General Local Models

Key: 2LH901
Hamiltonian: The 2–local Hamiltonian $$H = \displaystyle\sum_{e\in E(G)} H_e + \displaystyle\sum_{v\in V(G)}H_v$$
Problem: Extremal product state
Complexity: NP–complete
Ref: [KPT+24]
Conditionals:
  • 2–local interactions
  • $G$ representing an interaction graph
  • Constant magnitude interaction strengths
Reductions:
  • From Max–Cut$^L_W(G)$
Key: 2LH04
Hamiltonian: The 2–local Hamiltonian $$H = \displaystyle\sum_{e\in E(G)} H_e + \displaystyle\sum_{v\in V(G)}H_v$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [OT08]
Conditionals:
  • 2–local interactions
  • $G$ representing a spatially sparse graph
Reductions:
  • From 5–local Hamiltonian on a spatially sparse graph
  • To 2–local Hamiltonian on planar graph
Gadgets:
  • 3–to–2 local
Key: 2LH05
Hamiltonian: The 2–local Hamiltonian $$H = \displaystyle\sum_{e\in E(G)} H_e + \displaystyle\sum_{v\in V(G)}H_v$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [OT08]
Conditionals:
  • 2–local interactions
  • $G$ represents a planar graph
  • Pauli degree $\leq 3$
Reductions:
  • From 2–local Hamiltonian on a spatially sparse graph
  • To 2–local Hamiltonian on 2D square lattice
Gadgets:
  • Subdivision, Fork, Cross
Key: 2LH06
Hamiltonian: The 2–local Hamiltonian $$H = \displaystyle\sum_{e\in E(\Lambda)} H_e + \displaystyle\sum_{v\in V(\Lambda)}H_v$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [OT08]
Conditionals:
  • 2–local interactions
  • $\Lambda$ represents a 2D square lattice
  • Pauli degree $\leq 3$
Reductions:
  • From 2–local Hamiltonian on planar graph via embedding on a square lattice of sufficient granularity
  • To Heisenberg Hamiltonian with external field on spatially sparse graph
Key: 2LH07
Hamiltonian: The 2–local Real Hamiltonian $$H = \displaystyle\sum_{j=1}^m H_j$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [BL08]
Conditionals:
  • 2–local interactions
  • Real coefficients and gates
Reductions:
  • From 2–local Hamiltonian
  • To an instance of the (x–z/x–z)–Hamiltonian on some interaction graph and an instance of the ($[\![$xz$]\!]$*/x–z)–Hamiltonian on some interaction graph
Key: 2LH15
Hamiltonian: The 2–local Hamiltonian $$H = \displaystyle\sum_{e\in E(G)} H_e + \displaystyle\sum_{v\in V(G)}H_v$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [KKR05]
Conditionals:
  • 2–local interactions
  • $G$ representing an interaction graph
Reductions:
  • From the 3–local Hamiltonian
  • To the 2–local Hamiltonian on a spatially sparse lattice and the 2–local real Hamiltonians
Gadgets:
  • 3–to–2 local
Key: 3LH13
Hamiltonian: The 3–local Hamiltonian $$H = \displaystyle\sum_{j=1}^m H_j$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [KR03]
Conditionals:
  • $m = O\big(\mathsf{poly}(n)\big)$
  • Each $H_j$ acts on at most 3 of the $n$ qubits
Reductions:
  • From the 5–local Hamiltonian
  • To the 3–local Hamiltonian on a spatially sparse lattice and the 3–local real Hamiltonian
Key: 3LH14
Hamiltonian: The 3–local Hamiltonian $$H = \displaystyle\sum_{j=1}^m H_j$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [W23]
Conditionals:
  • 3–local interactions
  • A spatially sparse interaction graph
Reductions:
  • From the 3–local Hamiltonian or the 5–local Hamiltonian on a spatially sparse graph
  • To the 2–local Hamiltonian on a spatially sparse lattice
Key: kLH10
Hamiltonian: The 5–local Hamiltonian $$H = \displaystyle\sum_{j=1}^m H_j$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [OT08]
