Hamiltonian: The Fermi–Hubbard Model $$H = \displaystyle\sum_{\{i,j\}\in E(\Lambda)}\displaystyle\sum_{\sigma}t_{ij} c_{i\sigma}^\dagger c_{j\sigma} + U\displaystyle\sum_{i\in V(\Lambda)}n_{i\uparrow}n_{i\downarrow} + \displaystyle\sum_{i\in V(\Lambda)} \boldsymbol{B}_i \cdot \boldsymbol{S}_i $$

Problem: Ground state energy

Complexity: QMA–complete

Ref: [SV09]

Conditionals:

• $U\gg t$
• $U/t$ grows polynomially in system size
• Half filling
• Nearest neighbour interactions
• $\Lambda$ is a 2D square lattice
• External fields of the form: $-\displaystyle\sum_{i\in\Lambda} \boldsymbol{B}_i \cdot \boldsymbol{S}_i$

Reductions:

• From HaF–Heisenberg model on a 2D square lattice

Hamiltonian: The Fermi–Hubbard Model $$H = \displaystyle\sum_{\{i,j\}\in E(G)}\displaystyle\sum_{\sigma}t_{ij} c_{i\sigma}^\dagger c_{j\sigma} + U\displaystyle\sum_{i\in V(G)}n_{i\uparrow}n_{i\downarrow}$$

Problem: Ground state energy

Complexity: QMA–complete

Ref: [OIWF21]

Conditionals:

• $|t| \leq \sqrt{\mathsf{poly}(n) U}$
• $U/t$ grows polynomially in system size
• Half filling
• $\Lambda$ represents an interaction graph

Reductions:

• From HaF–Heisenberg model on a weighted interaction graph (IaF–Heisenberg Hamiltonian)
• To the Electronic Structure Hamiltonian