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

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. 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 $\epsilon>0$, it is QCMA-hard to approximate the optimal depth of a VQA ansatz within multiplicative factor $N^{1-\epsilon}$, 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].