This work identifies a necessary condition for any variational quantum approach to reach the exact ground state. Briefly, the norms of the projections of the input and the ground state onto each group module must match, implying that module weights of the solution state have to be known in advance in order to reach the exact ground state. An exemplary case is provided by matchgate circuits applied to problems whose solutions are classical bit strings, since all computational basis states share the same module-wise weights. Combined with the known classical simulability of quantum circuits for which observables lie in a small linear subspace, this implies that certain problems admit a classical surrogate for exact solution with each step taking $O(n^5)$ time. The Maximum Cut problem serves as an illustrative example.

翻译：本研究识别了任何变分量子方法达到精确基态的一个必要条件。简言之，输入态与基态在每个群模上的投影范数必须匹配，这意味着为达到精确基态，需预先知道解态的模权重。以匹配门电路应用于解为经典比特串的问题为例，所有计算基态共享相同的模权重。结合已知的量子电路经典可模拟性（其可观测量位于小线性子空间内），这暗示某些问题可在每步$O(n^5)$时间复杂度内通过经典替代求解精确解。最大割问题作为示例加以说明。