2–Local Pauli Models

Key: 2LPH18
Hamiltonian: The ($[\![$xz$]\!]$*/x–z) Hamiltonian $$H=\displaystyle\sum_{\{i,j\}\in E(G)} Q_{ij} X_iZ_j + T_{ij} Z_iX_j + \displaystyle\sum_{i\in V(G)} f_i X_i + h_i Z_i$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [BL08]
Conditionals:
  • Real coefficients
  • $G$ is the interaction graph
Reductions:
  • From the 2–local real Hamiltonian
  • To the ($[\![$xz$]\!]$*/x–z) Hamiltonian on a 2D square lattice
Gadgets:
  • ZZXX and ZX
Key: 2LPH19
Hamiltonian: The ($[\![$xz$]\!]$*/x–z) Hamiltonian $$H=\displaystyle\sum_{\{i,j\}\in E(\Lambda)} Q_{ij} X_iZ_j + T_{ij} Z_iX_j + \displaystyle\sum_{i\in V(\Lambda)} f_i X_i + h_i Z_i$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [CM16]
Conditionals:
  • Real coefficients
  • $\Lambda$ representing a square lattice
Reductions:
  • From the ($[\![$xz$]\!]$*/x–z) Hamiltonian
Key: 2LPH20
Hamiltonian: The (x–z/x–z) Hamiltonian $$H=\displaystyle\sum_{\{i,j\}\in E(G) } J_{ij} X_iX_j + L_{ij} Z_iZ_j + \displaystyle\sum_{i\in V(G)} f_i X_i + h_i Z_i$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [BL08]
Conditionals:
  • Real coefficients
  • $G$ is the interaction graph
Reductions:
  • From the 2–local real Hamiltonian
  • To the (x–z/x–z) Hamiltonian on a 2D square lattice
Gadgets:
  • ZZXX
Key: 2LPH20
Hamiltonian: The (x–y/x–z) Hamiltonian $$H=\displaystyle\sum_{\{i,j\}\in E(G) } J_{ij} X_iX_j + K_{ij} Y_iY_j + \displaystyle\sum_{i\in V(G)} f_i X_i + h_i Z_i$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [CBBK15]
Conditionals:
  • Real coefficients
  • $G$ is the interaction graph
Reductions:
  • From the (x–z/x–z) Hamiltonian
Gadgets:
  • $YY$ gadget
Key: 2LPH21
Hamiltonian: The (x–z/x–z) Hamiltonian $$H=\displaystyle\sum_{\{i,j\}\in E(\Lambda)} J_{ij} X_iX_j + L_{ij} Z_iZ_j + \displaystyle\sum_{i\in V(\Lambda)} f_i X_i + h_i Z_i$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [CM16]
Conditionals:
  • Real coefficients
  • $\Lambda$ representing a square lattice
Reductions:
  • From the (x–z/x–z) Hamiltonian
Key: 2LPH100
Hamiltonian: The weighted (xyz/.) Hamiltonian $$H=\displaystyle\sum_{\{i,j\}\in E(G)} J_{ij}\big(\alpha X_iX_j + \beta Y_iY_j + \gamma Z_iZ_j\big)$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [PM15], [CM16]
Conditionals:
  • $(\alpha+\beta,\alpha+\gamma,\beta+\gamma)\gt0$
  • $G$ is the interaction graph
Reductions:
  • From the (x–y–z/x–y–z) Hamiltonian
  • To the weighted (xyz/.) Hamiltonian on a 2D square lattice
Key: 2LPH101
Hamiltonian: The (x–y–z/x–y–z) Hamiltonian $$H=\displaystyle\sum_{i,j\in\Lambda} J_{ij}X_iX_j + K_{ij}Y_iY_j + L_{ij}Z_iZ_j + \displaystyle\sum_{i\in\Lambda} f_iX_i + g_iY_i + h_iZ_i$$
Problem: Ground state energy
Complexity: QMA–hard
Ref: [Wai23]
Conditionals:
  • Coefficients $\leq \mathsf{poly}(n)$
  • $G$ is the interaction graph
Reductions:
  • From the Parent Pauli Hamiltonian
  • To the weighted (xyz/.) Hamiltonian
Key: 2LPH102
Hamiltonian: The weighted (xyz/.) Hamiltonian $$H=\displaystyle\sum_{\{i,j\}\in E(\Lambda)} J_{ij}\big(\alpha X_iX_j + \beta Y_iY_j + \gamma Z_iZ_j\big)$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [PM15],[CM16]
Conditionals:
  • $(\alpha+\beta,\alpha+\gamma,\beta+\gamma)\gt0$
  • $\Lambda$ representing a 2D square lattice
Reductions:
  • From the weighted (xyz/.) Hamiltonian
  • To the weighted (xyz/.) Hamiltonian on a 2D triangular lattice, the IaF–weighted (xyz/.) Hamiltonian on a 2D square lattice
