Robust quantum gates for stochastic time-varying noise

TL;DR

This paper presents a systematic method for designing robust quantum gates that can withstand both low- and high-frequency stochastic time-varying noise, advancing the development of fault-tolerant quantum computing.

Contribution

The authors develop a versatile approach to generate smooth, noise-resilient pulses for quantum gates, applicable across various system and noise models.

Findings

Method effectively suppresses noise across frequency spectrum

Produces single smooth pulses for robust gate operations

Outperforms filter-transfer-function method in simulations

Abstract

How to effectively construct robust quantum gates for time-varying noise is a very important but still outstanding problem. Here we develop a systematic method to find pulses for quantum gate operations robust against both low- and high-frequency (comparable to the qubit transition frequency) stochastic time-varying noise. Our approach, taking into account the noise properties of quantum computing systems, can output single smooth pulses in the presence of multi-sources of noise. Furthermore, our method can be applied to different system models and noise models, and will make essential steps toward constructing high-fidelity and robust quantum gates for fault-tolerant quantum computation. Finally, we discuss and compare the gate operation performance by our method with that by the filter-transfer-function method.

Figures

Equations

$$\mathcal{I}\equiv 1-\frac{1}{4^n}{\left|\mathrm{Tr}\left[U_T^{†}U(t_f)\right]\right|}^2,$$

$$\mathcal{I}$$

$$J_1$$

$$J_2$$

$$⟨\mathcal{I}⟩=J_1+⟨J_2⟩+⟨ϵ⟩+⟨\mathcal{O}({\tilde{\mathcal{H}}}_N^m,m\ge 3)⟩.$$

$$⟨J_2⟩$$

$$C_{OU}(t_1,t_2)={\sigma }_{OU}^2\exp \left(-{\gamma }_{OU}\left|t_1-t_2\right|\right)$$

$$S_{OU}(\omega )=\frac{2{\sigma }_{OU}^2{\gamma }_{OU}}{({\gamma }_{OU}^2+{\omega }^2)}.$$

$${\mathcal{H}}_I(t)={\omega }_0\frac{Z}{2}+{\Omega }_X(t)\frac{X}{2},$$

$${\mathcal{H}}_N(t)={\beta }_Z(t){\omega }_0\frac{Z}{2}+{\beta }_X(t){\Omega }_X(t)\frac{X}{2}.$$

$${\Omega }_X(t)=\sum _{k=1}^{k_{\max }}{a}_k\sin \left(m_k\pi \frac{t}{t_f}\right),$$

$${\mathcal{H}}_I(t)={\omega }_0\frac{Z_1}{2}+{\Omega }_{X_1}(t)\frac{X_1}{2}+{\omega }_0\frac{Z_2}{2}+{\Omega }_{X_2}(t)\frac{X_2}{2}+J(t)\frac{Z_1{Z}_2}{2},$$

$$⟨J_2⟩$$

$$S(\omega )={\sigma }^2\left[\frac{2\gamma }{{\gamma }^2+{\omega }^2}+\frac{\gamma }{{\gamma }^2+(5\gamma -\omega {)}^2}+\frac{\gamma }{{\gamma }^2+(5\gamma +\omega {)}^2}\right]$$

$$\mathcal{I}=J_1$$

$$-\frac{2}{4^n}\mathrm{Re}\{\mathrm{Tr}[U_T^{†}{U}_I(t_f){]}^{\star }\cdot \mathrm{Tr}[U_T^{†}{U}_I(t_f)\cdot ({\Psi }_1+{\Psi }_2+\cdots )]\}$$

$$-\frac{1}{4^n}{\left|\mathrm{Tr}[U_T^{†}{U}_I(t_f)\cdot ({\Psi }_1+{\Psi }_2+\cdots )]\right|}^2.$$

