反铁磁相互作用和二维晶格的复杂性
The complexity of antiferromagnetic interactions and 2D lattices
摘要：
量子哈密顿量的最小本征值的估计可以形式化为局部哈密顿量问题。我们研究了局部哈密顿问题的自然特殊情况，其中相同的2-局部相互作用，具有不同的权重，应用于每对量子位。首先，我们考虑反铁磁/铁磁相互作用，在哈密顿量中的项的权重被限制为所有是相同的符号。我们表明，对称的2-本地相互作用，没有1-本地部分，问题是QMA完全或StoqMA。特别是反铁磁海森堡和反铁磁XY相互作用被证明是QMA完成。我们还证明了反铁磁横场伊辛模型的StoqMA完备性。其次，我们研究了在相互作用项只能选择在特定图上的限制下的局部哈密顿问题。我们证明，几乎所有的QMA-完全2-本地相互作用时，限制到一个二维正方形晶格仍然QMA-完全。最后，我们考虑这两个限制在同一时间，并发现，除了反铁磁海森堡相互作用，所有的相互作用，这是QMA-完全与正系数保持QMA-完全时，限制到一个二维三角形晶格。
英文摘要：
Estimation of the minimum eigenvalue of a quantum Hamiltonian can be formalised as the Local Hamiltonian problem. We study the natural special case of the Local Hamiltonian problem where the same 2-local interaction, with differing weights, is applied across each pair of qubits. First we consider antiferromagnetic/ferromagnetic interactions, where the weights of the terms in the Hamiltonian are restricted to all be of the same sign. We show that for symmetric 2-local interactions with no 1-local part, the problem is either QMA-complete or in StoqMA. In particular the antiferromagnetic Heisenberg and antiferromagnetic XY interactions are shown to be QMA-complete. We also prove StoqMA-completeness of the antiferromagnetic transverse field Ising model. Second, we study the Local Hamiltonian problem under the restriction that the interaction terms can only be chosen to lie on a particular graph. We prove that nearly all of the QMA-complete 2-local interactions remain QMA-complete when restricted to a 2D square lattice. Finally we consider both restrictions at the same time and discover that, with the exception of the antiferromagnetic Heisenberg interaction, all of the interactions which are QMA-complete with positive coefficients remain QMA-complete when restricted to a 2D triangular lattice.
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