Hamiltonian: The Electronic Structure Hamiltonian $$ H = \displaystyle\sum_{i,j \in [n]}\displaystyle\sum_{\sigma} t_{ij} c_{i\sigma}^\dagger c_{j\sigma} + \frac{1}{2} \displaystyle\sum_{i,j,k,l \in [n]}\displaystyle\sum_{\sigma,\tau} u_{ijkl}c_{i\sigma}^\dagger c_{j\tau}^\dagger c_{k\sigma} c_{l\sigma} $$

Problem: Ground state energy

Complexity: QMA–complete

Ref: [OIWF21]

Conditionals:

• Fixed–basis set
• Fixed particle number, $n$

Reductions:

• From the Fermi–Hubbard Hamiltonian

Hamiltonian: The Electronic Structure Hamiltonian $$ H = \displaystyle\sum_{i,j \in [n]}\displaystyle\sum_{\sigma} t_{ij} c_{i\sigma}^\dagger c_{j\sigma} + \frac{1}{2} \displaystyle\sum_{i,j,k,l \in [n]}\displaystyle\sum_{\sigma,\tau} u_{ijkl}c_{i\sigma}^\dagger c_{j\tau}^\dagger c_{k\sigma} c_{l\sigma} $$

Problem: Lowest–energy Slater Determinant

Complexity: NP–complete

Ref: [OIWF21]

Conditionals:

• Fixed–basis set
• Fixed particle number, $n$
• Inverse polynomial precision

Reductions:

• To a diagonal Hamiltonian