$$U_I(t_f)=e^{i\varphi }{U}_T(I+U_{ϵ}).$$

$$\mathcal{I}$$

$$ϵ(U_{ϵ},{\Psi }_j)=$$

$$-\frac{1}{2^{n-1}}\mathrm{Re}\{\mathrm{Tr}[U_{ϵ}({\Psi }_1+{\Psi }_2+\cdots )]\}$$

$$-\frac{2}{4^n}\mathrm{Re}\{\mathrm{Tr}[U_{ϵ}{]}^{\star }\cdot \mathrm{Tr}[{\Psi }_1+{\Psi }_2+\cdots ]\}$$

$$-\frac{2}{4^n}\mathrm{Re}\{\mathrm{Tr}[U_{ϵ}({\Psi }_1+{\Psi }_2+\cdots )]\cdot \mathrm{Tr}{\left[{\Psi }_1+{\Psi }_2+\cdots \right]}^{\star }\}$$

$$-\frac{2}{4^n}\mathrm{Re}\{\mathrm{Tr}[U_{ϵ}{]}^{\star }\cdot \mathrm{Tr}[U_{ϵ}({\Psi }_1+{\Psi }_2+\cdots )]\}$$

$$-\frac{1}{4^n}{\left|\mathrm{Tr}[U_{ϵ}({\Psi }_1+{\Psi }_2+\cdots )]\right|}^2,$$

$$J_3$$

$$J_4$$

$${\Psi }_q=(-i{)}^q{\int }_0^{t_f}d{t}_1{\int }_0^{t_1}d{t}_2\cdots {\int }_0^{t_{q-1}}d{t}_q{\tilde{\mathcal{H}}}_N(t_1){\tilde{\mathcal{H}}}_N(t_2)\cdots {\tilde{\mathcal{H}}}_N(t_q)$$

$${\Psi }_q$$

$$\times \{{\beta }_Z(t_1){\beta }_Z(t_2)\cdots {\beta }_Z(t_q)\}\{{\bar{R}}_Z(t_1){\bar{R}}_Z(t_2)\cdots {\bar{R}}_Z(t_q)\}.$$

Full text

I Introduction

To realize practical quantum computation, a set of high-fidelity universal quantum gates robust against noise in the qubit system is prerequisite. Constructing control pulses to operate quantum gates which meet this requirement is an important and timely issue. Quantum gates in open quantum systems have been investigated by various methods such as dynamical decoupling methods Khodjasteh and Viola (2009a); *KhodjastehViola2009a; *KhodjastehLidarViola2010; *KhodjastehBluhmViola2012; West et al. (2010); Souza et al. (2012); Fanchini et al. (2007); *FanchiniNapolitanoCiakmakEtAl2015; Chaudhry and Gong (2012); Xu et al. (2012) and optimal control methods Clausen et al. (2010); *ClausenBenskyKurizki2012; Wenin and Pötz (2008); Rebentrost et al. (2009); Hwang and Goan (2012); *TaiLinGoan2014; *HuangGoan2014; *ChouHuangGoan2015. For classical noise, there are many robust control methods such as composite pulses Tycko (1983); Levitt (1986); Wimperis (1994); Cummins et al. (2003); Brown et al. (2004); Vandersypen and Chuang (2005); Ichikawa et al. (2011); *BandoIchikawaKondoEtAl2013; Ichikawa et al. (2013); Merrill and Brown (2014), soft uniaxial positive control for orthogonal drift error (SUPCODE) Wang et al. (2012); Kestner et al. (2013); Wang et al. (2014a, b); Rong et al. (2014); Yang and Wang (2016), sampling-based learning control method Chen et al. (2014); Dong et al. (2015, 2016), inhomogeneous control methods Li and Khaneja (2006); Lee et al. (2013), analytical method Barnes et al. (2015), single-shot pulse method Daems et al. (2013), optimal control methods Kosut et al. (2013); Anderson et al. (2015); Gordon et al. (2008), invariant-based inverse engineering method Ruschhaupt et al. (2012); Lu et al. (2013), and filter-transfer-function (FTF) methods Green et al. (2013); Soare et al. (2014); Ball and Biercuk (2015); Paz-Silva and Viola (2014). However, in most of these methods Tycko (1983); Levitt (1986); Wimperis (1994); Cummins et al. (2003); Brown et al. (2004); Vandersypen and Chuang (2005); Ichikawa et al. (2011); *BandoIchikawaKondoEtAl2013; Ichikawa et al. (2013); Merrill and Brown (2014); Wang et al. (2012); Kestner et al. (2013); Wang et al. (2014a, b); Rong et al. (2014); Yang and Wang (2016); Chen et al. (2014); Dong et al. (2015, 2016); Li and Khaneja (2006); Lee et al. (2013); Barnes et al. (2015); Daems et al. (2013); Kosut et al. (2013); Anderson et al. (2015), noise is assumed to be quasi-static, i.e., is time-independent within the gate operation time but can vary between different gates. We call these robust control strategies the quasi-static-noise (QSN) methods. But this QSN assumption is not always valid Oliver (2014). The robust performance of control pulses obtained by the QSN methods under time-dependent noise (e.g., $1/f^{\alpha}$ noise with $\alpha\apprge 1$) Kabytayev et al. (2014); Wang et al. (2014a, b); Yang and Wang (2016) have been investigated, and it was found that they can still work well for relatively low-frequency non-Markovian noise.

