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

Authors Lennart Bittel , Sevag Gharibian , Martin Kliesch

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Author Details

Lennart Bittel
  • Institute for Theoretical Physics, Heinrich Heine University Düsseldorf, Germany
Sevag Gharibian
  • Department of Computer Science, and Institute for Photonic Quantum Systems, Universität Paderborn, Germany
Martin Kliesch
  • Institute for Theoretical Physics, Heinrich Heine University Düsseldorf, Germany
  • Institute for Quantum-Inspired and Quantum Optimization, Technische Universtiät Hamburg, Germany


We thank Ashley Montanaro for helpful discussions.

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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • 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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