Design method of dynamical decoupling sequences integrated with optimal control theory

A method for synthesizing dynamical decoupling (DD) sequences is presented, which can tailor these sequences to a given set of qubits, environments, instruments, and available resources using partial information of the system. The key concept behind the generation of the DD sequences involves not only extricating the strong dependence on the coupling strengths according to the “optimal control,” but also exploiting the “refocus” technique used conventionally to obtain DD sequences. The concept is a generalized one that harmonizes optimal control and designing of DD sequences.

Figure 1

Decoupling fidelities against (a) flip-angle error and (b) phase-orthogonality error for a single qubit coupled to four bath TLSs with ${\mathrm{UDD}}_{12}$, ${\mathrm{QDD}}_3$, and KDD sequences, and the sequence generated in this example. The waveforms of these sequences are displayed in the top panel of the figure. (c) Decoupling fidelities against coupling strengths to four TLSs denoted by $\parallel {\widehat{\mathcal{H}}}_{Q-B}\parallel$. Coupling strengths to TLSs were randomly generated for each point in the graph.

Figure 2

DD performance as evaluated for an $n$-qubit chain system coupled through dipolar interactions with (a) optimized pulse sequence and (b) MREV16 sequence. Fidelity $F$ is normalized so that $\overline{F}=1-(1-F)/\parallel {\widehat{\mathcal{H}}}_{Q-Q}\parallel$ because $\parallel {\widehat{\mathcal{H}}}_{Q-Q}\parallel$ becomes larger as the number of qubits increases.

Figure 3

Decoupling imperfection under resource restrictions in terms of amplitude of $\parallel \widehat{V}\left(t\right)T\parallel$ and bandwidth of $\widehat{V}\left(t\right)$. Imperfection is investigated with linear combination of zero- and first-order cost functions defined in Sec. 4, so that it is the reciprocal of DD performance. Regions in which the maximum amplitude or the bandwidth is excessively supplied are labeled as I and II, respectively, and the region where both resources are sufficiently supplied is labeled as III. All axes are in logarithmic scale.

Figure 4

System-modulation matrices $c_{\alpha x}\left(t\right)$ ($\alpha =x,y,z$) in regions I–III, represented in Cartesian coordinates. $c_{\alpha x}\left(t\right)$ are displayed as functions of $t\in [0,T]$.