Stochastic time-dependent noise is treated in the FTF method Green et al. (2013); Soare et al. (2014); Ball and Biercuk (2015) in which the area of the filter-transfer function in the frequency region, where the noise power spectral density (PSD) is non-negligible, is minimized. However, in this approach only the the filter-transfer function overlapping with the noise PSD in the preset frequency region is considered, but the detailed information of the distribution of the noise PSD is not included in the optimization cost function. Here we develop an optimal control method in time domain by choosing the ensemble average gate infidelity (error) as our cost function for optimization. As a result, the noise correlation function (CF) or equivalently the detailed noise PSD distribution appears naturally in our chosen optimization cost function. Therefore our method can have better robust performance against noise in a general case. The idea of our method is simple, and our method is not limited to particular system models, noise models, and noise CF’s. We demonstrate our robust control method for classical noise in this paper, but our method can be easily generalized to the case of quantum noise by replacing the ensemble average for classical noise with the trace over the degrees of freedom of the quantum noise (environment) Paz-Silva and Viola (2014). In other words, our method can be applied to systems with both classical noise and quantum noise present simultaneously.

II Ensemble average Infidelity and optimization method

We first introduce our robust control method here, and then compare it with the QSN method and the FTF method. We consider a total Hamiltonian $\mathcal{H}(t)=\mathcal{H}_{I}(t)+\mathcal{H}_{N}(t)$, where $\mathcal{H}_{I}(t)$ is the ideal system Hamiltonian and $\mathcal{H}_{N}(t)$ is the noise Hamiltonian. If a system is ideal, i.e., $\mathcal{H}_{N}(t)=0$, then its propagator is $U_{I}(t)=\mathcal{T}_{+}\exp[-i\int_{0}^{t}\mathcal{H}_{I}(t^{\prime})dt^{\prime}]$ (throughout this paper we set $\hbar=1$), where $\mathcal{T}_{+}$ is the time-ordering operator. However, in reality there may be many sources of noise present in the system, so $\mathcal{H}_{N}(t)=\sum_{j}\beta_{j}(t)H_{N_{j}}(t)$, where $\beta_{j}(t)$ is the strength of the $j$-th stochastic time-varying noise and $H_{N_{j}}(t)$ is the corresponding system coupling operator term. The propagator for a realistic system is then $U(t)=U_{I}(t)\cdot\mathcal{T}_{+}\exp[-i\int_{0}^{t}\tilde{\mathcal{H}}_{N}(t^{\prime})dt^{\prime}]$. Here $\tilde{\mathcal{H}}_{N}(t)=\Sigma_{j}\beta_{j}(t)R_{j}(t)$, is the noise Hamiltonian in the interaction picture transformed by $U_{I}(t)$ and $R_{j}(t)\equiv U_{I}^{\dagger}(t)H_{N_{j}}(t)U_{I}(t)$. Suppose that $U_{T}$ is our target gate and gate operation time is $t_{f}$. The gate infidelity (error) $\mathcal{I}$ for an $n$-qubit gate can be defined as

