Diagnosing Barren Plateaus with Tools from Quantum Optimal Control

Martin Larocca1,2, Piotr Czarnik2, Kunal Sharma3,2, Gopikrishnan Muraleedharan2, Patrick J. Coles2, and M. Cerezo4,5

1Departamento de Física “J. J. Giambiagi” and IFIBA, FCEyN, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
3Hearne Institute for Theoretical Physics and Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA USA
4Information Sciences, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
5Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

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Variational Quantum Algorithms (VQAs) have received considerable attention due to their potential for achieving near-term quantum advantage. However, more work is needed to understand their scalability. One known scaling result for VQAs is barren plateaus, where certain circumstances lead to exponentially vanishing gradients. It is common folklore that problem-inspired ansatzes avoid barren plateaus, but in fact, very little is known about their gradient scaling. In this work we employ tools from quantum optimal control to develop a framework that can diagnose the presence or absence of barren plateaus for problem-inspired ansatzes. Such ansatzes include the Quantum Alternating Operator Ansatz (QAOA), the Hamiltonian Variational Ansatz (HVA), and others. With our framework, we prove that avoiding barren plateaus for these ansatzes is not always guaranteed. Specifically, we show that the gradient scaling of the VQA depends on the degree of controllability of the system, and hence can be diagnosed through the dynamical Lie algebra $\mathfrak{g}$ obtained from the generators of the ansatz. We analyze the existence of barren plateaus in QAOA and HVA ansatzes, and we highlight the role of the input state, as different initial states can lead to the presence or absence of barren plateaus. Taken together, our results provide a framework for trainability-aware ansatz design strategies that do not come at the cost of extra quantum resources. Moreover, we prove no-go results for obtaining ground states with variational ansatzes for controllable system such as spin glasses. Our work establishes a link between the existence of barren plateaus and the scaling of the dimension of $\mathfrak{g}$.

In this work, we provide a novel framework for diagnosing the presence or absence of Barren Plateaus (BPs) in variational quantum algorithms and quantum machine learning models. Our work leverages tools from quantum control theory to connect the scaling of the cost-function gradients with the dimension of the so-called dynamical Lie algebra (DLA), the Lie closure of the generators of the parametrized quantum circuit. Our results greatly improve our understanding of the BP phenomenon, allowing us to predict their happening in a wide range of scenarios that were not covered by previous literature. Taken together, this work provides novel strategies for an active trainability-aware design of quantum neural network architectures, and showcases the importance of the DLA in variational quantum computing.

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[2] M. Cerezo, Guillaume Verdon, Hsin-Yuan Huang, Lukasz Cincio, and Patrick J. Coles, "Challenges and opportunities in quantum machine learning", Nature Computational Science 2 9, 567 (2022).

[3] Ryan LaRose, Eleanor Rieffel, and Davide Venturelli, "Mixer-phaser Ansätze for quantum optimization with hard constraints", arXiv:2107.06651, Quantum Machine Intelligence 4 2, 17 (2022).

[4] Matthias C. Caro, Hsin-Yuan Huang, M. Cerezo, Kunal Sharma, Andrew Sornborger, Lukasz Cincio, and Patrick J. Coles, "Generalization in quantum machine learning from few training data", Nature Communications 13 1, 4919 (2022).

[5] Martin Larocca, Nathan Ju, Diego García-Martín, Patrick J. Coles, and M. Cerezo, "Theory of overparametrization in quantum neural networks", arXiv:2109.11676.

[6] Samson Wang, Piotr Czarnik, Andrew Arrasmith, M. Cerezo, Lukasz Cincio, and Patrick J. Coles, "Can Error Mitigation Improve Trainability of Noisy Variational Quantum Algorithms?", arXiv:2109.01051.

[7] Christiane P. Koch, Ugo Boscain, Tommaso Calarco, Gunther Dirr, Stefan Filipp, Steffen J. Glaser, Ronnie Kosloff, Simone Montangero, Thomas Schulte-Herbrüggen, Dominique Sugny, and Frank K. Wilhelm, "Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe", arXiv:2205.12110.

