Exact Krylov#

QCANT provides a Krylov subspace routine exposed as QCANT.exact_krylov().

What it does#

The exact Krylov routine:

  • builds an electronic Hamiltonian using PennyLane quantum chemistry tooling,

  • prepares a reference state (Hartree-Fock by default or a provided statevector),

  • generates a Krylov basis from successive applications of the Hamiltonian, and

  • diagonalizes the Hamiltonian projected into that basis.

The basis construction can use a power basis (krylov_method="exact") or the Lanczos recurrence (krylov_method="lanczos"), which orthonormalizes the basis during construction.

Basic usage#

import numpy as np
import QCANT

symbols = ["H", "H"]
geometry = np.array([[0.0, 0.0, 0.0], [0.0, 0.0, 1.5]])

energies, basis_states, min_history = QCANT.exact_krylov(
   symbols=symbols,
   geometry=geometry,
   n_steps=3,
   active_electrons=2,
   active_orbitals=2,
   basis="sto-3g",
   charge=0,
   spin=0,
   krylov_method="exact",
   return_min_energy_history=True,
)

print(energies)
print(basis_states.shape)
print(min_history.shape)

Options#

  • krylov_method: "exact" (power basis) or "lanczos" (orthonormal recurrence).

  • basis_threshold: drop amplitudes below this threshold after each basis update and re-normalize the state (use 0.0 to disable).

  • use_sparse: use a sparse Hamiltonian representation for state updates.

  • return_min_energy_history: return the minimum energy after each basis step.

Outputs#

The function returns (energies, basis_states):

  • energies: eigenvalues obtained by diagonalizing the Hamiltonian in the generated basis

  • basis_states: statevectors after each step, shape (n_steps+1, 2**n_qubits)

If return_min_energy_history=True, the function returns (energies, basis_states, min_energy_history) where min_energy_history has shape (n_steps,).

Notes#

  • krylov_method="exact" does not orthogonalize during basis construction; orthogonalization is performed via the overlap matrix during projection.

  • krylov_method="lanczos" builds an orthonormal basis by construction.