felcu

Fourier-extension coefficients for LCU block encodings of matrix functions.

felcu computes coefficients for representing matrix functions via Linear Combination of Unitaries (LCU), using the Fourier extensions of P. Brearley, T. L. Howarth and B. Adcock, Fourier extensions for matrix-function block encodings with error-independent subnormalization bounds, arXiv:2601.18024. It supports five problems:

Problem	Function
identity (or action)	$f(\mathcal H) = \mathcal H$
inversion	$f(\mathcal H) = \mathcal H^{-1}$
exponentiation	$f(\mathcal H) = e^{\mathcal H}$
square_root	$f(\mathcal H) = \sqrt{\mathcal H}$ ($\mathcal H$ positive definite)
logarithm	$f(\mathcal H) = \log \mathcal H$ ($\mathcal H$ positive definite)

A matrix or spectral bounds are required. Every fit is centred on the spectrum,

$$ f(\mathcal H) = g\bigl(\tau(\mathcal H - \mu I)\bigr), \qquad g(x) \approx g_m(x) = \sum_{k=-m}^{m} c_k e^{ikx}, $$

so the block encoding is a linear combination of Hamiltonian simulations $e^{ik\tau\mathcal H}$. Identity and inversion also work on singular values, for non-Hermitian matrices, through the Hermitian dilation ($\mu = 0$, odd series). Inversion handles definite and indefinite spectra.

The coefficients come from one of the paper's three strategies, chosen with method=:

Method	Convergence	Subnormalisation $\alpha_m = \sum_k \lvert c_k\rvert$
sobolev (default)	near-exponential, down to the target	$O(\lVert f(\mathcal H)\rVert)$
reflected	algebraic, $O(m^{-1})$	$\lVert f(\mathcal H)\rVert$ (optimal) for convex $f$
arcsine_taylor	exponential, $O(\sin^m(\pi/\eta))$	bounded, larger than optimal

Installation

pip install .

Or, for development (editable install plus test dependencies):

pip install -e ".[dev]"

Quick start

from felcu import solve_fe_coefficients, evaluate_fourier_series, fitted_domain_segments
import numpy as np

# Fit exp(H) for a Hermitian matrix with eigenvalues in [0.1, 2.0]
result = solve_fe_coefficients(
    "exponentiation",
    sigma_min=0.1,
    sigma_max=2.0,
    target_error=1e-10,
)

print(f"Converged: {result.converged}")
print(f"Truncation order m = {result.m}")
print(f"Achieved L∞ error: {result.achieved_linf_error:.2e}")

# Evaluate the fitted series
for xs in fitted_domain_segments(result):
    y_fit = evaluate_fourier_series(xs, result.a0, result.ak, result.bk)

Documentation

Full documentation is at https://qaaat.github.io/felcu/:

• Notation: the paper's symbols and their names in the code
• Problems and constructions: spectra, scaling, bases, indefinite inversion
• Method: the Sobolev-regularised, reflected and arcsine-Taylor extensions
• Tuning the solver: every parameter
• Examples: the scripts in examples/
• API reference

To build it locally:

pip install -e ".[docs]"
mkdocs serve

Citation

If you use felcu in your work, please cite the paper:

P. Brearley, T. L. Howarth and B. Adcock, Fourier extensions for matrix-function block encodings with error-independent subnormalization bounds, arXiv:2601.18024 (2026).

GitHub's "Cite this repository" button gives the same reference in BibTeX and APA formats, from CITATION.cff.

License

felcu is released under the MIT License.