real_fourier_pwc_signal

Graph.real_fourier_pwc_signal(duration, segment_count, initial_coefficient_lower_bound=-1, initial_coefficient_upper_bound=1, fixed_frequencies=None, optimizable_frequency_count=None, randomized_frequency_count=None, *, name=None)

Create a piecewise-constant signal constructed from Fourier components.

Use this function to create a signal defined in terms of Fourier (sine/cosine) basis signals that can be optimized by varying their coefficients and, optionally, their frequencies.

Parameters

duration (float) – The total duration $\tau$ of the signal.
segment_count (int) – The number of segments $N$ to use for the piecewise-constant approximation.
initial_coefficient_lower_bound (float , optional) – The lower bound $c_\mathrm{min}$ on the initial coefficient values. Defaults to -1.
initial_coefficient_upper_bound (float , optional) – The upper bound $c_\mathrm{max}$ on the initial coefficient values. Defaults to 1.
fixed_frequencies (np.ndarray or None , optional) – A 1D array object containing the fixed non-zero frequencies $\{f_m\}$ to use for the Fourier basis. If provided, must be non-empty and specified in the inverse units of duration (for example if duration is in seconds, these values must be given in Hertz).
optimizable_frequency_count (int or None , optional) – The number of non-zero frequencies $M$ to use, if the frequencies can be optimized. Defaults to 0.
randomized_frequency_count (int or None , optional) – The number of non-zero frequencies $M$ to use, if the frequencies are to be randomized but fixed. Defaults to 0.
name (str or None , optional) – The name of the node.

Returns

The optimizable, real-valued, piecewise-constant signal built from the appropriate Fourier components.

WARNING

You must provide exactly one of fixed_frequencies, optimizable_frequency_count, or randomized_frequency_count.

Notes

This function sets the basis signal frequencies $\{f_m\}$ depending on the chosen mode:

For fixed frequencies, you provide the frequencies directly.
For optimizable frequencies, you provide the number of frequencies $M$ , and this function creates $M$ unbounded optimizable variables $\{f_m\}$ , with initial values in the ranges $\{[(m-1)/\tau, m/\tau]\}$ .
For randomized frequencies, you provide the number of frequencies $M$ , and this function creates $M$ randomized constants $\{f_m\}$ in the ranges $\{[(m-1)/\tau, m/\tau]\}$ .

After this function creates the $M$ frequencies $\{f_m\}$ , it produces the signal

\alpha^\prime(t) = v_0 + \sum_{m=1}^M [ v_m \cos(2\pi t f_m) + w_m \sin(2\pi t f_m) ],

where $\{v_m,w_m\}$ are (unbounded) optimizable variables, with initial values bounded by $c_\mathrm{min}$ and $c_\mathrm{max}$ . This function produces the final piecewise-constant signal $\alpha(t)$ by sampling $\alpha^\prime(t)$ at $N$ equally spaced points along the duration $\tau$ , and using those sampled values as the constant segment values.

You can use the signals created by this function for chopped random basis (CRAB) optimization ¹.

References

[1] P. Doria, T. Calarco, and S. Montangero, Phys. Rev. Lett. 106, 190501 (2011).

Examples

See the “Fourier-basis optimization on a qutrit” example in the How to optimize controls using arbitrary basis functions user guide.