device_builder.utils.chebyshev

Helper functions for computing chebyshev moments.

def get_chebyshev_moments( operator: Callable[[numpy.ndarray, numpy.ndarray], numpy.ndarray], spectral_bounds: tuple[float, float], state: numpy.ndarray, number_of_chebyshev_moments: int, batch_size: int) -> device_builder.models.chebyshev.ChebyshevMoments:

Compute Chebyshev moments for the given batch size.

NOTE: The spectral bounds of the operator must be in the interval (-1, 1). The spectral bounds of the original linear operator are passed via the spectral_bounds argument.

Arguments:

operator (Callable[[xp.ndarray, xp.ndarray], xp.ndarray]): Linear operator for Chebyshev iteration.
spectral_bounds (tuple[float, float]): Spectral bounds of the original linear operator.
state (xp.ndarray): State for which to compute Chebyshev moments.
number_of_chebyshev_moments (int): Number of Chebyshev moments to calculate.
batch_size (int): Number of random states to generate.

Returns:

ChebyshevMoments: Model holding reference to Chebyshev moments and supporting information.

def get_density_of_states( chebyshev_moments: device_builder.models.chebyshev.ChebyshevMoments, energy_window: tuple[float, float], n_grid_points: int, n_grid_points_batch: int, max_n_chebyshev_moments: Optional[int] = None, eta: float = 0.0) -> tuple[float, numpy.ndarray, numpy.ndarray]:

Get density of states from given Chebyshev moments.

The computation of the DOS is batched to reduce the memory footprint of the get_dos function.

NOTE: Memory consumption scales: chebyshev_moments.size x n_grid_points_batch

Arguments:

chebyshev_moments (ChebyshevMoments): Pre-computed Chebyshev moments.
energy_window (tuple[float, float]): Energy window to compute DOS.
n_grid_points (int): Target for number of grid points.
n_grid_points_batch (int): Number of grid points for each batch.
max_n_chebyshev_moments (Optional[int]): Maximum number of Chebyshev moments to be used for computing the DOS. Default is None, which means all Chebyshev moments are used.
eta (float): Broadening parameter via shifting energies into complex plane. Defaults to 0.0, meaning no additional broadening.

Returns:

tuple[float, np.ndarray, np.ndarray]: Tuple of float: Integration weight, np.ndarray: Energy grid points, np.ndarray: Density of states on grid points.

def average_chebyshev_moments( chebyshev_moments: list[device_builder.models.chebyshev.ChebyshevMoments]) -> device_builder.models.chebyshev.ChebyshevMoments:

Average Chebyshev moments from a list of ChebyshevMoments.

Arguments:

chebyshev_moments (list[ChebyshevMoments]): List of ChebyshevMoments to be averaged.

Returns:

ChebyshevMoments: Average Chebyshev moments.

def get_dos( chebyshev_moments: numpy.ndarray, window: tuple[float, float], chebyshev_window: tuple[float, float], n_grid_points: int, eta: float) -> tuple[float, numpy.ndarray, numpy.ndarray]:

Returns the density of states (DOS) in given energy window.

NOTE: The input requires precomputed Chebyshev moments of the Hamiltonian for which the DOS is computed.

NOTE: At present, both, the energy window and the energy window scaled into the Chebyshev window are required input.

NOTE: If 'bare' Chebyshev moments are used the DOS is normalized to 1. You can use 'pre-scaled' Chebyshev moments to enforce a different normalization condition.

Arguments:

chebyshev_moments (np.ndarray): Precomputed Chebyshev moments.
window (tuple[float, float]): Energy window for which to compute the DOS.
chebyshev_window (tuple[float, float]): Energy window scaled into Chebyshev interval.
n_grid_points (int): Number of discretization points for energy window.
eta (float): Broadening parameter via shifting energies into complex plane.

Returns:

tuple[float, np.ndarray, np.ndarray]: Tuple of float: energy spacing between grid points, np.ndarray: discretized energy window (equidistant grid), np.ndarray: DOS on grid points.