Intractability of Electronic Structure in a Fixed Basis

Finding the ground-state energy of electrons subject to an external electric field is a fundamental problem in computational chemistry. While the theory of QMA-completeness has been instrumental in understanding the complexity of finding ground states in many-body quantum systems, prior to this work it has been unknown whether or not the special form of the Hamiltonian for the electronic structure of molecules can be exploited to find ground states efficiently or whether the problem remains hard for this special case. We prove that the electronic-structure problem, when restricted to a fixed single-particle basis and a fixed number of electrons, is QMA-complete. In our proof, the local Hamiltonian is encoded in the choice of spatial orbitals used to discretize the electronic-structure Hamiltonian. In contrast, Schuch and Verstraete have proved hardness for the electronic-structure problem with an additional site-specific external magnetic field, but without the restriction to a fixed basis, by encoding a local Hamiltonian on qubits in the site-specific magnetic field. We also show that estimation of the energy of the lowest-energy Slater-determinant state (i.e., the Hartree-Fock state) is nondeterministic polynomial time (NP)-complete for the electronic-structure Hamiltonian in a fixed basis.