[8] Nic Ezzell, Elliott M. Ball, Aliza U. Siddiqui, Mark M. Wilde, Andrew T. Sornborger, Patrick J. Coles, and Zoë Holmes, "Quantum Mixed State Compiling", arXiv:2209.00528.

[9] Louis Schatzki, Andrew Arrasmith, Patrick J. Coles, and M. Cerezo, "Entangled Datasets for Quantum Machine Learning", arXiv:2109.03400.

[10] Stefan H. Sack, Raimel A. Medina, Alexios A. Michailidis, Richard Kueng, and Maksym Serbyn, "Avoiding Barren Plateaus Using Classical Shadows", PRX Quantum 3 2, 020365 (2022).

[11] Pejman Jouzdani, Calvin W. Johnson, Eduardo R. Mucciolo, and Ionel Stetcu, "An Alternative Approach to Quantum Imaginary Time Evolution", arXiv:2208.10535.

[12] Martín Larocca, Frédéric Sauvage, Faris M. Sbahi, Guillaume Verdon, Patrick J. Coles, and M. Cerezo, "Group-Invariant Quantum Machine Learning", PRX Quantum 3 3, 030341 (2022).

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[14] Supanut Thanasilp, Samson Wang, Nhat A. Nghiem, Patrick J. Coles, and M. Cerezo, "Subtleties in the trainability of quantum machine learning models", arXiv:2110.14753.

[15] Frederic Sauvage, Martin Larocca, Patrick J. Coles, and M. Cerezo, "Building spatial symmetries into parameterized quantum circuits for faster training", arXiv:2207.14413.

[16] Junyu Liu, Khadijeh Najafi, Kunal Sharma, Francesco Tacchino, Liang Jiang, and Antonio Mezzacapo, "An analytic theory for the dynamics of wide quantum neural networks", arXiv:2203.16711.

[17] Yanzhu Chen, Linghua Zhu, Chenxu Liu, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "How Much Entanglement Do Quantum Optimization Algorithms Require?", arXiv:2205.12283.

[18] Annie E. Paine, Vincent E. Elfving, and Oleksandr Kyriienko, "Quantum Kernel Methods for Solving Differential Equations", arXiv:2203.08884.

[19] Andi Gu, Angus Lowe, Pavel A. Dub, Patrick J. Coles, and Andrew Arrasmith, "Adaptive shot allocation for fast convergence in variational quantum algorithms", arXiv:2108.10434.

[20] Alejandro Sopena, Max Hunter Gordon, Diego García-Martín, Germán Sierra, and Esperanza López, "Algebraic Bethe Circuits", arXiv:2202.04673.

[21] Adam Callison and Nicholas Chancellor, "Hybrid quantum-classical algorithms in the noisy intermediate-scale quantum era and beyond", Physical Review A 106 1, 010101 (2022).

[22] Massimiliano Incudini, Francesco Martini, and Alessandra Di Pierro, "Structure Learning of Quantum Embeddings", arXiv:2209.11144.

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

[24] Daniel Bultrini, Samson Wang, Piotr Czarnik, Max Hunter Gordon, M. Cerezo, Patrick J. Coles, and Lukasz Cincio, "The battle of clean and dirty qubits in the era of partial error correction", arXiv:2205.13454.

[25] Antonio Anna Mele, Glen Bigan Mbeng, Giuseppe Ernesto Santoro, Mario Collura, and Pietro Torta, "Avoiding barren plateaus via transferability of smooth solutions in Hamiltonian Variational Ansatz", arXiv:2206.01982.

[26] Nishant Jain, Brian Coyle, Elham Kashefi, and Niraj Kumar, "Graph neural network initialisation of quantum approximate optimisation", arXiv:2111.03016.

[27] Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, and Dacheng Tao, "Gaussian initializations help deep variational quantum circuits escape from the barren plateau", arXiv:2203.09376.

[28] Roeland Wiersema and Nathan Killoran, "Optimizing quantum circuits with Riemannian gradient flow", arXiv:2202.06976.

