Parameters
• duration (float) – The total duration of the sequence $$\tau$$ (in seconds).

• 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 $$X_{\pi/2}$$ rotation is added at the start and end of the sequence. Defaults to False.

• name (string, optional) – Name of the sequence. Defaults to None.

Returns

Return type

DynamicDecouplingSequence

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 $$X_{\pi}$$ pulse, therefore has timing offset defined by

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

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 $$j$$-th inner sequence has timing offset defined by

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

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

References

1

J. R. West, B. H. Fong, and D. A. Lidar, Physical Review Letters 104, 130501 (2010).