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,人们发现,所谓的引导当地汉密尔顿问题 -- -- 估计一个美元-当地汉密尔顿人最小的egen值,如果提供量子状态的描述,保证与真正的地面状态有重大重叠 -- -- 当所需要的精确度在系统大小为美元时是反向的多元体积时,以美元=Geq 6美元完成BQP,即使指导国与地面国家的重叠接近恒定的美元(left) (\frac{1unhp{poly}-\Omega\left(fleft) (\frac{1unhop{poly}}{(n){right}\right)\right)$。我们从三个方面改进了这一结果:显示,当汉密尔密尔顿是2个地方时,它仍然为BQP-f-form。 (二) 指导国与目标天国之间的重叠程度相当于1美元\mega\left(mac{l){lack}(n_right)$),当一个人对估计动力状态感兴趣时,而不是地表可能。