[TABLE]

where ${\rm Tr}$ denotes a trace over the $n$-qubit system state space. If noise strength is not too strong, we can expand the propagator $U(t_{f})$ in terms of $\tilde{\mathcal{H}}_{N}(t)$ by Dyson series Dyson (1949) into the form $U(t_{f})=U_{I}(t_{f})\cdot[I+\Psi_{1}+\Psi_{2}+\cdots]$, where the first two terms of $\Psi_{j}$ are $\Psi_{1}=-i\int_{0}^{t_{f}}\tilde{\mathcal{H}}_{N}(t^{\prime})dt^{\prime}$, and $\Psi_{2}=-\int_{0}^{t_{f}}dt_{1}\int_{0}^{t_{1}}dt_{2}\tilde{\mathcal{H}}_{N}(t_{1})\tilde{\mathcal{H}}_{N}(t_{2})$. Substituting the expanded $U(t_{f})$ into $\mathcal{I}$ in Eq. (1), the expanded infidelity $\mathcal{I}$ (see Appendix A) takes the form

[TABLE]

Here $J_{1}$ is the definition of gate infidelity for the ideal system, $J_{2}$ is the lowest-order contribution of the noise to the gate infidelity, $\epsilon$ (detailed form shown in Appendix A) denotes an extra contribution that is correlated to $J_{1}$ and the Dyson expansion terms $\Psi_{j}$, and $\mathcal{O}(\tilde{\mathcal{H}}_{N}^{m},m\geq 3)$ represents other higher-order terms excluding $\epsilon$. If noise strength is not too strong such that $|\Psi_{j+1}|\ll|\Psi_{j}|$, the extra contribution $\epsilon$ will become negligible when $J_{1}$ is getting small (see discussion in Appendix A). The symbol Re in Eq. (4) denotes taking the real part of the quantity it acts on. Because noise $\beta_{j}(t)$ is stochastic, we denote the ensemble average of the infidelity over the different noise realizations as

[TABLE]

Here

[TABLE]

where $C_{jk}(t_{1},t_{2})=\left\langle\beta_{j}(t_{1})\beta_{k}(t_{2})\right\rangle$ is the CF for noise $\beta_{j}(t_{1})$ and $\beta_{k}(t_{2})$. The first-order noise term proportional to ${\rm Re}[{\rm Tr}(\Psi_{1})]$ vanishes due to the fact that ${\rm Tr}(\Psi_{1})$ is purely imaginary rather than the assumption of $\left\langle\beta_{j}(t)\right\rangle=0$ (see Appendix A). If different sources of noise are independent, $C_{jk}(t_{1},t_{2})=0$ for $j\neq k$, and if noise Hamiltonian $\mathcal{H}_{N}(t)$ is traceless, the second term in Eq. (6) vanishes. The ideal Hamiltonian $\mathcal{H}_{I}(t)$ is a function of the control field $\Omega(t),$ that is $\mathcal{H}_{I}(t)=\mathcal{H}_{I}(\Omega(t))$, and the control field $\Omega(t)$ is chosen to be a function of a set of control parameters $[a_{1},a_{2},\cdots]$. Then $U_{I}(t)$ and each term of the ensemble average infidelity $\left\langle\mathcal{I}\right\rangle$ in Eq. (5) are also a function of the control parameter set $[a_{1},a_{2},\cdots]$. Our goal is to search the optimal parameter set $[a_{1},a_{2},\cdots]$ that minimizes the ensemble average infidelity $\left\langle\mathcal{I}\right\rangle$. If the noise strength or fluctuation is not large, then the dominant noise contribution to $\left\langle\mathcal{I}\right\rangle$ is from $\left\langle J_{2}\right\rangle$ as the higher order terms $\langle\mathcal{O}(\tilde{\mathcal{H}}_{N}^{m},m\geq 3)\rangle$ can be neglected (see Appendix B). $J_{1}$ can generally be made sufficiently small so that the extra term $\left\langle\epsilon\right\rangle$ in $\left\langle\mathcal{I}\right\rangle$ of Eq. (5) can be safely ignored. So we concentrate on the minimization of $\left\langle\mathcal{I}\right\rangle\cong J_{1}+\left\langle J_{2}\right\rangle$ for obtaining the optimal control parameter set. We will, however, use the full-order ensemble average infidelity $\left\langle\mathcal{I}\right\rangle$ (described later) to exam the performance of the optimal control parameter set found this way.

