# filter_function

Graph.filter_function(control_hamiltonian, noise_operator, frequencies, sample_count=100, projection_operator=None, *, name=None)

Evaluate the filter function for a control Hamiltonian and a noise operator at the given frequency elements.

### Parameters

• control_hamiltonian (Pwc) – The control Hamiltonian $H_\mathrm{c}(t)$
• noise_operator (Pwc) – The noise operator $N(t)$
• frequencies (list or tuple or np.ndarray) – The elements in the frequency domain at which to return the values of the filter function.
• sample_count (int or None , optional) – The number of points in time, $M$
• projection_operator (np.ndarray or None , optional) – The projection operator $P$
• name (str or None , optional) – The name of the node.

### Returns

The filter function.

### Return type

FilterFunction

Graph.frequency_domain_noise_operator : Control-frame noise operator in the frequency domain.

## Notes

The filter function is defined as 1:

F(f) = \frac{1}{\mathrm{Tr}(P)} \mathrm{Tr} \left( P \mathcal{F} \left\\{ \tilde N^\prime(t) \right\\}(f) \mathcal{F} \left\\{ \tilde N^\prime(t) \right\\}(f)^\dagger P \right),

with the control-frame noise operator in the frequency domain

\mathcal{F} \left\\{ \tilde N^\prime(t) \right\\}(f) = \int_0^\tau e^{-i 2\pi f t} \tilde N^\prime(t) \mathrm{d}t,

where

$\tilde N^\prime(t) = \tilde N(t) - \frac{\mathrm{Tr}\left( P \tilde N(t) P \right)}{\mathrm{Tr}(P)} \mathbb{I}$

is the traceless control-frame noise operator in the time domain,

$\tilde N(t) = U_c^\dagger(t) N(t) U_c(t)$

is the control-frame noise operator in the time domain, and $U_c(t)$