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Documentation Index

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haiqu.sdk.skqd.siam_hamiltonian(norb, t, U, V, mu)

Build single-impurity Anderson model Hamiltonian tensors in site basis. 
H = -t * sum_{<i,j> in bath, sigma} (c^dag_i c_j + h.c.)
  - V * sum_sigma (c^dag_d c_1 + h.c.)
  + mu * n_d
  + U * n_d_up * n_d_down
The impurity is placed on site 0.
  • Parameters:
    • norb (int) — Number of spatial orbitals (impurity + bath sites). Must be >= 2.
    • t (float) — Hopping parameter between bath sites.
    • U (float) — On-site interaction strength at the impurity.
    • V (float) — Impurity-bath hybridization.
    • mu (float) — Chemical potential at the impurity.
  • Returns: (h1e, h2e) — one-body tensor shape (norb, norb) and two-body tensor shape (norb, norb, norb, norb).
  • Return type: tuple[ndarray, ndarray]

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haiqu.sdk.skqd.build_siam_momentum_basis_krylov_circuits(norb, krylov_dim, dt, h1e, h2e, impurity_index)

Build Krylov subspace circuits for SIAM in momentum basis. The ground state of the Anderson impurity model is significantly sparser in momentum basis — fewer determinants carry significant weight — making SQD sampling far more effective than in site basis. The Trotter step applies a single CPhase on the impurity orbital only (since U is localized there in momentum basis).
  • Parameters:
    • norb (int) — Number of spatial orbitals.
    • krylov_dim (int) — Dimension of the Krylov subspace (number of time-evolution steps + 1).
    • dt (float) — Time step for Trotter evolution.
    • h1e (ndarray) — One-body Hamiltonian tensor in momentum basis, shape (norb, norb).
    • h2e (ndarray) — Two-body Hamiltonian tensor in momentum basis, shape (norb, norb, norb, norb).
    • impurity_index (int) — Index of the impurity orbital in momentum ordering.
  • Returns: List of quantum circuits representing the Krylov subspace.
  • Return type: list[QuantumCircuit]