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.

  • 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.


The noise operator in the frequency domain.

Return type


See also


The filter function.


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

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


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

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. If sample_count is set, the Fourier integral is approximated as

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

where \(\tau\) is the duration of the control Hamiltonian.



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).