量子可满足性阈值的界
Bounds on the quantum satisfiability threshold
摘要：
量子k-SAT是决定是否存在垂直于一组向量的n量子比特状态的问题，其中每个向量位于k量子比特的希尔伯特空间中。等价地，问题是决定特定类型的局部哈密顿量是否具有能量为零的基态。我们考虑随机量子k-SAT公式与n个变量和m = αn条款，并询问在什么值的α这些公式不再是可满足的。我们证明了随机量子3-SAT的阈值至多为3.594。相比之下，统计物理学的令人信服的论据表明，经典的3-SAT阈值是α \approx 4.267。对于较大的k，我们表明，量子阈值是一个常数因子小于经典的。我们的边界工作，通过确定的通用秩的满足子空间的某些小工具，然后使用微分方程的技术来分析各种算法，分区超图到这些小工具的集合。我们使用微分方程建立上界的可满足性阈值似乎是新颖的，我们的技术可以适用于各种经典问题。
英文摘要：
Quantum k-SAT is the problem of deciding whether there is a n-qubit state which is perpendicular to a set of vectors, each of which lies in the Hilbert space of k qubits. Equivalently, the problem is to decide whether a particular type of local Hamiltonian has a ground state with zero energy. We consider random quantum k-SAT formulas with n variables and m = αn clauses, and ask at what value of αthese formulas cease to be satisfiable. We show that the threshold for random quantum 3-SAT is at most 3.594. For comparison, convincing arguments from statistical physics suggest that the classical 3-SAT threshold is α\approx 4.267. For larger k, we show that the quantum threshold is a constant factor smaller than the classical one. Our bounds work by determining the generic rank of the satisfying subspace for certain gadgets, and then using the technique of differential equations to analyze various algorithms that partition the hypergraph into a collection of these gadgets. Our use of differential equation to establish upper bounds on a satisfiability threshold appears to be novel, and our techniques may apply to various classical problems as well.
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