new_walsh_sequence

qctrlopencontrols.new_walsh_sequence(duration, paley_order, pre_post_rotation=False, name=None)

Creates the Walsh sequence.

Parameters

duration (float) – Total duration of the sequence $\tau$ (in seconds).
paley_order (int) – The paley order $k$ of the Walsh sequence.
pre_post_rotation (bool , optional) – If True, a $X_{\pi/2}$ rotation is added at the start and end of the sequence. Defaults to False.
name (string , optional) – Name of the sequence. Defaults to None.

Notes

The Walsh sequence is defined by the switching function $y(t)$ given by a Walsh function. To define the Walsh sequence, we first introduce the Rademacher function ¹, which is defined as

R_j(x) := {\rm sgn}\left[\sin(2^j \pi x)\right] \;, \quad\; x \in [0, 1]\;, \; j \geq 0 \;.

The $j$ -th Rademacher function $R_j(x)$ is thus a periodic square wave switching $2^{j-1}$ times between $\pm 1$ over the interval $[0, 1]$ . The Walsh function of Paley order $k$ is denoted ${\rm PAL}_k(x)$ and defined as

{\rm PAL}_k(x) = \Pi_{j = 1}^m R_j(x)^{b_j} \;, \quad\; x \in [0, 1] \;.

where $(b_m, b_{m-1}, \cdots, b_1)$ is the binary representation of $k$ . That is

k = b_m 2^{m-1} + b_{m-1}2^{m-2} + \cdots + b_12^0 \;,

where $m = m(k)$ indexes the most significant binary bit of $k$ .

The $k$ -th order Walsh sequence ² is then defined by

y(t) = {\rm PAL}_k(t / \tau) \;

with offset times $\{t_j / \tau\}$ defined at the switching times of the Walsh function.

References

[1] H. Rademacher, Math. Ann. 87, 112–138 (1922).

[2] H. Ball and M. J Biercuk, EPJ Quantum Technol. 2, 11 (2015).

Parameters

Returns

Return type

Notes

References