Recently it was shown that the so-called guided local Hamiltonian problem -- estimating the smallest eigenvalue of a $k$-local Hamiltonian when provided with a description of a quantum state ('guiding state') that is guaranteed to have substantial overlap with the true groundstate -- is BQP-complete for $k \geq 6$ when the required precision is inverse polynomial in the system size $n$, and remains hard even when the overlap of the guiding state with the groundstate is close to a constant $\left(\frac12 - \Omega\left(\frac{1}{\mathop{poly}(n)}\right)\right)$. We improve upon this result in three ways: by showing that it remains BQP-complete when i) the Hamiltonian is 2-local, ii) the overlap between the guiding state and target eigenstate is as large as $1 - \Omega\left(\frac{1}{\mathop{poly}(n)}\right)$, and iii) when one is interested in estimating energies of excited states, rather than just the groundstate. Interestingly, iii) is only made possible by first showing that ii) holds.

翻译：近期研究表明，所谓的引导式局部哈密顿问题——即在已知一个与真实基态具有显著重叠的量子态（"引导态"）描述时，估算$k$-局部哈密顿量最小本征值的问题——在所需精度关于系统规模$n$呈逆多项式时，对$k \geq 6$情形属于BQP完全问题，且即使引导态与基态的重叠接近常数$\left(\frac12 - \Omega\left(\frac{1}{\mathop{poly}(n)}\right)\right)$时仍保持困难性。我们从三个方面改进了这一结果：通过证明该问题在以下条件下仍然保持BQP完全性——i）哈密顿量为2-局部，ii）引导态与目标本征态的重叠可达$1 - \Omega\left(\frac{1}{\mathop{poly}(n)}\right)$，以及iii）关注点从基态能量估算扩展到激发态能量估算。值得注意的是，条件iii）的成立必须以条件ii）的证明为前提。