felcu¶

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

felcu computes the coefficients needed to block encode a matrix function \(f(\mathcal H)\) as a Linear Combination of Unitaries (LCU). It rescales the spectrum to \(\mathcal G = \tau(\mathcal H - \mu I)\) and approximates the fitted function \(g(x) = f(\mu + x/\tau)\) on the fitted set by a Fourier extension

\[ g_m(x) = \sum_{k=-m}^{m} c_k e^{ikx} = a_0 + \sum_{k=1}^{m} \bigl[ a_k \sin(kx) + b_k \cos(kx) \bigr], \]

so that \(f(\mathcal H) \approx g_m(\mathcal G) = \sum_k c_k e^{ik\mathcal G}\), a linear combination of Hamiltonian simulations. Three strategies for choosing the coefficients are available through method=, each trading convergence against the subnormalisation \(\alpha_m = \sum_k \lvert c_k\rvert\):

"sobolev" (the default): a Sobolev-regularised fit, which converges near-exponentially down to the target tolerance while keeping \(\alpha_m\) at the optimal scale \(O(\|f(\mathcal H)\|)\);
"reflected": reflected extensions, with optimal \(\alpha\) but algebraic convergence;
"arcsine_taylor": arcsine-Taylor expansions, with exponential convergence at a larger \(\alpha\).

See Method for how they compare.

The methods are those of P. Brearley, T. L. Howarth and B. Adcock, Fourier extensions for matrix-function block encodings with error-independent subnormalization bounds, arXiv:2601.18024; these docs follow its notation.

Five problems are supported:

Problem	Function	Spectral data
`identity` (or `action`)	\(f(\mathcal H) = \mathcal H\)	eigenvalues (Hermitian) or singular values
`inversion`	\(f(\mathcal H) = \mathcal H^{-1}\)	eigenvalues (definite or indefinite) or singular values
`exponentiation`	\(f(\mathcal H) = e^{\mathcal H}\)	eigenvalues
`square_root`	\(f(\mathcal H) = \sqrt{\mathcal H}\)	positive-definite eigenvalue interval
`logarithm`	\(f(\mathcal H) = \log\mathcal H\)	positive-definite eigenvalue interval

A matrix or spectral bounds are required. Every fit is centred on the spectrum, \(f(\mathcal H) = g(\tau(\mathcal H - \mu I))\). Singular values, for non-Hermitian matrices, are fitted through the Hermitian dilation, whose spectrum is centred at \(\mu = 0\), by an odd series. See Problems and constructions.

At a glance¶

from felcu import solve_fe_coefficients

result = solve_fe_coefficients(
    "exponentiation",
    sigma_min=0.1,
    sigma_max=2.0,
    target_error=1e-10,
)

result.m            # truncation order
result.a0, result.ak, result.bk
result.complex_coefficients   # [c_{-m}, ..., c_m]
result.circuit_coefficients   # [c_0, 0, c_1, c_-1, ..., c_m, c_-m] for the compressed circuit
result.alpha        # subnormalisation alpha_m

Where next¶

Getting started: install, a first fit, and how to apply the result to a matrix.
Notation: the paper's symbols and their names in the code.
Problems and constructions: spectra, scaling, bases and the two-interval fit of inversion.
Method: what the solver actually computes.
Tuning the solver: every parameter and when to change it.
Examples: short scripts, one idea each, plus plots.
API reference.

Citation¶

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

@misc{brearley2026fourier,
  title         = {Fourier extensions for matrix-function block encodings
                   with error-independent subnormalization bounds},
  author        = {Brearley, Peter and Howarth, Thomas L. and Adcock, Ben},
  year          = {2026},
  eprint        = {2601.18024},
  archivePrefix = {arXiv},
  url           = {https://arxiv.org/abs/2601.18024},
}

The repository's CITATION.cff holds the same reference, and GitHub's "Cite this repository" button exports it in other formats.

License¶

felcu is released under the MIT License.