{"status":"public","intvolume":"       264","year":"2023","author":[{"first_name":"Lennart","last_name":"Bittel","full_name":"Bittel, Lennart"},{"orcid":"0000-0002-9992-3379","first_name":"Sevag","full_name":"Gharibian, Sevag","last_name":"Gharibian","id":"71541"},{"last_name":"Kliesch","full_name":"Kliesch, Martin","first_name":"Martin"}],"_id":"34138","title":"The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate","department":[{"_id":"623"},{"_id":"7"}],"volume":264,"external_id":{"arxiv":["2211.12519"]},"user_id":"71541","citation":{"chicago":"Bittel, Lennart, Sevag Gharibian, and Martin Kliesch. “The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate.” In <i>Proceedings of the 38th Computational Complexity Conference (CCC)</i>, 264:34:1-34:24. Leibniz International Proceedings in Informatics (LIPIcs), 2023. <a href=\"https://doi.org/10.4230/LIPIcs.CCC.2023.34\">https://doi.org/10.4230/LIPIcs.CCC.2023.34</a>.","short":"L. Bittel, S. Gharibian, M. Kliesch, in: Proceedings of the 38th Computational Complexity Conference (CCC), 2023, p. 34:1-34:24.","bibtex":"@inproceedings{Bittel_Gharibian_Kliesch_2023, series={Leibniz International Proceedings in Informatics (LIPIcs)}, title={The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate}, volume={264}, DOI={<a href=\"https://doi.org/10.4230/LIPIcs.CCC.2023.34\">10.4230/LIPIcs.CCC.2023.34</a>}, number={34}, booktitle={Proceedings of the 38th Computational Complexity Conference (CCC)}, author={Bittel, Lennart and Gharibian, Sevag and Kliesch, Martin}, year={2023}, pages={34:1-34:24}, collection={Leibniz International Proceedings in Informatics (LIPIcs)} }","apa":"Bittel, L., Gharibian, S., &#38; Kliesch, M. (2023). The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate. <i>Proceedings of the 38th Computational Complexity Conference (CCC)</i>, <i>264</i>(34), 34:1-34:24. <a href=\"https://doi.org/10.4230/LIPIcs.CCC.2023.34\">https://doi.org/10.4230/LIPIcs.CCC.2023.34</a>","ieee":"L. Bittel, S. Gharibian, and M. Kliesch, “The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate,” in <i>Proceedings of the 38th Computational Complexity Conference (CCC)</i>, 2023, vol. 264, no. 34, p. 34:1-34:24, doi: <a href=\"https://doi.org/10.4230/LIPIcs.CCC.2023.34\">10.4230/LIPIcs.CCC.2023.34</a>.","mla":"Bittel, Lennart, et al. “The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate.” <i>Proceedings of the 38th Computational Complexity Conference (CCC)</i>, vol. 264, no. 34, 2023, p. 34:1-34:24, doi:<a href=\"https://doi.org/10.4230/LIPIcs.CCC.2023.34\">10.4230/LIPIcs.CCC.2023.34</a>.","ama":"Bittel L, Gharibian S, Kliesch M. The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate. In: <i>Proceedings of the 38th Computational Complexity Conference (CCC)</i>. Vol 264. Leibniz International Proceedings in Informatics (LIPIcs). ; 2023:34:1-34:24. doi:<a href=\"https://doi.org/10.4230/LIPIcs.CCC.2023.34\">10.4230/LIPIcs.CCC.2023.34</a>"},"language":[{"iso":"eng"}],"series_title":"Leibniz International Proceedings in Informatics (LIPIcs)","type":"conference","date_updated":"2023-07-10T14:33:00Z","publication":"Proceedings of the 38th Computational Complexity Conference (CCC)","abstract":[{"lang":"eng","text":"Variational Quantum Algorithms (VQAs), such as the Quantum Approximate\r\nOptimization Algorithm (QAOA) of [Farhi, Goldstone, Gutmann, 2014], have seen\r\nintense study towards near-term applications on quantum hardware. A crucial\r\nparameter for VQAs is the depth of the variational ansatz used - the smaller\r\nthe depth, the more amenable the ansatz is to near-term quantum hardware in\r\nthat it gives the circuit a chance to be fully executed before the system\r\ndecoheres. This potential for depth reduction has made VQAs a staple of Noisy\r\nIntermediate-Scale Quantum (NISQ)-era research.\r\n  In this work, we show that approximating the optimal depth for a given VQA\r\nansatz is intractable. Formally, we show that for any constant $\\epsilon>0$, it\r\nis QCMA-hard to approximate the optimal depth of a VQA ansatz within\r\nmultiplicative factor $N^{1-\\epsilon}$, for $N$ denoting the encoding size of\r\nthe VQA instance. (Here, Quantum Classical Merlin-Arthur (QCMA) is a quantum\r\ngeneralization of NP.) We then show that this hardness persists even in the\r\n\"simpler\" setting of QAOAs. To our knowledge, this yields the first natural\r\nQCMA-hard-to-approximate problems. To achieve these results, we bypass the need\r\nfor a PCP theorem for QCMA by appealing to the disperser-based NP-hardness of\r\napproximation construction of [Umans, FOCS 1999]."}],"date_created":"2022-11-24T08:07:56Z","publication_status":"published","issue":"34","doi":"10.4230/LIPIcs.CCC.2023.34","page":"34:1-34:24"}