High-order dynamical decoupling in the weak-coupling regime

Abstract

We introduce a high-order dynamical decoupling (DD) scheme for arbitrary system-bath interactions in the weak-coupling regime. Given any decoupling group $\mathcal G$ that averages the interaction to zero, our construction yields pulse sequences whose length scales as $\mathcal{O}(|\mathcal G| K)$, while canceling all error terms linear in the system-bath coupling strength up to order $K$ in the total evolution time. As a corollary, for an $n$-qubit system with $k$-local system-bath interactions, we obtain an $\mathcal{O}(n^{k-1}K)$-pulse sequence, a significant improvement over existing schemes with $\mathcal{O}(\exp(n))$ pulses (for $k=\mathcal{O}(1)$). The construction is obtained via a mapping to the continuous necklace-splitting problem, which asks how to cut a multi-colored interval into pieces that give each party the same share of every color. We provide explicit pulse sequences for suppressing general single-qubit decoherence, prove that the pulse count is asymptotically optimal, and verify the predicted error scaling in numerical simulations. For the same number of pulses, we observe that our sequences outperform the state-of-the-art Quadratic DD in the weak-coupling regime. We also note that the same construction extends to suppress slow, time-dependent Hamiltonian noise.

Figures (7)

• Figure 1: (a) Discrete necklace-splitting example with two agents ($q=2$) and three colors ($K=3$): by choosing two cuts, each agent receives the same number of red, blue, and green beads. (b) Continuous necklace-splitting picture for dynamical decoupling: the first three time-moments $\int_{0}^{1}\tau^{m}d\tau$ ($K=3$ in this example) are partitioned by the cuts and assigned to the $|\mathcal{G}|$ agents ($q=|\mathcal{G}|$) so that each agent receives the same moments. In the DD construction, the cuts mark the pulse times, $\mathcal{G}$ denotes the decoupling group for the given noise model, and the time-moments correspond to the error terms, whose equal distribution ensures that averaging over the group cancels these contributions. The necklace-splitting theorem guarantees that there exist at most $(q-1)K$ such cuts.
• Figure 2: Errors for optimized DD sequences at different coupling strengths (from left to right): $J=10^{-3}$, $J=10^{-4}$, and $J=10^{-5}$. Here $\beta=1$, and $T$ denotes the total evolution time. As $J$ decreases, the error scaling increasingly follows the expected order $K$ (indicated by shaded lines). A crossover is observed where the slope transitions from $K$ to $2$, consistent with the scaling $\mathcal{O}(JT^{K+1})+\mathcal{O}(J^{2}T^{2})$.
• Figure 3: Comparison between our DD sequences and QDD for the single-qubit general decoherence model. We show the average trace distance error of the reduced system state versus $T$ for several small coupling strengths $J$, comparing our $K=5$ DD (16 pulses) with QDD at $K_{\rm QDD}=3$ (16 pulses). We observe that our sequence achieves smaller errors as $J$ decreases and for larger $T$, where higher-order cancellation is more effective. Each data point is averaged over 100 random initial product states.
• Figure S1: Comparison between our order-$K$ decoupling sequence and QDD in the same single-qubit model used in the main text with $\beta=1$ and $J=10^{-5}$. The plot shows the trace-distance error between the actual and ideal system states as a function of total evolution time $T$.
• Figure S2: Comparison between our DD sequences and QDD for the single-qubit general decoherence model. We show the average trace distance error of the reduced system state versus $T$ for several small coupling strengths $J$, comparing our $K=3$ DD (10 pulses) with QDD at $K_{\rm QDD}=2$ (8 pulses). Each data point is averaged over 100 random initial product states.

Theorems & Definitions (7)

• Theorem 1: Existence of an order-$K$ moment-cancelling sequence with at most $(|\mathcal{G}|-1)K$ pulses
• Lemma 2: $\Omega(K)$ lower bound
• Theorem S1: Existence of an order-$K$ moment-cancelling sequence with at most $(|\mathcal{G}|-1)K$ pulses; Restatement of Theorem \ref{['thm:existence']}
• Lemma S2: Necklace-splitting for continuous densities, Theorem 1.2 in alon1987splitting
• proof : Proof of Theorem \ref{['thm:existence_apx']}
• Lemma S3: $\Omega(K)$ lower bound; Restatement of Lemma \ref{['lem:lowerbound']}
• proof