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Physical Review A

Optimal experimental dynamical decoupling of both longitudinal and transverse relaxations

Xing-Long Zhen, Feihao Zhang, Guanru Feng, Hang Li, Gui‐Lu Long

Published Feb 2, 2016• 26 citations
Publisher

Abstract

Both longitudinal and transverse relaxations exist in the practical environment. Their simultaneous eliminations are extremely demanding in real applications. Previous experimental work has focused mainly on the suppression of transverse relaxation. In this paper we investigate the performance of three important dynamical decoupling schemes---quadratic dynamical decoupling, periodic dynamical decoupling, and concatenated dynamical decoupling---in an environment with hybrid errors. We propose a method to engineer arbitrary environment by modulating the control field. The technique developed here is universal and can be applied to other quantum information processing systems. Three-dimensional filter functions technique is utilized to analyze the fidelity decay of a one-qubit state protected by dynamical decoupling sequences. This enables us to quantitatively compare the performance of different dynamical decoupling sequences and demonstrate the superiority of quadratic dynamical decoupling in experiments for the first time. Our work reveals that quadratic dynamical decoupling is optimal conditioned on the appropriate noise properties. The difference of constructing dynamical decoupling sequences with various Pauli pulses is also investigated.

DOI
10.1103/physreva.93.022304

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Both longitudinal and transverse relaxations exist in the practical environment. Their simultaneous eliminations are extremely demanding in real applications. Previous experimental work has focused mainly on the suppression of transverse relaxation. In this paper we investigate the performance of three important dynamical decoupling schemes---quadratic dynamical decoupling, periodic dynamical decoupling, and concatenated dynamical decoupling---in an environment with hybrid errors. We propose a method to engineer arbitrary environment by modulating the control field. The technique developed here is universal and can be applied to other quantum information processing systems. Three-dimensional filter functions technique is utilized to analyze the fidelity decay of a one-qubit state protected by dynamical decoupling sequences. This enables us to quantitatively compare the performance of different dynamical decoupling sequences and demonstrate the superiority of quadratic dynamical decoupling in experiments for the first time. Our work reveals that quadratic dynamical decoupling is optimal conditioned on the appropriate noise properties. The difference of constructing dynamical decoupling sequences with various Pauli pulses is also investigated.
In addition to the traditional two-level system, the three-level system serves as another important elemental building block for the manipulation of qubits. However, the quantum information processing in the three-level system is also subject to the decoherence induced by the interaction between the quantum system and its environment or by the crosstalk between different qutrits. In this work, we construct a sequence of physically feasible dynamical decoupling operators for the three-level system to mitigate not only the transverse dephasing between the excited state and ground states but also the longitudinal relaxation among them. Combining the Hamiltonian engineering and our constructed dynamical decoupling sequence, we further realize the dynamical decoupling protection of qutrit-based quantum gates. Our scheme can effectively enhance the fidelity of three-level-based quantum gates through filtering out the environmental noises, which may provide a new horizon to improve the accuracy of three-level-based quantum manipulation.
One of the most significant hurdles to be overcome on the path to practical quantum information processors is dealing with quantum errors. Dynamical decoupling is a particularly promising approach that complements conventional quantum error correction by eliminating some correlated errors without the overhead of additional qubits. In practice, the control pulses used for decoupling are imperfect and thus introduce errors which can accumulate after many pulses. These instrumental errors can destroy the quantum state. Here we examine several dynamical decoupling sequences, and their concatenated variants, using electron spin resonance of donor electron spins in a 28^{28}28Si crystal. All of the sequences cancel phase noise arising from slowly fluctuating magnetic fields in our spectrometer, but only those sequences based upon alternating πππ-rotations about the X- and Y-axes in the rotating frame (XYXY sequences) demonstrate the ability to store an arbitrary quantum state. By comparing the experimental results with a detailed theoretical analysis we demonstrate that the superior performance of XYXY sequences arises from the fact that they are self-correcting for the dominant instrumental pulse errors in magnetic resonance experiments. We further find that concatenated sequences perform better than the periodic variants, maintaining near 100% fidelities for spin states even after several hundred control pulses. Intuitively, one would expect the instrumental error to increase with the number of pulses in the sequence but we show that the dominant first-order error does not increase when concatenating the XYXY sequence.
Two recent developments in quantum control, concatenation and optimization of pulse intervals, are combined to yield a strategy to suppress unwanted couplings in quantum systems to high order. Longitudinal relaxation and transverse dephasing can be suppressed so that systems with a small splitting between their energy levels can be kept isolated from their environment. The required number of pulses grows exponentially with the desired order but is only the square root of the number needed if only concatenation is used. An approximate scheme even brings the number down to polynomial growth. The approach is expected to be useful for quantum information and for high-precision nuclear magnetic resonance.
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.
Abstract Dynamical decoupling (DD) is a quantum error-mitigation technique based on applying sequences of control pulses to suppress decoherence. Here we perform a numerical study to compare different DD protocols, namely Carr-Purcell-Meiboom-Gill (CPMG), Periodic Dynamical Decoupling (PDD), Concatenated Dynamical Decoupling (CDD) and Uhrig Dynamical Decoupling (UDD) applied to three commonly used noise power spectral densities: (i) white, (ii) 1/f, (iii) 1/f 2 and their hybrid combinations. By ignoring qubit relaxation, we simulate these protocols focusing purely on dephasing noise. We then consider independent multi-axial noise with uniaxial control pulses. The effective coherence times T2,eff are extracted for each case and compared with the baseline free induction decay T ∗ 2 [6]. In all these simulations, the time-domain decoherence profiles follow a stretched exponential form ∼ exp −(t/T2,eff) β with the effective coherence times dependent on the noise spectrum considered, the strength of the noise, and the DD sequence employed. An exception is observed for the UDD under specific noise conditions where it cannot be described by a simple stretched exponential decay form.
Over the past decade we have seen an explosion of demonstrations of quantum coherence in atomic, optical, and condensed matter systems. These developments have placed a new emphasis on the production of robust and optimal quantum control techniques in the presence of environmental noise. We discuss the use of dynamical decoupling as a form of open-loop quantum control capable of suppressing the effects of dephasing in quantum coherent systems. We introduce the concept of dynamical decoupling pulse-sequence construction as a filter-design problem, making connections with filter design from control theory and electrical engineering in the analysis of pulse-sequence performance for the preservation of the phase degree of freedom in a quantum superposition. A detailed mathematical description of how dephasing and the suppression of dephasing can be reduced to a linear control problem is provided, and used as motivation and context for studies of the filtration properties of various dynamical decoupling sequences. Our work then takes this practical perspective in addressing both "standard" sequences derived from nuclear magnetic resonance and novel optimized sequences developed in the context of quantum information. Additionally, we review new techniques for the numerical construction of optimized pulse sequences using the filter-design perspective. We show how the filter-design perspective permits concise comparisons of the relative capabilities of these sequences and reveals the physics underlying their functionality. The use of this new analytical framework allows us to derive new insights into the performance of these sequences and reveals important limiting issues, such as the effect of digital clocking on optimized sequence performance.