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oai:drops-oai.dagstuhl.de:18304 2024-03-06T11:01:33Z ddc:004 open_access

The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate Bittel, Lennart Gharibian, Sevag Kliesch, Martin Variational quantum algorithms (VQA) Quantum Approximate Optimization Algorithm (QAOA) circuit depth minimization Quantum-Classical Merlin-Arthur (QCMA) hardness of approximation hybrid quantum algorithms 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Lennart Bittel and Sevag Gharibian and Martin Kliesch 2023 Is Part Of LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CCC.2023.34 urn:nbn:de:0030-drops-183045 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.34 eng https://creativecommons.org/licenses/by/4.0/legalcode