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.

DOI: 10.4230/lipics.ccc.2023.34

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2^n

-dimensional Hilbert space. We define a discretized quantum exhaustive search that works well for small size problems. We provide an example of an efficient partial discretized quantum exhaustive search for larger-size problems, in order to extend classical tools to the quantum computing domain, for near future and far future goals. Our method enables obtaining intuition on NP-complete and NP-hard problems as well as on Quantum Merlin Arthur (QMA)-complete and QMA-hard problems. We demonstrate our ideas in many simple cases, providing the energy landscape for various problems and presenting two types of energy curves via VQAs.

\mathsf{NP}

-hard, limiting the prospect of general guarantees for efficient training. In this Letter, we prove that even the weaker task of finding a local minimum of such VQA training objectives is strongly

\mathsf{NP}

-hard, including when the objective admits efficient classical evaluation. Moreover, we show that this hardness persists even for the task of finding a parameter vector within

\ell_p

-distance strictly less than

π/2

of some local minimizer, for every

p\geq 1

. Our central technical result is that approximating a local minimizer of a Hermitian trigonometric polynomial is strongly

\mathsf{NP}

-hard. By explicitly constructing quantum circuits whose training objectives reproduce these hard instances, we obtain a polynomial-time reduction to VQA training. Our results establish a fundamental computational barrier to variational quantum training: even reaching the vicinity of a local minimum remains hard in the worst case.