qctrlopencontrols.new_quadratic_sequence(duration, inner_offset_count, outer_offset_count, pre_post_rotation=False, name=None)

### Parameters

• duration (float) – The total duration of the sequence $\tau$
• inner_offset_count (int) – Number of inner $Z_{\pi}$ pulses $n_1$
• outer_offset_count (int) – Number of outer $X_{\pi}$ pulses $n_2$
• pre_post_rotation (bool , optional) – If True, a $X_{\pi/2}$False.
• name (string , optional) – Name of the sequence. Defaults to None.

### Return type

DynamicDecouplingSequence

new_uhrig_sequence

## Notes

The quadratic sequence 1 is parameterized by duration $\tau$, number of inner offsets $n_1$, and number of outer offsets $n_2$. The outer sequence consists of $n_2$ pulses of type $X_{\pi}$, which partition the time-domain into $n_2+1$ sub-intervals on which inner sequences consisting of $n_1$ pulses of type $Z_{\pi}$ are nested. The total number of offsets is $n = n_1 + n_2(n_1 + 1)$

The pulse times for outer sequence $(X_{\pi}^1, \cdots, X_{\pi}^{n_2})$ are defined according to the Uhrig sequence for $t \in [0, \tau]$. The $j$-th $X_{\pi}$

$t_x^j = \tau \sin^2 \left[ \frac{j \pi}{2(n_2 + 1)} \right] \;,$

where $j = 1, \cdots, n_2$. On each sub-interval defined by the outer sequence, an inner sequence $(Z_{\pi}^1, \cdots, Z_{\pi}^{n_1})$ is implemented. The pulse times for the inner sequences are also defined according to the Uhrig sequence. The $k$-th pulse of the $j$

$t_z(k, j) = (t_x^j - t_x^{j - 1}) \sin^2 \left[ \frac{k \pi} {2 (n_1 + 1)} \right] + t_{x}^{j - 1} \;,$

where $k = 1, \cdots, n_1$ and $j = 1, \cdots, n_2 + 1$