## `eigenkit.chemistry.exact`

Exact classical reference: dense matrix and ground-state energy of a qubit `Observable`. For validating VQE on small systems (exponential cost — keep `n_qubits` small).

### `dense_hamiltonian(observable, n_qubits, offset=0.0)`

The `2**n_qubits × 2**n_qubits` matrix of `observable + offset * I`.

Qubit 0 is the most significant factor of the tensor product, which matches the ordering of basis states in the simulators.

Parameters:

| Name         | Type         | Description                                                                                                                                                 | Default    |
| ------------ | ------------ | ----------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------- |
| `observable` | `Observable` | The qubit operator.                                                                                                                                         | *required* |
| `n_qubits`   | `int`        | The number of qubits.                                                                                                                                       | *required* |
| `offset`     | `float`      | The constant (identity) term that the mapping returned separately, added back to the energy. For a molecule, it includes the nuclear repulsion, in Hartree. | `0.0`      |

Returns:

| Type      | Description                           |
| --------- | ------------------------------------- |
| `ndarray` | The matrix, as a complex NumPy array. |

### `exact_ground_state_energy(observable, n_qubits, offset=0.0)`

The lowest eigenvalue of `observable + offset * I`, by dense diagonalisation.

Parameters:

| Name         | Type         | Description                                                                                                                                                 | Default    |
| ------------ | ------------ | ----------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------- |
| `observable` | `Observable` | The qubit operator.                                                                                                                                         | *required* |
| `n_qubits`   | `int`        | The number of qubits.                                                                                                                                       | *required* |
| `offset`     | `float`      | The constant (identity) term that the mapping returned separately, added back to the energy. For a molecule, it includes the nuclear repulsion, in Hartree. | `0.0`      |

Returns:

| Type    | Description              |
| ------- | ------------------------ |
| `float` | The ground-state energy. |