Conditionals:
  • 5–local interactions
  • A spatially sparse interaction graph
Reductions:
  • From 5–local Hamiltonian on a generic graph
  • To 2–local Hamiltonian on a spatially sparse graph
Key: kLH11
Hamiltonian: The $O(\log n)$–local Hamiltonian $$H = \displaystyle\sum_{j=1}^m H_j$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [KSV02]
Conditionals:
  • $m = O\big(\mathsf{poly}(n)\big)$
  • Each $H_j$ acts on at most $O(\log n)$ of the $n$ qubits
Reductions:
  • To the 5–local Hamiltonian
Key: kLH12
Hamiltonian: The 5–local Hamiltonian $$H = \displaystyle\sum_{j=1}^m H_j$$
Problem: Ground state energy
Complexity: QMA–complete
Ref: [KSV02]
Conditionals:
  • $m = O\big(\mathsf{poly}(n)\big)$
  • Each $H_j$ acts on at most 5 of the $n$ qubits
Reductions:
  • From the $O(\log n)$–local Hamiltonian
  • To the 5–local Hamiltonian on a spatially sparse lattice and the 3–local Hamiltonian
Key: GLH01
Hamiltonian: The 5–local Hamiltonian $$H = \displaystyle\sum_{j=1}^m H_j$$
Problem: Ground state energy (Guided)
Complexity: BQP–hard
Ref: [CFG+23]
Conditionals:
  • A (semi–classical) guiding state such that $||\Pi_- \ket{s}|| \geq \delta^2$
  • $\delta \in (0, 1 - \Omega(1/\mathsf{poly}(n)))$
  • $a,b \in [0,1]$ such that $b-a \in \Omega(1/\mathsf{poly}(n))$
  • $m = O\big(\mathsf{poly}(n)\big)$
Key: GLH02
Hamiltonian: The 2–local Hamiltonian $$H = \displaystyle\sum_{j=1}^m H_j$$
Problem: Ground state energy (Guided)
Complexity: BQP–hard
Ref: [CFG+23]
Conditionals:
  • A (semi–classical) guiding state such that $||\Pi_- \ket{s}|| \geq \delta^2$
  • $\delta \in (0, 1 - \Omega(1/\mathsf{poly}(n)))$
  • $a,b \in [0,1]$ such that $b-a \in \Omega(1/\mathsf{poly}(n))$
  • $m = O\big(\mathsf{poly}(n)\big)$
Reductions:
  • From the $5$–local Hamiltonian (guided) problem
  • To $2$–local Hamiltonians such as the (xy/.) and Heisenberg Hamiltonians
Key: PFP-2LH-001
Hamiltonian: General $2$-local Hamiltonian $$H = \sum_{e\in E(G)} H_e$$
Problem: Partition function
Complexity: FPTAS
Ref: [MH20]
Conditionals:
  • $2$–local interactions
  • $||H_e|| \leq 1$
  • The interaction graph $G$ has maximum degree $\Delta$
  • $\beta \in \mathbb{C}$
  • $\beta \leq 1 / (e^4 \Delta)$
Key: PFP-kLH-001
Hamiltonian: General $k$-local Hamiltonian $$H = \sum_{e\in E(G)} H_e$$
Problem: Partition function
Complexity: FPTAS
Ref: [MM24]
Conditionals:
  • $2$–local interactions
  • $||H_e|| \leq 1$
  • The interaction (hyper)graph $G$ has maximum degree $\Delta$
  • $\beta \in \mathbb{C}$
  • $\beta \leq 1 / (e^4 \Delta{k \choose 2})$
Key: PFP-GLH-001
Hamiltonian: Stable quantum perturbations of a classical spin system Hamiltonian $$H = H_{\Phi} + \lambda H_{\Psi}$$
Problem: Partition function
Complexity: FPTAS
Ref: [MH23]
Conditionals:
  • $H_{\Phi}$ is diagonal in a basis indexed by a classical spin system
  • $H_{\Psi}$ is local
  • $|\lambda|$ is small and $|\lambda|\leq \lambda^{\star}$
  • Stable quantum perturbation of classical spin system
  • $G$ is a finite induced subgraph of $\mathbb{Z}^\nu$
  • $\beta \geq \beta^{\star}$
Key: PFP-GLH-002
Hamiltonian: The Ferromagnetic $q$-state Potts Model $$H(\sigma) = \sum_{\{i,j\} \in E(G)} \delta{\sigma_i \neq \sigma_j}$$
Problem: Partition function
Complexity: FPRAS
Ref: [BCHPT19]
Conditionals:
  • $G$ is $\mathbb{T}_n^d$ where $d\geq 2$
  • $q \geq q_0(d)$
  • $\beta \in \mathbb{R}^+$