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Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer

Wierichs, David and Gogolin, Christian and Kastoryano, Michael (2020) Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer. Physical Review Research, 2 (4). Art. No. 043246. ISSN 2643-1564. doi:10.1103/physrevresearch.2.043246.

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We compare the bfgs optimizer, adam and NatGrad in the context of vqes. We systematically analyze their performance on the qaqa ansatz for the transverse field Ising and the XXZ model as well as on overparametrized circuits with the ability to break the symmetry of the Hamiltonian. The bfgs algorithm is frequently unable to find a global minimum for systems beyond about 20 spins and adam easily gets trapped in local minima or exhibits infeasible optimization durations. NatGrad on the other hand shows stable performance on all considered system sizes, rewarding its higher cost per epoch with reliability and competitive total run times. In sharp contrast to most classical gradient-based learning, the performance of all optimizers decreases upon seemingly benign overparametrization of the ansatz class, with bfgs and adam failing more often and more severely than NatGrad. This does not only stress the necessity for good ansatz circuits but also means that overparametrization, an established remedy for avoiding local minima in machine learning, does not seem to be a viable option in the context of vqes. The behavior in both investigated spin chains is similar, in particular the problems of bfgs and adam surface in both systems, even though their effective Hilbert space dimensions differ significantly. Overall our observations stress the importance of avoiding redundant degrees of freedom in ansatz circuits and to put established optimization algorithms and attached heuristics to test on larger system sizes. Natural gradient descent emerges as a promising choice to optimize large vqes.

Item Type:Article
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URLURL TypeDescription Paper
Wierichs, David0000-0002-0983-7136
Kastoryano, Michael0000-0001-5233-7957
Additional Information:© 2020 Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Received 14 May 2020; accepted 23 October 2020; published 17 November 2020. We would like to thank Chae-Yeun Park, David Gross, Gian-Luca Anselmetti, and Thorben Frank for helpful discussions. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy-Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1-390534769. The authors would like to thank Covestro Deutschland AG, Kaiser Wilhelm Allee 60, 51373 Leverkusen, for the support with computational resources. The work was conducted while all three authors were affiliated with the Institute for Theoretical Physics of the University of Cologne.
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Deutsche Forschungsgemeinschaft (DFG)EXC 2004/1-390534769
Issue or Number:4
Record Number:CaltechAUTHORS:20201117-104742100
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:106699
Deposited By: Tony Diaz
Deposited On:17 Nov 2020 18:54
Last Modified:16 Nov 2021 18:56

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