Effect of barren plateaus on gradient-free optimization

Andrew Arrasmith1, M. Cerezo1,2, Piotr Czarnik1, Lukasz Cincio1, and Patrick J. Coles1

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
2Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA

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Barren plateau landscapes correspond to gradients that vanish exponentially in the number of qubits. Such landscapes have been demonstrated for variational quantum algorithms and quantum neural networks with either deep circuits or global cost functions. For obvious reasons, it is expected that gradient-based optimizers will be significantly affected by barren plateaus. However, whether or not gradient-free optimizers are impacted is a topic of debate, with some arguing that gradient-free approaches are unaffected by barren plateaus. Here we show that, indeed, gradient-free optimizers do not solve the barren plateau problem. Our main result proves that cost function differences, which are the basis for making decisions in a gradient-free optimization, are exponentially suppressed in a barren plateau. Hence, without exponential precision, gradient-free optimizers will not make progress in the optimization. We numerically confirm this by training in a barren plateau with several gradient-free optimizers (Nelder-Mead, Powell, and COBYLA algorithms), and show that the numbers of shots required in the optimization grows exponentially with the number of qubits.

Quantum machine learning and variational quantum algorithms provide the potential for near-term, practical applications of quantum computers. However, these techniques require the use of a classical optimization loop that can, in some cases, prove prohibitive. In particular, some of these optimization landscapes present barren plateaus, where the gradient of the cost function is exponentially suppressed except within an exponentially small region. As a result, optimization on a barren plateau landscape requires computational resources that scale exponentially with the number of qubits used.

Perhaps because of the mention of gradients in the definition of a barren plateau, a number of researchers have postulated that a gradient free optimizer might be able to beat the resource scaling. This thought turns out to be incorrect. Here we provide a proof that no gradient free strategy (including models that build surrogate gradients) can escape this resource scaling in barren plateau landscapes.

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[3] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, "Noisy intermediate-scale quantum (NISQ) algorithms", arXiv:2101.08448.

[4] Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J. Coles, "Noise-Induced Barren Plateaus in Variational Quantum Algorithms", arXiv:2007.14384.

[5] Michael Broughton, Guillaume Verdon, Trevor McCourt, Antonio J. Martinez, Jae Hyeon Yoo, Sergei V. Isakov, Philip Massey, Ramin Halavati, Murphy Yuezhen Niu, Alexander Zlokapa, Evan Peters, Owen Lockwood, Andrea Skolik, Sofiene Jerbi, Vedran Dunjko, Martin Leib, Michael Streif, David Von Dollen, Hongxiang Chen, Shuxiang Cao, Roeland Wiersema, Hsin-Yuan Huang, Jarrod R. McClean, Ryan Babbush, Sergio Boixo, Dave Bacon, Alan K. Ho, Hartmut Neven, and Masoud Mohseni, "TensorFlow Quantum: A Software Framework for Quantum Machine Learning", arXiv:2003.02989.

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[21] Yidong Liao, Min-Hsiu Hsieh, and Chris Ferrie, "Quantum Optimization for Training Quantum Neural Networks", arXiv:2103.17047.

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