[29] Xiaozhen Ge, Re-Bing Wu, and Herschel Rabitz, "The Optimization Landscape of Hybrid Quantum-Classical Algorithms: from Quantum Control to NISQ Applications", arXiv:2201.07448.

[30] Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, and Dacheng Tao, "Toward Trainability of Deep Quantum Neural Networks", arXiv:2112.15002.

[31] John Napp, "Quantifying the barren plateau phenomenon for a model of unstructured variational ansätze", arXiv:2203.06174.

[32] Ayush Asthana, Chenxu Liu, Oinam Romesh Meitei, Sophia E. Economou, Edwin Barnes, and Nicholas J. Mayhall, "Minimizing state preparation times in pulse-level variational molecular simulations", arXiv:2203.06818.

[33] Alicia B. Magann, Kenneth M. Rudinger, Matthew D. Grace, and Mohan Sarovar, "Feedback-based quantum optimization", arXiv:2103.08619.

[34] Supanut Thanasilp, Samson Wang, M. Cerezo, and Zoë Holmes, "Exponential concentration and untrainability in quantum kernel methods", arXiv:2208.11060.

[35] Xinbiao Wang, Junyu Liu, Tongliang Liu, Yong Luo, Yuxuan Du, and Dacheng Tao, "Symmetric Pruning in Quantum Neural Networks", arXiv:2208.14057.

[36] Kishor Bharti, Tobias Haug, Vlatko Vedral, and Leong-Chuan Kwek, "Noisy intermediate-scale quantum algorithm for semidefinite programming", Physical Review A 105 5, 052445 (2022).

[37] Zeyi Tao, Jindi Wu, Qi Xia, and Qun Li, "LAWS: Look Around and Warm-Start Natural Gradient Descent for Quantum Neural Networks", arXiv:2205.02666.

[38] Manas Sajjan, Junxu Li, Raja Selvarajan, Shree Hari Sureshbabu, Sumit Suresh Kale, Rishabh Gupta, Vinit Singh, and Sabre Kais, "Quantum Machine Learning for Chemistry and Physics", arXiv:2111.00851.

[39] Saad Yalouz, Bruno Senjean, Filippo Miatto, and Vedran Dunjko, "Encoding strongly-correlated many-boson wavefunctions on a photonic quantum computer: application to the attractive Bose-Hubbard model", arXiv:2103.15021.

[40] L. C. G. Govia, C. Poole, M. Saffman, and H. K. Krovi, "Freedom of the mixer rotation axis improves performance in the quantum approximate optimization algorithm", Physical Review A 104 6, 062428 (2021).

[41] Owen Lockwood, "Optimizing Quantum Variational Circuits with Deep Reinforcement Learning", arXiv:2109.03188.

[42] Enrico Fontana, Ivan Rungger, Ross Duncan, and Cristina Cîrstoiu, "Efficient recovery of variational quantum algorithms landscapes using classical signal processing", arXiv:2208.05958.

[43] Alicia B. Magann, Kenneth M. Rudinger, Matthew D. Grace, and Mohan Sarovar, "Lyapunov control-inspired strategies for quantum combinatorial optimization", arXiv:2108.05945.

[44] Chiara Leadbeater, Louis Sharrock, Brian Coyle, and Marcello Benedetti, "F-Divergences and Cost Function Locality in Generative Modelling with Quantum Circuits", Entropy 23 10, 1281 (2021).

[45] Ioannis Kolotouros, Ioannis Petrongonas, and Petros Wallden, "Adiabatic quantum computing with parameterized quantum circuits", arXiv:2206.04373.

[46] James Sud, Stuart Hadfield, Eleanor Rieffel, Norm Tubman, and Tad Hogg, "A Parameter Setting Heuristic for the Quantum Alternating Operator Ansatz", arXiv:2211.09270.

The above citations are from Crossref's cited-by service (last updated successfully 2022-11-30 01:53:48) and SAO/NASA ADS (last updated successfully 2022-11-30 01:53:49). The list may be incomplete as not all publishers provide suitable and complete citation data.