Electronic Structure in a Fixed Basis Is QMA-Complete

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Determines that ground-state energy in a fixed single-particle basis with fixed particle number and inverse-polynomial precision is QMA-complete, and that optimizing the lowest-energy Slater determinant is NP-complete. Reduction chain: encode antiferromagnetic Heisenberg → Fermi-Hubbard using Jordan-Wigner and perturbation theory, then build Gaussian-based composite orbitals so the electronic Hamiltonian approximates the Hubbard model. Parameters (Gaussian exponents, inter-orbital distances, on-site penalty) are chosen so approximation errors are bounded; construction omits nuclear point potentials so cusps are not captured.

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1. 0:00Intro
2. 1:19k-Local Hamiltonian (on qudits)
3. 6:49Electronic structure Hamiltonian in a fixed basis
4. 7:44Proof overview
5. 8:30Antiferromagnetic Heisenberg model
6. 10:32Jordan Wigner
7. 12:53Hubbard to Heisenberg recap
8. 13:23Orbitals
9. 18:17Product states and Slater determinants
10. 19:53Proof of NP-hardness of Slater-determinant
11. 20:50Outlook
12. 24:50Carving out a Hubbard model