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The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate

Authors: Lennart Bittel, Sevag Gharibian, and Martin Kliesch

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Variational Quantum Algorithms (VQAs), such as the Quantum Approximate Optimization Algorithm (QAOA) of [Farhi, Goldstone, Gutmann, 2014], have seen intense study towards near-term applications on quantum hardware. A crucial parameter for VQAs is the depth of the variational ansatz used - the smaller the depth, the more amenable the ansatz is to near-term quantum hardware in that it gives the circuit a chance to be fully executed before the system decoheres. In this work, we show that approximating the optimal depth for a given VQA ansatz is intractable. Formally, we show that for any constant ε > 0, it is QCMA-hard to approximate the optimal depth of a VQA ansatz within multiplicative factor N^(1-ε), for N denoting the encoding size of the VQA instance. (Here, Quantum Classical Merlin-Arthur (QCMA) is a quantum generalization of NP.) We then show that this hardness persists in the even "simpler" QAOA-type settings. To our knowledge, this yields the first natural QCMA-hard-to-approximate problems.

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Lennart Bittel, Sevag Gharibian, and Martin Kliesch. The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 34:1-34:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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  author =	{Bittel, Lennart and Gharibian, Sevag and Kliesch, Martin},
  title =	{{The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{34:1--34:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-183045},
  doi =		{10.4230/LIPIcs.CCC.2023.34},
  annote =	{Keywords: Variational quantum algorithms (VQA), Quantum Approximate Optimization Algorithm (QAOA), circuit depth minimization, Quantum-Classical Merlin-Arthur (QCMA), hardness of approximation, hybrid quantum algorithms}
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