Finding the ground state energy of electrons subject to an external electric field is a fundamental problem in computational chemistry. We prove that this electronic-structure problem, when restricted to a fixed single-particle basis and fixed number of electrons, is QMA-complete. Schuch and Verstraete have shown hardness for the electronic-structure problem with an additional site-specific external magnetic field, but without the restriction to a fixed basis. In their reduction, a local Hamiltonian on qubits is encoded in the site-specific magnetic field. In our reduction, the local Hamiltonian is encoded in the choice of spatial orbitals used to discretize the electronic-structure Hamiltonian. As a step in their proof, Schuch and Verstraete show a reduction from the antiferromagnetic Heisenberg Hamiltonian to the Fermi-Hubbard Hamiltonian. We combine this reduction with the fact that the antiferromagnetic Heisenberg Hamiltonian is QMA-hard to observe that the Fermi-Hubbard Hamiltonian on generic graphs is QMA-hard, even when all the hopping coefficients have the same sign. We then reduce from Fermi-Hubbard by showing that an instance of Fermi-Hubbard can be closely approximated by an instance of the Electronic-Structure Hamiltonian in a fixed basis. Finally, we show that estimating the energy of the lowest-energy Slater-determinant state (i.e., the Hartree-Fock state) is NP-complete for the Electronic-Structure Hamiltonian in a fixed basis.
翻译:寻找受外部电场影响的电子的地面状态能量是计算化学的一个根本问题。 我们证明, 当局限于固定的单粒基和固定电子数量时, 电子结构问题是QMA完成的。 Schuch 和 Verstraete 对电子结构问题表现出了硬性, 增加了一个特定地点的外部磁场, 但没有限制于固定的磁场。 在减少时, 一个本地的汉密尔顿人正在特定地点的磁场中编码。 在我们的减少中, 本地的汉密尔顿人正在选择用于将电子结构汉密尔顿人分解的空间轨道。 作为证据的一个步骤, Schuch 和 Verstraete 显示了从抗黄磁性海森堡到Fermi-Hubbard 汉密尔密尔密尔密尔顿的减硬性。 我们把这种减员与一个事实结合起来, 在特定地点的磁场域域域域域域域域中, Fermi- Hubbarian 在通用的平面图中选择了空间轨道, 也就是Sermi-hnial- 的精度, 的精度, 的精度的精度的精度在Sr-h-h-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r- 的精度的精度的精度的精度上显示一个固定的精度的精度的精度的精度的精度的精度的精度的精度, 的精度上, 的精度上, 的精度在S-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-r-l-l-l-l-l