We use two-step optimization to achieve this goal. The first step is called the $J_{1}$ optimization in which $J_{1}$ is the cost function. The gate infidelities $J_{1}$ in an ideal unitary system with gate-operation-controllability and a sufficient number of control parameters can be made as low as one wishes, limited only by the machine precision of the computation. So using an ensemble of random control parameter sets as initial guesses, we obtain after the $J_{1}$ optimization an ensemble of optimized control parameters sets all with very low values of $J_{1}$. The second step is called the $J_{1}+\left\langle J_{2}\right\rangle$ optimization. We take $J_{1}+\left\langle J_{2}\right\rangle$ as a cost function and randomly choose some optimized control parameter sets in the first optimization step as initial guesses to run the optimal control algorithm. After the $J_{1}+\left\langle J_{2}\right\rangle$ optimization, we obtain an ensemble of control parameter sets with low values of $J_{1}+\left\langle J_{2}\right\rangle$, and then choose the lowest one as the optimal control parameter set. The purpose of using the two-step optimization is to improve optimization efficiency. If we run $J_{1}+\left\langle J_{2}\right\rangle$ optimization directly from an ensemble of random control parameter sets, we need more optimization iterations to achieve the goal, and the success rate is relatively low compared with the two-step optimization. Besides, the $J_{1}+\left\langle J_{2}\right\rangle$ optimization enables us to know separately the optimized values of $J_{1}$ and $\left\langle J_{2}\right\rangle$. When $\left\langle J_{2}\right\rangle$ can be minimized to a very small value as in the case of static or low-frequency noise, one has to use a small time step for simulation to make $J_{1}$ smaller than $\left\langle J_{2}\right\rangle$. However, for high-frequency noise, $\left\langle J_{2}\right\rangle$ is hard to be minimized to a very small value, and one can instead choose a suitable larger time step to make $J_{1}$ just one or two orders of magnitude smaller than $\left\langle J_{2}\right\rangle$, saving substantially the optimization time especially for multi-qubits and multi-sources of noise. We use the gradient-free and model-free Nelder-Mead (NM) algorithm Nelder and Mead (1965) in both the $J_{1}$ and $J_{1}+\left\langle J_{2}\right\rangle$ optimization steps. However, the NM algorithm may be stuck in local traps in the $J_{1}+\left\langle J_{2}\right\rangle$ parameter space topography. To overcome this problem, we use repeating-NM algorithm in the $J_{1}+\left\langle J_{2}\right\rangle$ optimization step. The control parameter set from the first $J_{1}+\left\langle J_{2}\right\rangle$ optimization may lie in a local trap. Therefore, we add random fluctuations to this control parameter set and try to pull it out of the trap. Then we use this shifted control parameter set as an initial guess to run the second $J_{1}+\left\langle J_{2}\right\rangle$ optimization. We repeat the same procedure many times until the values of $J_{1}+\left\langle J_{2}\right\rangle$ can not be improved (reduced) anymore, and then output the corresponding control parameter set. Our optimization method employing the gradient-free and model-free NM algorithm is quite general, capable of dealing with different forms or structures of the ideal system Hamiltonian $\mathcal{H}_{I}(t)$, control field $\Omega(t)$, noise Hamiltonian $\mathcal{H}_{N}(t)$, and noise CF $C_{jk}(t_{1},t_{2})$ for a few qubit systems.

