After nearly two decades of research, the question of a quantum PCP theorem for quantum Constraint Satisfaction Problems (CSPs) remains wide open. As a result, proving QMA-hardness of approximation for ground state energy estimation has remained elusive. Recently, it was shown [Bittel, Gharibian, Kliesch, CCC 2023] that a natural problem involving variational quantum circuits is QCMA-hard to approximate within ratio N^(1-eps) for any eps > 0 and N the input size. Unfortunately, this problem was not related to quantum CSPs, leaving the question of hardness of approximation for quantum CSPs open. In this work, we show that if instead of focusing on ground state energies, one considers computing properties of the ground space, QCMA-hardness of computing ground space properties can be shown. In particular, we show that it is (1) QCMA-complete within ratio N^(1-eps) to approximate the Ground State Connectivity problem (GSCON), and (2) QCMA-hard within the same ratio to estimate the amount of entanglement of a local Hamiltonian's ground state, denoted Ground State Entanglement (GSE). As a bonus, a simplification of our construction yields NP-completeness of approximation for a natural k-SAT reconfiguration problem, to be contrasted with the recent PCP-based PSPACE hardness of approximation results for a different definition of k-SAT reconfiguration [Karthik C.S. and Manurangsi, 2023, and Hirahara, Ohsaka, STOC 2024].

翻译：经过近二十年的研究，量子约束可满足性问题（CSP）的量子PCP定理问题仍然悬而未决。因此，证明基态能量估计的近似计算具有QMA难度这一目标始终未能实现。最近的研究[Bittel, Gharibian, Kliesch, CCC 2023]表明，涉及变分量子电路的自然问题在近似比为N^(1-eps)（其中eps > 0，N为输入规模）时是QCMA难解的。然而，该问题与量子CSP并无直接关联，使得量子CSP的近似计算复杂性仍为开放问题。本研究中，我们证明若将关注点从基态能量转移至基态空间的性质计算，则能够证明基态空间性质计算的QCMA难度。具体而言，我们证明：（1）在近似比为N^(1-eps)时，基态连通性问题（GSCON）的近似计算是QCMA完全的；（2）在相同近似比下，估计局部哈密顿量基态的纠缠量（记为基态纠缠问题GSE）是QCMA难解的。作为额外成果，我们通过简化构造得到一个自然k-SAT重构问题的近似计算NP完全性，这与近期基于PCP的、针对不同k-SAT重构定义所获得的PSPACE难度近似结果[Karthik C.S. and Manurangsi, 2023, 以及Hirahara, Ohsaka, STOC 2024]形成了鲜明对比。