# frequency_domain_noise_operator

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

Create a control-frame noise operator in the frequency domain for a control Hamiltonian and a noise operator at the given frequencies.

### 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 noise operator in the frequency domain.

### Return type

Tensor

Graph.filter_function : The filter function.

## Notes

The control-frame noise operator in the frequency domain is defined as the Fourier transform of the operator in the time domain 1:

\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 $U_c(t)$

\mathcal{F} \left\\{ \tilde N^\prime(t) \right\\}(f) \approx \sum_{n=0}^{M-1} \frac{\tau}{M} e^{-i 2\pi f n \tau/M} \langle \tilde N^\prime (n\tau/M) \rangle,

where $\tau$