We construct classical algorithms computing an approximation of the ground state energy of an arbitrary $k$-local Hamiltonian acting on $n$ qubits. We first consider the setting where a good ``guiding state'' is available, which is the main setting where quantum algorithms are expected to achieve an exponential speedup over classical methods. We show that a constant approximation (i.e., an approximation with constant relative accuracy) of the ground state energy can be computed classically in $\mathrm{poly}\left(1/\chi,n\right)$ time and $\mathrm{poly}(n)$ space, where $\chi$ denotes the overlap between the guiding state and the ground state (as in prior works in dequantization, we assume sample-and-query access to the guiding state). This gives a significant improvement over the recent classical algorithm by Gharibian and Le Gall (SICOMP 2023), and matches (up a to polynomial overhead) both the time and space complexities of quantum algorithms for constant approximation of the ground state energy. We also obtain classical algorithms for higher-precision approximation. For the setting where no guided state is given (i.e., the standard version of the local Hamiltonian problem), we obtain a classical algorithm computing a constant approximation of the ground state energy in $2^{O(n)}$ time and $\mathrm{poly}(n)$ space. To our knowledge, before this work it was unknown how to classically achieve these bounds simultaneously, even for constant approximation. We also discuss complexity-theoretic aspects of our results and their implications for the quantum PCP conjecture.

翻译：我们构建了经典算法，用于计算作用于n个量子比特的任意k-局域哈密顿量基态能量的近似值。我们首先考虑存在良好"引导态"的情形，这是量子算法预期能实现相对于经典方法指数级加速的主要场景。我们证明，基态能量的常数逼近（即具有常数相对精度的近似）可以在$\mathrm{poly}\left(1/\chi,n\right)$时间和$\mathrm{poly}(n)$空间内通过经典计算获得，其中$\chi$表示引导态与基态的重叠度（与先前去量子化研究相同，我们假设对引导态具有采样查询访问权限）。这相较于Gharibian和Le Gall（SICOMP 2023）近期提出的经典算法有显著改进，并且在时间与空间复杂度上（至多多项式开销）与量子算法对基态能量常数逼近的结果相匹配。我们还获得了更高精度逼近的经典算法。对于未给定引导态的情形（即局域哈密顿量问题的标准版本），我们获得了在$2^{O(n)}$时间和$\mathrm{poly}(n)$空间内计算基态能量常数逼近的经典算法。据我们所知，在本工作之前，即使对于常数逼近，如何通过经典方法同时实现这些界限仍是未知的。我们还讨论了研究结果的复杂性理论意义及其对量子PCP猜想的启示。