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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Abstract

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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[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).

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[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).

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[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.