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 Hc(t)H_\mathrm{c}(t).
  • noise_operator (Pwc) – The noise operator N(t)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, MM, 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 or None , optional) – The projection operator PP. Defaults to the identity matrix.
  • name (str or None , optional) – The name of the node.

Returns

The filter function.

Return type

FilterFunction

SEE ALSO

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

N~′(t)=N~(t)−Tr(PN~(t)P)Tr(P)I \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,

N~(t)=Uc†(t)N(t)Uc(t) \tilde N(t) = U_c^\dagger(t) N(t) U_c(t)

is the control-frame noise operator in the time domain, and Uc(t)U_c(t) is the time evolution induced by the control Hamiltonian.

References

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