Iterative rotation scheme for robust dynamical decoupling

The loss of quantum information due to interactions with external degrees of freedom, which is known as decoherence, remains one of the main obstacles for large-scale implementations of quantum computing. Accordingly, different measures are being explored for reducing its effect. One of them is dynamical decoupling (DD) which offers a practical solution because it only requires the application of control pulses to the system qubits. Starting from basic DD sequences, more sophisticated schemes were developed that eliminate higher-order terms of the system-environment interaction and are also more robust against experimental imperfections. A particularly successful scheme, called concatenated DD (CDD), gives a recipe for generating higher-order sequences by inserting lower-order sequences into the delays of a generating sequence. Here, we show how this scheme can be improved further by converting some of the pulses to virtual (and thus ideal) pulses. The resulting scheme, called ${\left(XY4\right)}^n$, results in lower power deposition and is more robust against pulse imperfections than the original CDD scheme.

Figure 1

CDD${}_2$ and ${\left(XY4\right)}^2$ pulse sequence schemes. The black solid boxes represent the DD $\pi$ pulses of the inner sequences with their respective phases. The gray (blue) boxes are the $\pi$ pulses of the generating sequence for CDD, while for ${\left(XY4\right)}^2$ they are virtual and appear as a transparent white stripe of zero duration. The toggling frame Hamiltonians are represented by the respective signs of the different terms proportional to the ${\widehat{S}}_x$, ${\widehat{S}}_y$, and ${\widehat{S}}_z$ components. Sign changes during the pulses are represented by diagonal lines \ or /. In the CDD${}_2$ scheme, the terms marked by circles are compensated only at the end of the complete cycle, but the ${\left(XY4\right)}^2$ scheme compensates all terms over the basic four-pulse cycle. The toggling frame Hamiltonians of the free evolution interaction for ${\left(XY4\right)}^2$ are the same for all blocks of the inner sequence, i.e., equal to those of the $XY4$ sequence. The toggling frame Hamiltonian of the flip-angle error terms of ${\left(XY4\right)}^2$ is equal to that of the CDD scheme with ideal pulses.

Figure 2

Decays of coherence under the influence of different DD sequences. (a) Normalized spin signal decay of the echo trains of different CDD sequences, the Hahn echo decay [51], and the free evolution (FID). The CDD decay curves are plotted for their optimal delays between pulses that are given when the curves of panel (b) have a maximum. (b) Decay times of different CDD sequences for different delays between pulses. The optimal delay time is defined when the decay is a maximum. Inset: dependence as a function of the CDD order matching with theoretical predictions [33]. ${\tau }_B=110\mu$s is the bath correlation time [20].

Figure 3

Normalized spin signal decays for ${\left(XY4\right)}^2$ and CDD${}_2$ for different delays $\tau$.

Figure 4

Decay times for ${\left(XY4\right)}^n$ and CDD${}_n$ of order 2 and 3 as a function of the duty cycle.

Figure 5

Normalized spin signal after one cycle of different DD sequences as a function of the pulse length of the DD pulses and the delay between them. The labels (a) and (s) refer to the asymmetric and symmetric versions of the sequences.

Figure 6

Normalized spin signal after one cycle of different DD sequences as a function of the offset frequency of the DD pulses and the delay between them. The labels (a) and (s) refer to the asymmetric and symmetric versions of the sequences.

Figure 7

Normalized spin signal after one cycle for the symmetric (s) and asymmetric (a) version of ${\left(XY4\right)}^2$ as a function of the offset frequency of the DD pulses and the delay between them. Both sequences have the same number of pulses and cycle time.

Figure 8

Normalized spin signal after about 100 pulses for different DD sequences as a function of the pulse length of the DD pulses and the delay between them. All sequences have 100 pulses except ${\left(XY4\right)}^2$, which contains 96. The labels (a) and (s) refer to the asymmetric and symmetric version of the sequences.

Figure 9

Normalized spin signal after about 100 pulses for different DD sequences as a function of the offset frequency of the DD pulses and the delay between them. All sequences have 100 pulses except ${\left(XY4\right)}^2$, which contains 96. The labels (a) and (s) refer to the asymmetric and symmetric version of the sequences.