Product states, unentangled tensor products of single qubits, are a ubiquitous ansatz in quantum computation, including for state-of-the-art Hamiltonian approximation algorithms. A natural question is whether we should expect to efficiently solve product state problems on any interesting families of Hamiltonians. We completely classify the complexity of finding minimum-energy product states for Hamiltonians defined by any fixed set of allowed 2-qubit interactions. Our results follow a line of work classifying the complexity of solving Hamiltonian problems and classical constraint satisfaction problems based on the allowed constraints. We prove that estimating the minimum energy of a product state is in P if and only if all allowed interactions are 1-local, and NP-complete otherwise. Equivalently, any family of non-trivial two-body interactions generates Hamiltonians with NP-complete product-state problems. Our hardness constructions only require coupling strengths of constant magnitude. A crucial component of our proofs is a collection of hardness results for a new variant of the Vector Max-Cut problem, which should be of independent interest. Our definition involves sums of distances rather than squared distances and allows linear stretches. A corollary of our classification is a new proof that optimizing product states in the Quantum Max-Cut model (the quantum Heisenberg model) is NP-complete.

翻译：乘积态（即单量子比特非纠缠张量积）是量子计算中普遍使用的拟设，包括最先进的哈密顿量近似算法。一个自然的问题是：我们是否应当期望在任何有趣的哈密顿量族上有效求解乘积态问题？我们完整分类了由任意固定允许的双量子比特相互作用定义的哈密顿量中寻找最小能量乘积态的复杂度。我们的结果沿袭了基于允许约束类别分类哈密顿量问题与经典约束满足问题复杂度的工作线索。我们证明：当且仅当所有允许的相互作用均为1-局域时，估计乘积态最小能量属于P类；否则为NP完全。等价地，任何非平凡的两体相互作用族都会生成具有NP完全乘积态问题的哈密顿量。我们的困难性构造仅需常数量级的耦合强度。证明的关键是一组针对向量最大割问题新变种的困难性结果，这本身应具有独立研究价值。我们的定义采用距离和（而非平方距离）并允许线性伸缩。该分类的一个推论是：优化量子最大割模型（量子海森堡模型）中的乘积态为NP完全的新证明。