> ## Documentation Index
> Fetch the complete documentation index at: https://docs.haiqu.ai/llms.txt
> Use this file to discover all available pages before exploring further.

# SIAM Model

### haiqu.sdk.skqd.siam\_hamiltonian(norb, t, U, V, mu)

Build single-impurity Anderson model Hamiltonian tensors in site basis.

```text theme={null}
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*]

### 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*]