# 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, optional) – The number of points in time, $$M$$, to sample the control-frame noise operator. These samples are used to calculate the approximate Fourier integral efficiently. If None the piecewise Fourier integral is calculated exactly. Defaults to 100.

• projection_operator (np.ndarray, optional) – The projection operator $$P$$. Defaults to None, in which case the identity operator is used.

• name (str, optional) – The name of the node.

Returns

The filter function.

Return type

FilterFunction

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 $$U_c(t)$$ is the time evolution induced by the control Hamiltonian.

References

1

H. Ball, M. J. Biercuk, A. R. R. Carvalho, J. Chen, M. Hush, L. A. De Castro, L. Li, P. J. Liebermann, H. J. Slatyer, and C. Edmunds, Quantum Sci. Technol. 6, 044011 (2021).