Rigorous performance bounds for quadratic and nested dynamical decoupling

We present rigorous performance bounds for the quadratic dynamical decoupling pulse sequence which protects a qubit from general decoherence, and for its nested generalization to an arbitrary number of qubits. Our bounds apply under the assumptions of instantaneous pulses and of bounded perturbing environment and qubit-environment Hamiltonians such as those realized by baths of nuclear spins in quantum dots. We prove that if the total sequence time is fixed then the trace-norm distance between the unperturbed and protected system states can be made arbitrarily small by increasing the number of applied pulses.

Figure 1

Illustration of QDD${}_{3,3}$. Time flows from left to right. The switching instants are distributed according to Eqs. (3) and (4). The pulses ${\pi }_z$ and ${\pi }_x$ are rotations about the $z$ and $x$ axes, respectively, by the angle $\pi$. The total sequence time is $T$.

Figure 2

The upper bound $L_{\alpha }$ for $\parallel A_x\parallel =\parallel A_y\parallel =\parallel A_z\parallel$ according to Eqs. (43) in the isotropic case, i.e., ${\eta }_x={\eta }_y={\eta }_z$, as a function of $\varepsilon$ for various numbers of pulses $N_1=N_2=N\in \{2,6,16,34\}$, at fixed values of ${\eta }_{\alpha }$. In each panel the curves become steeper as $N$ increases.

Figure 3

The upper bounds for $\parallel A_x\parallel$ (dotted lines), $\parallel A_y\parallel$ (dashed lines) and $\parallel A_z\parallel$ (filled lines) according to Eq. (43). Here $N_2$ is 10. The values of $N_1$ in the four panels going from left to right and top to bottom are $2,10,18$, and 34. The value of ${\eta }_z={10}^{-2}$.

Figure 4

The upper bounds for $\parallel A_x\parallel$ (dotted lines), $\parallel A_y\parallel$ (dashed lines) and $\parallel A_z\parallel$ (filled lines) according to Eqs. (43). We take $N_2$ to be 9 in this case. The values of $N_1$ in the four panels going from left to right and top to bottom are $3,10,19$, and 34. The value of ${\eta }_z={10}^{-2}$ as in the previous case.

Figure 5

The upper bound ${\Delta }_{d_{\min }}$ for ${\sum }_{\vec{\mu }\in \mathcal{K}}\parallel A_{\vec{\mu }}\parallel$ as a function of $\varepsilon$ for various pulse numbers $black{d}_{\min }\in \{5,10,20,40\}$, at fixed values of $\eta$, and for $m=10$. In each panel the curves become steeper as $d_{\min }$ increases.