The ensemble infidelity $\left\langle\mathcal{I}\right\rangle$ we use to show the robust performance of the gate as the noise strength varies is calculated using the full evolution of the total system-noise Hamiltonian and many realizations of the noise without any other approximation. By inputting the optimal control parameter set obtained by the optimization strategy into the total system-noise Hamiltonian $\mathcal{H}(t)=\mathcal{H}_{I}(t)+\mathcal{H}_{N}(t)$ to obtain numerically the full propagator for a single noise realization, we can calculate the gate infidelity $\mathcal{I}$ using Eq. (1) for the noise realization. The procedure is repeated for many different noise realizations. Then we take an ensemble average of the infidelities over the different noise realizations to obtain $\left\langle\mathcal{I}\right\rangle$.

In principle, we could deal with any given form of the noise correlation function (or equivalently the noise PSD) to inset into Eq. (6) for the $J_{1}+\left\langle J_{2}\right\rangle$ optimization. But as a particular example, we choose the Ornstein-Uhlenbeck (OU) process $\beta_{OU}(t)$ to simulate stochastic time-varying noise Finch and Yt (2004). Studying the influence of and developing robust strategies against time-dependent noise is an important subject of research in quantum control problems both theoretically and experimentally Soare et al. (2014); Ball and Biercuk (2015); Green et al. (2013); Oliver (2014); Kabytayev et al. (2014). If the initial noise $\beta_{OU}(t=0)$ is a normal distribution with zero mean and with standard deviation $\sigma_{OU}$, then the noise CF of the OU process $\beta_{OU}(t)$ is

[TABLE]

with the noise correlation time $\tau\sim(1/\gamma_{OU})$, and the corresponding noise PSD is Lorentzian

[TABLE]

Lorentzian PSDs of spin noise resulting in a fluctuating magnetic field at the location of the qubits in InGaAs semiconductor quantum dots have been measured experimentally Kuhlmann et al. (2013); Li et al. (2012). Generally, small $\gamma_{OU}$ corresponds to low-frequency or quasi-static noise; large $\gamma_{OU}$ corresponds to high-frequency noise. The noise $\beta_{OU}(t)$ can be simulated through the formula $\beta_{OU}(t+dt)=\left(1-\gamma_{OU}dt\right)\beta_{OU}(t)+\sigma_{OU}\sqrt{2\gamma_{OU}}dW(t)$, where $W(t)$ is a Wiener process Finch and Yt (2004). Figures 1(c), (d), and (e) show the different realizations of the noise $\beta_{OU}(t)$ with $\sigma_{OU}=10^{-3}$ for different values of $\gamma_{OU}/\omega_{0}=10^{-7}$, $10^{-3}$, and $10^{-1}$, respectively, where $\omega_{0}$ is the typical system frequency. We note here that the particular choice of the OU noise should by no means diminish the value of our work and the power of our method. Any given or experimentally measured well-behaved noise PSD or noise correlation function can be dealt with. We will demonstrate later that our method can also work effectively for another form of noise PSD different from that of the OU noise when we compare the performance of our method with the FTF method. The reason to use the OU noise in the system-noise Hamiltonian here is that it is relatively easy to simulate its stochastic noise realizations in the time domain. Therefore, we can calculate the full-order ensemble average infidelity $\left\langle\mathcal{I}\right\rangle$ to show that our $J_{1}+\left\langle J_{2}\right\rangle$ optimization that minimizes the second-order noise contribution to the average infidelity $\left\langle\mathcal{I}\right\rangle$ can indeed work rather well for not too strong a noise fluctuation.

