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比。我们需要确定由$M$个随机$k$-局域秩一投影子之和构成的$N$量子比特哈密顿量的零能量基态。已知当且仅当底层因子图存在子句覆盖二聚体构型时，零能量的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。我们通过复分析与代数方法相结合的方式重新审视并完整证明了这一结果，其中代数方法基于布赫贝格尔算法处理具有随机系数的复多项式方程组。我们还讨论了数值实验，这些实验从乘积态不能张成整个零能量基态空间的意义上，探究了PRODSAT相中纠缠态的存在性。