qiskit_addon_sqd.fermion.optimize_orbitals

optimize_orbitals(bitstring_matrix, /, hcore, eri, k_flat, *, open_shell=False, spin_sq=0.0, num_iters=10, num_steps_grad=10000, learning_rate=0.01, max_davidson=100)[source]

Optimize orbitals to produce a minimal ground state.

The process involves iterating over 3 steps:

For num_iters iterations:

  • Rotate the integrals with respect to the parameters, k_flat

  • Diagonalize and approximate the groundstate energy and wavefunction amplitudes

  • Optimize k_flat using gradient descent and the wavefunction amplitudes found in Step 2

Refer to Sec. II A 4 for more detailed discussion on this orbital optimization technique.

Parameters:

  • bitstring_matrix (tuple[ndarray, ndarray] | ndarray) –

    A set of configurations defining the subspace onto which the Hamiltonian will be projected and diagonalized. This is a 2D array of bool representations of bit values such that each row represents a single bitstring. The spin-up configurations should be specified by column indices in range (N, N/2], and the spin-down configurations should be specified by column indices in range (N/2, 0], where N is the number of qubits.

    (DEPRECATED) The configurations may also be specified by a length-2 tuple of sorted 1D arrays containing unsigned integer representations of the determinants. The two lists should represent the spin-up and spin-down orbitals, respectively.

  • hcore (ndarray) – Core Hamiltonian matrix representing single-electron integrals

  • eri (ndarray) – Electronic repulsion integrals representing two-electron integrals

  • k_flat (ndarray) – 1D array defining the orbital transform. This array will be reshaped to be of shape (# orbitals, # orbitals) before being used as a similarity transform operator on the orbitals. Thus len(k_flat)=# orbitals**2.

  • open_shell (bool) – A flag specifying whether configurations from the left and right halves of the bitstrings should be kept separate. If False, CI strings from the left and right halves of the bitstrings are combined into a single set of unique configurations and used for both the alpha and beta subspaces.

  • spin_sq (float) – Target value for the total spin squared for the ground state

  • num_iters (int) – The number of iterations of orbital optimization to perform

  • max_davidson (int) – The maximum number of cycles of Davidson’s algorithm to perform during diagonalization.

  • num_steps_grad (int) – The number of steps of gradient descent to perform during each optimization iteration

  • learning_rate (float) – The learning rate to use during gradient descent

Returns:

  • The groundstate energy found during the last optimization iteration

  • An optimized 1D array defining the orbital transform

  • Average orbital occupancy

Return type:

A tuple containing