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Non-trivial symmetries in quantum landscapes and their resilience to quantum noise

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Enrico Fontana1,2,3, M. Cerezo1,4, Andrew Arrasmith1, Ivan Rungger5, and Patrick J. Coles1

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
2Department of Computer and Information Sciences, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK
3National Physical Laboratory, Teddington TW11 0LW, UK
4Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA
5National Physical Laboratory, Teddington, UK

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Abstract

Very little is known about the cost landscape for parametrized Quantum Circuits (PQCs). Nevertheless, PQCs are employed in Quantum Neural Networks and Variational Quantum Algorithms, which may allow for near-term quantum advantage. Such applications require good optimizers to train PQCs. Recent works have focused on quantum-aware optimizers specifically tailored for PQCs. However, ignorance of the cost landscape could hinder progress towards such optimizers. In this work, we analytically prove two results for PQCs: (1) We find an exponentially large symmetry in PQCs, yielding an exponentially large degeneracy of the minima in the cost landscape. Alternatively, this can be cast as an exponential reduction in the volume of relevant hyperparameter space. (2) We study the resilience of the symmetries under noise, and show that while it is conserved under unital noise, non-unital channels can break these symmetries and lift the degeneracy of minima, leading to multiple new local minima. Based on these results, we introduce an optimization method called Symmetry-based Minima Hopping (SYMH), which exploits the underlying symmetries in PQCs. Our numerical simulations show that SYMH improves the overall optimizer performance in the presence of non-unital noise at a level comparable to current hardware. Overall, this work derives large-scale circuit symmetries from local gate transformations, and uses them to construct a noise-aware optimization method.

In this work, we study the cost landscape for parametrized quantum circuits (PQCs), which are employed in quantum neural networks and variational quantum algorithms. We unravel the presence of an exponentially large symmetry in the PQCs landscape, yielding an exponentially large degeneracy of the cost function minima. We then study the resilience of these symmetries under quantum noise, and show that while they are conserved under unital noise, non-unital channels can break these symmetries and lift the degeneracy of minima. Based on these results, we introduce an optimization method called Symmetry-based Minima Hopping (SYMH), which exploits the underlying symmetries in PQCs.

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Cited by

[1] Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H. Booth, and Jonathan Tennyson, “The Variational Quantum Eigensolver: a review of methods and best practices”, arXiv:2111.05176.

[2] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles, “Variational Quantum Algorithms”, arXiv:2012.09265.

[3] Taylor L. Patti, Khadijeh Najafi, Xun Gao, and Susanne F. Yelin, “Entanglement devised barren plateau mitigation”, Physical Review Research 3 3, 033090 (2021).

[4] 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.

[5] Johannes Herrmann, Sergi Masot Llima, Ants Remm, Petr Zapletal, Nathan A. McMahon, Colin Scarato, François Swiadek, Christian Kraglund Andersen, Christoph Hellings, Sebastian Krinner, Nathan Lacroix, Stefania Lazar, Michael Kerschbaum, Dante Colao Zanuz, Graham J. Norris, Michael J. Hartmann, Andreas Wallraff, and Christopher Eichler, “Realizing quantum convolutional neural networks on a superconducting quantum processor to recognize quantum phases”, Nature Communications 13, 4144 (2022).

[6] Andrew Arrasmith, Zoë Holmes, M. Cerezo, and Patrick J. Coles, “Equivalence of quantum barren plateaus to cost concentration and narrow gorges”, Quantum Science and Technology 7 4, 045015 (2022).

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

[8] Tobias Haug, Kishor Bharti, and M. S. Kim, “Capacity and Quantum Geometry of Parametrized Quantum Circuits”, PRX Quantum 2 4, 040309 (2021).

[9] Dmitry A. Fedorov, Bo Peng, Niranjan Govind, and Yuri Alexeev, “VQE method: a short survey and recent developments”, Materials Theory 6 1, 2 (2022).

[10] M. Bilkis, M. Cerezo, Guillaume Verdon, Patrick J. Coles, and Lukasz Cincio, “A semi-agnostic ansatz with variable structure for quantum machine learning”, arXiv:2103.06712.

[11] Enrico Fontana, Nathan Fitzpatrick, David Muñoz Ramo, Ross Duncan, and Ivan Rungger, “Evaluating the noise resilience of variational quantum algorithms”, Physical Review A 104 2, 022403 (2021).

[12] Tobias Stollenwerk and Stuart Hadfield, “Diagrammatic Analysis for Parameterized Quantum Circuits”, arXiv:2204.01307.

[13] Kosuke Ito, Wataru Mizukami, and Keisuke Fujii, “Universal noise-precision relations in variational quantum algorithms”, arXiv:2106.03390.

[14] 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.

[15] Joonho Kim and Yaron Oz, “Quantum Energy Landscape and VQA Optimization”, arXiv:2107.10166.

[16] Kun Wang, Zhixin Song, Xuanqiang Zhao, Zihe Wang, and Xin Wang, “Detecting and quantifying entanglement on near-term quantum devices”, npj Quantum Information 8, 52 (2022).

The above citations are from SAO/NASA ADS (last updated successfully 2022-09-16 22:19:27). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2022-09-16 22:19:25).

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  • Source: https://quantum-journal.org/papers/q-2022-09-15-804/
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