The $k$-QSAT problem is a quantum analog of the famous $k$-SAT constraint satisfaction problem. We must determine the zero energy ground states of a Hamiltonian of $N$ qubits consisting of a sum of $M$ random $k$-local rank-one projectors. It is known that product states of zero energy exist with high probability if and only if the underlying factor graph has a clause-covering dimer configuration. This means that the threshold of the PRODSAT phase is a purely geometric quantity equal to the dimer covering threshold. We revisit and fully prove this result through a combination of complex analysis and algebraic methods based on Buchberger's algorithm for complex polynomial equations with random coefficients. We also discuss numerical experiments investigating the presence of entanglement in the PRODSAT phase in the sense that product states do not span the whole zero energy ground state space.

翻译：$k$-QSAT问题是著名$k$-SAT约束满足问题的量子类似物。我们需要确定由$N$个量子比特构成的哈密顿量的零能基态，该哈密顿量由$M$个随机$k-局域秩一投影算子的和组成。已知当且仅当底层因子图具有子句覆盖的二聚体构型时，零能乘积态以高概率存在。这意味着PRODSAT相的阈值是一个纯几何量，等于二聚体覆盖阈值。我们通过复分析与基于布赫伯格算法的代数方法（针对随机系数的复多项式方程）的结合，重新审视并完整证明了这一结果。我们还讨论了数值实验，探讨PRODSAT相中纠缠的存在性——即乘积态并不张成整个零能基态空间。