Key: 2LPH112
Hamiltonian: The IaF–weighted (xyz/.) Hamiltonian $$H=\displaystyle\sum_{\{i,j\}\in E(G)} J_{ij}^{^{(\geq 0)}}\big(\alpha X_iX_j + \beta Y_iY_j + \gamma Z_iZ_j\big)$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [PM15]
Conditionals:
  • $J_{ij} \geq 0$
  • $(\alpha+\beta,\alpha+\gamma,\beta+\gamma)\gt0$
  • $\Lambda$ representing an interaction graph
Reductions:
  • From the weighted (xyz/.) Hamiltonian
  • To the IaF–weighted (xyz/.) Hamiltonian on a 2D triangular lattice
Key: 2LPH113
Hamiltonian: The IaF–weighted (xyz/.) Hamiltonian $$H=\displaystyle\sum_{\{i,j\}\in E(\Lambda)} J_{ij}^{^{(\geq 0)}}\big(\alpha X_iX_j + \beta Y_iY_j + \gamma Z_iZ_j\big)$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [PM15]
Conditionals:
  • $J_{ij} \geq 0$
  • $(\alpha+\beta,\alpha+\gamma,\beta+\gamma)\gt0$
  • $\Lambda$ representing a 2D triangular lattice
Reductions:
  • From the IaF–weighted (xyz/.) Hamiltonian
  • To the IaF–Heisenberg and IaF–(xy/.) Hamiltonians on a 2D triangular lattice
Key: 2LPH114
Hamiltonian: The IaF–weighted (xyz/.) Hamiltonian $$H=\displaystyle\sum_{\{i,j\}\in E(G)} J_{ij}^{^{(\geq 0)}}\big(\alpha X_iX_j + \beta Y_iY_j + \gamma Z_iZ_j\big)$$
Problem: Ground state energy
Complexity: StoqMA–complete
Ref: [PM15]
Conditionals:
  • $J_{ij} \geq 0$
  • $\alpha = -\beta \neq 0$, $(\alpha+\gamma,\beta+\gamma)\gt0$
  • $\Lambda$ representing an interaction graph
Reductions:
  • From the IaF–weighted (xyz/.) Hamiltonian
Key: 2LPH115
Hamiltonian: The IaF–weighted (xyz/.) Hamiltonian $$H=\displaystyle\sum_{\{i,j\}\in E(G)} J_{ij}^{^{(\geq 0)}}\big(\alpha X_iX_j + \alpha Y_iY_j - (|\alpha| + \epsilon) Z_iZ_j\big)$$
Problem: Ground state energy
Complexity: P
Ref: [PM15]
Conditionals:
  • $J_{ij} \geq 0$
  • $\epsilon \geq 0$
  • $\Lambda$ representing an interaction graph
Reductions:
  • From the IaF–weighted (xyz/.) Hamiltonian
  • To the ferromagnetic Quantum Ising Model and the ferromagnetic Heisenberg Model
Key: 2LPH301
Hamiltonian: The (xy/z) Hamiltonian $$H=\frac{1}{2}\displaystyle\sum_{\{i,j\}\in E(G) \atop i\neq j} X_iX_j + Y_iY_j - \frac{1}{2}\displaystyle\sum_{\{i,j\}\in E(G)\atop i=j} Z_i$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [CGW14]
Conditionals:
  • $G$ is a graph
  • $A(G)$ is $G$'s square $0$–$1$ symmetric adjacency matrix
Reductions:
  • From the IaF–weighted (xyz/.) Hamiltonian
  • To the IaF–Heisenberg and IaF–(xy/.) Hamiltonians on a 2D triangular lattice
Key: 2LPH501
Hamiltonian: The (xy/.) Hamiltonian $$H=\displaystyle\sum_{\{i,j\} \in E(G)} J_{ij} \big(X_iX_j + Y_iY_j \big)$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [CM16]
Conditionals:
  • $G$ is an interaction graph
Key: 2LPH502
Hamiltonian: The IaF–(xy/.) Hamiltonian $$H=\displaystyle\sum_{\{i,j\} \in E(G)} J_{ij}^{(\geq 0)} \big(X_iX_j + Y_iY_j \big)$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [PM15]
Conditionals:
  • $\Lambda$ representing a $2$D triangular lattice
Reductions:
  • From the (xy/.) Hamiltonian
Key: PFP-2LP-001
Hamiltonian: The (x-y/z) Hamiltonian $$H = \sum_{\{i,j\}\in E(G)} J_{ij}X_iX_j + K_{ij}Y_i Y_j + \sum_{i \in V(G)} h_i Z_i$$
Problem: Partition function
Complexity: FPTAS
Ref: [BG17]
Conditionals:
  • $–J_{ij} \in [0,1]$, $|K_{ij}| \in [0,1]$ and $|h_i| \in [0,1]$
  • $G$ is a graph
  • $\beta \in \mathbb{R}^+$