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：最近的研究表明，所谓引导式局部哈密顿问题——在给定一个与真实基态有显著重叠的量子态（"引导态"）描述的情况下，估计$k$-局部哈密顿量的最小本征值——当所需精度关于系统规模$n$呈多项式倒数时，对$k \geq 6$是BQP完全的，并且即使引导态与基态的重叠接近于常数$\left(\frac12 - \Omega\left(\frac{1}{\mathop{poly}(n)}\right)\right)$，该问题仍保持困难。我们在三个方面改进了这一结果：通过证明以下情况仍保持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$1 - \Omega\left(\frac{1}{\mathop{poly}(n)}\right)$；（iii）当关注的不是基态而是激发态能量估计时。有趣的是，(iii) 的实现仅当首先证明 (ii) 成立时才成为可能。