License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2023.34
URN: urn:nbn:de:0030-drops-183045
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18304/
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Bittel, Lennart ; Gharibian, Sevag ; Kliesch, Martin

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

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Abstract

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.

BibTeX - Entry

@InProceedings{bittel_et_al:LIPIcs.CCC.2023.34,
  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 =		{https://drops.dagstuhl.de/opus/volltexte/2023/18304},
  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}
}

Keywords: Variational quantum algorithms (VQA), Quantum Approximate Optimization Algorithm (QAOA), circuit depth minimization, Quantum-Classical Merlin-Arthur (QCMA), hardness of approximation, hybrid quantum algorithms
Collection: 38th Computational Complexity Conference (CCC 2023)
Issue Date: 2023
Date of publication: 10.07.2023

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