.. currentmodule:: seemps

.. _analysis_derivatives:

************************************
Function Differentiation
************************************

The approximation of differential operators is key to efficiently solve partial 
differential equations. 

Finite Differences
==================

The second-order finite difference method approximates the first and second derivatives of `f(x)` as

.. math::
  \frac{\partial f(x)}{\partial x} = \frac{f(x + \Delta x) - f(x - \Delta x)}{2 \Delta x} + O(\Delta x^2),

.. math::  
  \frac{\partial^2 f(x)}{\partial x^2} = \frac{f(x + \Delta x) - 2 f(x) + f(x - \Delta x)}{ \Delta x^2} + O(\Delta x^2).

This is translated into the quantum register representation using the displacement operator

.. math::
    \hat{\Sigma}^+|s\rangle = \left\{
    \begin{array}{ll}
      |s+1\rangle, & s < 2^{n} \\
      0 & \mbox{else.}
    \end{array}
  \right. \quad \hat{\Sigma}^- = \left(\hat{\Sigma}^+\right)^\dagger,

leading to

.. math::
    \ket{\partial_{x}f^{(n)}} \simeq \frac{1}{2\Delta{x}}\left(\hat{\Sigma}^+-\hat{\Sigma}^-\right)\ket{f^{(n)}}, 

.. math::
    \ket{\partial^2_{x}f^{(n)}} \simeq \frac{1}{\Delta{x}^2}\left(\hat{\Sigma}^+-2\mathbb{I}+\hat{\Sigma}^-\right)\ket{f^{(n)}}.

This approximation is improved with a smoother formula to avoid noise resilience following `Holoborodko <http://www.holoborodko.com/pavel/numerical-methods/numerical-derivative/smooth-low-noise-differentiators>`_.

An example on how to use these functions is shown in `Differentiation.ipynb <https://github.com/juanjosegarciaripoll/seemps2/blob/main/examples/Differentiation.ipynb>`_.

.. autosummary::
    :toctree: generated/

    ~seemps.analysis.finite_differences.smooth_finite_differences_mpo

Hermite Distributed Approximate Functionals (HDAF)
==================================================



Fourier approximation
=====================
Ref. :cite:t:`GarciaRipoll2021` shows that the quantum Fourier transform :math:`\mathcal{F}` can be used to construct
as differential operator as

.. math::
  D(-i\nabla) := \mathcal{F}^{-1} \sum_{\lbrace s \rbrace} D(p_s)\ket{s}\!\bra{s} \mathcal{F}.

This can be obtained combining the :func:`seemps.analysis.operators.p_to_n_mpo` function with SeeMPS's QFT :func:`seemps.qft.qft`.
However, the QFT is not yet optimally implemented for this task, since the bond dimension scales linearly with the number of sites.