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arXiv:1110.1011 · Physical Review A

Effects of time-reversal symmetry in dynamical decoupling

Alexandre M. Souza, Gonzalo A. Álvarez, Dieter Suter

Published Mar 7, 2012• 28 citations
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Abstract

Dynamical decoupling is a technique for preserving the coherence of quantum-mechanical states in the presence of a noisy environment. It uses sequences of inversion pulses to suppress the environmental perturbations by periodically refocusing them. It has been shown that different sequences of inversion pulses show vastly different performance, in particular also concerning the correction of experimental pulse imperfections. Here, we investigate specifically the role of time-reversal symmetry in the building blocks of the pulse sequence. We show that using time-symmetric building blocks often improves the performance of the sequence compared to sequences formed by time-asymmetric building blocks. Using quantum state tomography of the echoes generated by the sequences, we analyze the mechanisms that lead to loss of fidelity and show how they can be compensated by suitable concatenation of symmetry-related blocks of decoupling pulses.

DOI
10.1103/physreva.85.032306

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Dynamical decoupling is a technique for preserving the coherence of quantum-mechanical states in the presence of a noisy environment. It uses sequences of inversion pulses to suppress the environmental perturbations by periodically refocusing them. It has been shown that different sequences of inversion pulses show vastly different performance, in particular also concerning the correction of experimental pulse imperfections. Here, we investigate specifically the role of time-reversal symmetry in the building blocks of the pulse sequence. We show that using time-symmetric building blocks often improves the performance of the sequence compared to sequences formed by time-asymmetric building blocks. Using quantum state tomography of the echoes generated by the sequences, we analyze the mechanisms that lead to loss of fidelity and show how they can be compensated by suitable concatenation of symmetry-related blocks of decoupling pulses.
Quantum computers, which process information encoded in quantum mechanical systems, hold the potential to solve some of the hardest computational problems. A substantial obstacle for the further development of quantum computers is the fact that the lifetime of quantum information is usually too short to allow practical computation. A promising method for increasing the lifetime, known as dynamical decoupling (DD), consists of applying a periodic series of inversion pulses to the quantum bits. In the present review, we give an overview of this technique and compare different pulse sequences proposed earlier. We show that pulse imperfections, which are always present in experimental implementations, limit the performance of DD. The loss of coherence due to the accumulation of pulse errors can even exceed the perturbation from the environment. This effect can be largely eliminated by a judicious design of pulses and sequences. The corresponding sequences are largely immune to pulse imperfections and provide an increase of the coherence time of the system by several orders of magnitude.
Avoiding the loss of coherence of quantum mechanical states is an important prerequisite for quantum information processing. Dynamical decoupling (DD) is one of the most effective experimental methods for maintaining coherence, especially when one can access only the qubit system and not its environment (bath). It involves the application of pulses to the system whose net effect is a reversal of the system-environment interaction. In any real system, however, the environment is not static, and therefore the reversal of the system-environment interaction becomes imperfect if the spacing between refocusing pulses becomes comparable to or longer than the correlation time of the environment. The efficiency of the refocusing improves therefore if the spacing between the pulses is reduced. Here, we quantify the efficiency of different DD sequences in preserving different quantum states. We use 13C^{13}\mathrm{C}13C nuclear spins as qubits and an environment of 1H^{1}\mathrm{H}1H nuclear spins as the environment, which couples to the qubit via magnetic dipole-dipole couplings. Strong dipole-dipole couplings between the proton spins result in a rapidly fluctuating environment with a correlation time of the order of 100 $\ensuremath{\mu}$s. Our experimental results show that short delays between the pulses yield better performance if they are compared with the bath correlation time. However, as the pulse spacing becomes shorter than the bath correlation time, an optimum is reached. For even shorter delays, the pulse imperfections dominate over the decoherence losses and cause the quantum state to decay.
We consider the effect of broadband decoherence on the performance of refocusing sequences, having in mind applications of dynamical decoupling in concatenation with quantum error correcting codes as the first stage of coherence protection. Specifically, we construct cumulant expansions of effective decoherence operators for a qubit driven by a pulse of a generic symmetric shape and for several sequences of $\ensuremath{\pi}$ and $\ensuremath{\pi}/2$ pulses. While, in general, the performance of soft pulses in decoupling sequences in the presence of Markovian decoherence is worse than that of the ideal $\ensuremath{\delta}$ pulses, it can be substantially improved by shaping.
Active protection of quantum states is an essential prerequisite for the implementation of quantum computing. Dynamical decoupling (DD) is a promising approach that applies sequences of control pulses to the system in order to reduce the adverse effect of system-environment interactions. Since every hardware device has finite precision, the errors of the DD control pulses can themselves destroy the stored information rather than protect it. We experimentally compare the performance of different DD sequences in the presence of an environment that was chosen such that all relevant DD sequences can equally suppress its effect on the system. Under these conditions, the remaining decay of the qubits under DD allows us to compare very precisely the robustness of the different DD sequences with respect to imperfections of the control pulses.
The dynamical decoupling (DD) method is widely adopted to preserve coherence in different quantum systems. In the case of ideal pulses, its effects on the suppression of noise can be analytically described by the mathematical form of filter function. However, in practical experiments, the unavoidable pulse errors limit the efficiency of DD. In this paper, we study the effects of imperfect pulses on DD efficiency based on quantum trajectories. By directly generating a pseudo noise sequence correlated in time, we can explore the performance of DD with different pulse errors in the typical noise environment. It shows that, for the typical 1/ f noise environment, the phase error of operational pulses severely affects the performance of noise suppression, while the detuning and intensity errors have less influence. Also, we get the thresholds of these errors for efficient DD under the given experimental conditions. Our method can be widely applied to guide practical DD experimental implementation.
Dynamical decoupling can be used to preserve arbitrary quantum states despite undesired interactions with the environment, using control Hamiltonians affecting the system only. We present a system-independent analysis of dynamical decoupling based on leading order decoupling error estimates, valid for bounded-strength environments. Using as a key tool a renormalization transformation of the effective system-bath coupling Hamiltonian, we delineate the reliability domain of dynamical decoupling used for quantum state preservation, in a general setting for a single qubit. We specifically analyze and compare two deterministic dynamical decoupling schemes---periodic and concatenated---and distinguish between two limiting cases of fast versus slow environments. We prove that concatenated decoupling outperforms periodic decoupling over a wide range of parameters. These results are obtained for both ``ideal'' (zero-width) and realistic (finite-width) pulses This work extends and generalizes our earlier work [Phys. Rev. Lett. 95, 180501 (2005)].