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# Optimal Dynamical Decoupling Sequence for Ohmic Spectrum 

Yu Pan 1 , $^{2}$, Zai-Rong Xi$^{1}$, ∗ and Wei Cui 1 , 2 

$^{1}$Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China and 

$^{2}$Graduate University of Chinese Academy of Sciences, Beijing 100039, People’s Republic of China 

We investigate the optimal dynamical decoupling sequence for a qubit coupled to an ohmic envi- ronment. By analytically computing the derivatives of the decoherence function, the optimal pulse locations are found to satisfy a set of non-linear equations which can be easily solved. These equa- tions incorporates the environment information such as high-energy (UV) cutoff frequency ω$_{c}$ , giving a complete description of the decoupling process. The solutions explain previous experimental and theoretical results of locally optimized dynamical decoupling (LODD) sequence in high-frequency- dominated environment, which were obtained by purely numerical computation and experimental feedback. As shown in numerical comparison, these solutions outperform the Uhrig dynamical decoupling (UDD) sequence by one or more orders of magnitude in the ohmic case. 

Suppressing decoherence is one of the fundamental is- sues in the field of quantum information processing. De- coherence, which has been caused by the environmental noise, plagues almost all the implementations of quantum bit. To eliminate the unwanted coupling between a qubit and its environment, several schemes have been proposed and tested. Among them a promising one is dynamical decoupling [1–4], which restores the qubit coherence by applying delicately designed sequence of control pulses. 

For a qubit that can be modeled by a spin-1/2 par- ticle, the oldest dynamical decoupling sequence is peri- odic dynamical decoupling (PDD). Originated from pulse sequences widely used in nuclear magnetic resonance (NMR) [5], the PDD sequence consists of periodic and equidistant π pulses. To achieve better performance, there has been an extensive study in how to optimize the pulse locations [6–15]. One important progress is the powerful Uhrig DD (UDD) [10], which employs n pulses located at t$_{j}$ according to the simple rules 

where δ$_{j}$ = t$_{j}$/T and T is the total evolution time. UDD is first derived on spin-boson model and further proved to be universal in the sense that it can remove the qubit- bath coupling to n th order in generic environment [11]. 

Beyond UDD, another locally optimized dynamical de- coupling (LODD) sequence has drawn great attention [14]. LODD, along with its simplified version optimized noise-filtration dynamic decoupling (OFDD) [15], gener- ates the decoupling sequence by directly optimizing the decoherence function using numeric methods as well as experimental feedback. It has been shown to be able to suppress decoherence effect by orders of magnitude over 

PACS numbers: 

## I. INTRODUCTION 

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UDD for certain noise spectrum, especially for the one with a high frequency part and sharp high-energy (UV) cutoff. 

However, in spite of the great experimental success, an- alytical results about the LODD sequence is insufficient. Until recently S. Pasini and G. S. Uhrig has made an analytical progress in optimizing the decoherence func- tion for power law spectrum (PLODD) [13]. The power law spectrum ω α for α < 1 without UV cutoff is consid- ered. They minimize the decoherence function through expanding the function and separating, canceling diver- gences from the relevant terms and solving variation problems. Inspired by Pasini’s work, we try to analyze the LODD problem with respect to the ohmic spectrum S ( ω ) ∼ ω and a sharp UV cutoff. Ohmic noise is the ma- jor decoherence source often found in a qubit’s environ- ment, for example, the semiconducting quantum dot [16] and superconducting qubit [17]. Optimal performance pulse sequence is found analytically which entirely dif- fers from the UDD sequence in such environment. We call this kind of optimal sequence HLODD (LODD for ohmic spectrum) for short. 

We organize this paper as follows. In the second sec- tion we propose the optimization problem of the deco- herence function. In section III, we derive the analytical equations for the optimal pulse sequence. In the follow- ing section, we run a simulation to verify our results. Conclusions are put in section V. 

## II. OPTIMIZATION OF THE DECOHERENCE FUNCTION 

Given a two-level quantum system, when the environ- mental noise behaves quantum-mechanically, we use the long-established spin-boson model with pure dephasing 

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