Bittel, LennartLennartBittelGharibian, SevagSevagGharibianKliesch, MartinMartinKliesch2023-01-042023-01-042023-07-1038th Computational Complexity Conference (CCC 2023)978-3-95977-282-2http://hdl.handle.net/11420/14490Variational 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. This potential for depth reduction has made VQAs a staple of Noisy Intermediate-Scale Quantum (NISQ)-era research. 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 N¹⁻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 even in the "simpler" setting of QAOAs. To our knowledge, this yields the first natural QCMA-hard-to-approximate problems. To achieve these results, we bypass the need for a PCP theorem for QCMA by appealing to the disperser-based NP-hardness of approximation construction of [Umans, FOCS 1999].enhttps://creativecommons.org/licenses/by/4.0/Quantum PhysicsQuantum PhysicsComputer Science - Computational ComplexityComputer SciencesConference Paperhttps://doi.org/10.15480/882.1394610.4230/LIPIcs.CCC.2023.3410.15480/882.139462211.12519v1Conference Paper