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Cost function dependent barren plateaus in shallow parametrized quantum circuits

Variational quantum algorithms (VQAs) optimize the parameters θ of a parametrized quantum circuit V(θ) to minimize a cost function C. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two...

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Detalles Bibliográficos
Autores principales: Cerezo, M., Sone, Akira, Volkoff, Tyler, Cincio, Lukasz, Coles, Patrick J.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Nature Publishing Group UK 2021
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7979934/
https://www.ncbi.nlm.nih.gov/pubmed/33741913
http://dx.doi.org/10.1038/s41467-021-21728-w
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author Cerezo, M.
Sone, Akira
Volkoff, Tyler
Cincio, Lukasz
Coles, Patrick J.
author_facet Cerezo, M.
Sone, Akira
Volkoff, Tyler
Cincio, Lukasz
Coles, Patrick J.
author_sort Cerezo, M.
collection PubMed
description Variational quantum algorithms (VQAs) optimize the parameters θ of a parametrized quantum circuit V(θ) to minimize a cost function C. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming V(θ) is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining C in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when V(θ) is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining C with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of V(θ) is [Formula: see text] . Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation.
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spelling pubmed-79799342021-04-16 Cost function dependent barren plateaus in shallow parametrized quantum circuits Cerezo, M. Sone, Akira Volkoff, Tyler Cincio, Lukasz Coles, Patrick J. Nat Commun Article Variational quantum algorithms (VQAs) optimize the parameters θ of a parametrized quantum circuit V(θ) to minimize a cost function C. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming V(θ) is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining C in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when V(θ) is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining C with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of V(θ) is [Formula: see text] . Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation. Nature Publishing Group UK 2021-03-19 /pmc/articles/PMC7979934/ /pubmed/33741913 http://dx.doi.org/10.1038/s41467-021-21728-w Text en © The Author(s) 2021 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
spellingShingle Article
Cerezo, M.
Sone, Akira
Volkoff, Tyler
Cincio, Lukasz
Coles, Patrick J.
Cost function dependent barren plateaus in shallow parametrized quantum circuits
title Cost function dependent barren plateaus in shallow parametrized quantum circuits
title_full Cost function dependent barren plateaus in shallow parametrized quantum circuits
title_fullStr Cost function dependent barren plateaus in shallow parametrized quantum circuits
title_full_unstemmed Cost function dependent barren plateaus in shallow parametrized quantum circuits
title_short Cost function dependent barren plateaus in shallow parametrized quantum circuits
title_sort cost function dependent barren plateaus in shallow parametrized quantum circuits
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7979934/
https://www.ncbi.nlm.nih.gov/pubmed/33741913
http://dx.doi.org/10.1038/s41467-021-21728-w
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