发现量子变分算法达到精确基态的必要条件，揭示部分问题可经典模拟。
Reachability Constraints in Variational Quantum Circuits: Optimization within Polynomial Group Module
- 通过群模投影范数匹配，给出达到精确基态的必要条件。
- 证明某些问题可在O(n⁵)时间内用经典方法求解，如最大割问题。
- 适用于研究量子变分算法极限及可经典模拟的量子问题场景。
本文识别出任何变分量子方法达到精确基态的必要条件：输入态与基态在每个群模上的投影范数必须一致，这意味着要达到精确基态，解态的模权重必须预先已知。以匹配门电路为例，当问题解为经典比特串时，所有计算基态具有相同的模权重。结合可观测量位于小线性子空间时量子电路可经典模拟的已知结果，表明某些问题存在经典替代方案，每步耗时O(n⁵)。最大割问题为此类情形的典型例证。
原文摘要 · Abstract (English)
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.
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