Review Research 8 2, 023325 (2026).

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[13] Kasidit Srimahajariyapong, Supanut Thanasilp, and Thiparat Chotibut, "Connecting phases of matter to the flatness of the loss landscape in analog variational quantum algorithms", Communications Physics 9 1, 111 (2026).

[14] Julien Baglio, "Cross-platform hardware benchmark of style-based quantum GANs for data augmentation on superconducting and trapped-ion processors", AIP Advances 16 6, 065008 (2026).

[15] Manuel S. Rudolph, Sacha Lerch, Supanut Thanasilp, Oriel Kiss, Oxana Shaya, Sofia Vallecorsa, Michele Grossi, and Zoë Holmes, "Trainability barriers and opportunities in quantum generative modeling", npj Quantum Information 10 1, 116 (2024).

[16] Martín Larocca, Supanut Thanasilp, Samson Wang, Kunal Sharma, Jacob Biamonte, Patrick J. Coles, Lukasz Cincio, Jarrod R. McClean, Zoë Holmes, and M. Cerezo, "Barren plateaus in variational quantum computing", Nature Reviews Physics 7 4, 174 (2025).

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[18] Jonas Jäger, Florian J Kiwit, and Carlos A Riofrío, "Scaling quantum machine learning without tricks: full-resolution and diverse image generation", Quantum Science and Technology 11 3, 035042 (2026).

[19] Samantha V. Barron, Daniel J. Egger, Elijah Pelofske, Andreas Bärtschi, Stephan Eidenbenz, Matthis Lehmkuehler, and Stefan Woerner, "Provable bounds for noise-free expectation values computed from noisy samples", Nature Computational Science 4 11, 865 (2024).

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[21] Paolo Braccia, Pablo Bermejo, Lukasz Cincio, and M. Cerezo, "Computing exact moments of local random quantum circuits via tensor networks", Quantum Machine Intelligence 6 2, 54 (2024).

[22] Yuhan Yao and Yoshihiko Hasegawa, "Linking barren plateaus to effective parameters in deep rotation-gate-based parameterized quantum circuits", Physical Review A 112 6, 062443 (2025).

[23] Antonio Sannia, Francesco Tacchino, Ivano Tavernelli, Gian Luca Giorgi, and Roberta Zambrini, "Engineered dissipation to mitigate barren plateaus", npj Quantum Information 10 1, 81 (2024).

[24] Abhinav Anand and Kenneth R. Brown, "Hamiltonian-based graph-state ansatz for variational quantum algorithms", Physical Review A 111 1, 012437 (2025).

[25] Michael Ragone, Bojko N. Bakalov, Frédéric Sauvage, Alexander F. Kemper, Carlos Ortiz Marrero, Martín Larocca, and M. Cerezo, "A Lie algebraic theory of barren plateaus for deep parameterized quantum circuits", Nature Communications 15 1, 7172 (2024).

[26] Travis L. Scholten, Carl J. Williams, Dustin Moody, Michele Mosca, William Hurley, William J. Zeng, Matthias Troyer, and Jay M. Gambetta, "Assessing the Benefits and Risks of Quantum Computers", arXiv:2401.16317, (2024).

[27] Su Yeon Chang, Supanut Thanasilp, Bertrand Le Saux, Sofia Vallecorsa, and Michele Grossi, "Latent Style-based Quantum GAN for high-quality Image Generation", arXiv:2406.02668, (2024).

[28] Abhinav Deshpande, Marcel Hinsche, Khadijeh Najafi, Kunal Sharma, Ryan Sweke, and Christa Zoufal, "Dynamic parameterized quantum circuits: expressive and barren-plateau free", arXiv:2411.05760, (2024).

[29] Nikita A. Nemkov, Evgeniy O. Kiktenko, and Aleksey K. Fedorov, "Barren plateaus swamped with traps", Physical Review A 111 1, 012441 (2025).

[30] Shabnam Jabeen, Dmytro Kurdydyk, Aadi Palnitkar, Mihir Talati, Jeffrey Yan, and Jinghong Yang, "Quantum Machine Learning for State Tomography Using Classical Data", arXiv:2507.01246, (2025).

[31] Gabriele Agliardi and Enrico Prati, "Quantum data encoding as a distinct abstraction layer in the design of quantum circuits", Quantum Science and Technology 10 2, 025008 (2025).

[32] Samantha V. Barron, Daniel J. Egger, Elijah Pelofske, Andreas Bärtschi, Stephan Eidenbenz, Matthis Lehmkuehler, and Stefan Woerner, "Provable bounds for noise-free expectation values computed from noisy samples", arXiv:2312.00733, (2023).

[33] Paolo Braccia, Pablo Bermejo, Lukasz Cincio, and M. Cerezo, "Computing exact moments of local random quantum circuits via tensor networks", arXiv:2403.01706, (2024).

[34] Naipunnya Raj, Rajiv Sangle, Avinash Singh, and Krishna Kumar Sabapathy, "Quantum Generative Adversarial Autoencoders: Learning latent representations for quantum data generation", arXiv:2509.16186, (2025).

[35] Anastasja D. Helgesen, Jan-Åke Larsson, and Michael Felsberg, "Quantum Classification through Tournament Voting for Robust Single-Shot Inference", arXiv:2406.04944, (2024).

[36] N. A. Nemkov, "Statistical Models of Barren Plateaus and Anti-Concentration of Pauli Observables", Soviet Journal of Experimental and Theoretical Physics Letters 122 3, 190 (2025).

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