Enhanced Convergence and Robust Performance of Randomized Dynamical Decoupling

We demonstrate the advantages of randomization in coherent quantum dynamical control. For systems which are either time-varying or require decoupling cycles involving a large number of operations, we find that simple randomized protocols offer superior convergence and stability as compared to deterministic counterparts. In addition, we show how randomization may allow us to outperform purely deterministic schemes at long times, including combinatorial and concatenated methods. General criteria for optimally interpolating between deterministic and stochastic design are proposed and illustrated in explicit decoupling scenarios relevant to quantum information storage.

Figure 1

PDD (dashed lines) vs NRD (solid line) based on a nested pulse sequence for $H$ from Eq. (1) in 1D with $N=6$, $J_{ij}^{(a)}\equiv J|i-j{|}^{-3}$ in the logical-rotating frame. Main panel: average fidelity at $T_n=n|\mathcal{G}|\Delta t$. Free evolution: oscillating solid line. Inset: average fidelity within a cycle, $t_n=n\Delta t$, $\Delta t={10}^{-3}{J}^{-1}$. Dotted line: average over different path choices (RPD, see text). Average over ${10}^2$ realizations.

Figure 2

Deterministic vs randomized DD based on ${\mathcal{G}}_8$ for Hamiltonian $H$ as in Fig. 1, except $N=|\mathcal{G}|=8$. Data are averaged over ${10}^2$ control realizations, $\Delta t=0.05J^{-1}$. Note that at times $T_n=8n\Delta t$, as considered, the inner sequences characterizing SDD, CDD, and SRPD are not necessarily completed. For CDD, $\ell =5$ is achieved at $102.4J^{-1}$. Dashed and dot-dashed lines correspond to different inner PDD sequences for EMD. Free evolution: (black) solid line.

Figure 3

Deterministic vs randomized DD based on a $|\mathcal{G}|=4$ sequence for 1D nearest-neighbor couplings with $N=8$ in the logical-rotating frame. $\Delta (t)=\sum _{k=1}^5\sin (10\pi R_{k}Jt)$, $R_k$ uniformly random in $[0.9,1.1]$. Average fidelity at $T_n=n|\mathcal{G}|\Delta t$, ${10}^2$ realizations. PDD: down-triangles; SDD: up-triangles; CDD: squares; NRD: solid line; RPD: long-dashed line; SRPD: dot-dashed line. Free evolution: (black) short-dashed line. Main panel: $\Delta t=0.05J^{-1}$, $\tau /\Delta t\approx 2$. Inset: $\Delta t=0.025J^{-1}$—notice the sensitivity of deterministic protocols (including CDD) to changes in $\Delta t$.