III Results and demonstrations

III.1 Comparison with quasi-static-noise method

III.1.1 Single-qubit gates

We demonstrate as an example the implementation of single-qubit gates in the presence of time-varying noise using our method. The ideal system Hamiltonian for the qubit is

[TABLE]

where $X$ and $Z$ stand for the Pauli matrices, $\omega_{0}$ is the qubit transition frequency, and $\Omega_{X}(t)$ is the control field in the $X$ term. The noise Hamiltonian is written as

[TABLE]

We call $\beta_{Z}(t)$ the $Z$-noise and $\beta_{X}(t)$ the $X$-noise, and assume that they are independent OU noises with CF’s $C_{ZZ}(t_{1},t_{2})=\sigma_{ZZ}^{2}\exp\left(-\gamma_{ZZ}\left|t_{1}-t_{2}\right|\right)$ and $C_{XX}(t_{1},t_{2})=\sigma_{XX}^{2}\exp\left(-\gamma_{XX}\left|t_{1}-t_{2}\right|\right)$ as the form of Eq. (7). We choose the control pulse as a composite sine pulse expressed as

[TABLE]

where the set of the strengths of the single sine pulses is the control parameter set $[a_{k}]=[a_{1},a_{2},\cdots,a_{k_{{\rm max}}}]$ and $\{m_{k}\}$ is a set of integers, chosen depending on the nature of the system Hamiltonians and the target gates as well as the properties of the noise models. For each control pulse, we choose the number of control parameters $k_{{\rm max}}$ to range from $8$ to $20$ in our calculations.

We define below three optimization strategies, namely, the ideal-gate (IDG) strategy, quasi-static-noise (QSN) strategy, and time-varying-noise (TVN) strategy. The IDG strategy is to perform the first-step optimization ($J_{1}$ optimization) only and to show the performance of an ideal gate pulse in the presence of noise. The TVN strategy is our proposed method described earlier above, in which the actual $\gamma_{ZZ}$ and $\gamma_{XX}$ values are used in the noise CF’s of the cost function $\left\langle J_{2}\right\rangle$ for the second-step optimization. The QSN strategy uses the same optimization procedure as the TVN strategy, but with $\gamma_{ZZ}=\gamma_{XX}=0$ for the noise CF’s in the cost function $\left\langle J_{2}\right\rangle$. Thus it is regarded to represent the QSN methods. We choose the gate operation time $t_{f}=20/\omega_{0}$. After the optimizations of Hadamard gate, we plot the corresponding $J_{1}+\left\langle J_{2}\right\rangle$ values obtained from these three strategies versus $\gamma_{ZZ}$ in Fig. 1(a) for the $Z$-noise and versus $\gamma_{XX}$ in Fig. 1(b) for the $X$-noise. For low-frequency (quasi-static) noise ($\gamma_{ZZ}=\gamma_{XX}=10^{-7}\omega_{0}$), the performance of the TVN strategy and the QSN strategy are about the same but they are several orders of magnitude better in infidelity $J_{1}+\left\langle J_{2}\right\rangle$ value than the IDG strategy which does not take the noise into account at all. As the noise goes from the low frequency to high frequency ($\gamma_{ZZ}=\gamma_{XX}=10^{-1}\omega_{0}$), the TVN strategy taking account of the time-varying noise information in the cost function gets better and better (from a factor-level to an order-of-magnitude-level) improvement in $J_{1}+\left\langle J_{2}\right\rangle$ values than the QSN strategy in which noise is assumed to be quasi-static. In addition to the Hadamard gate, we perform calculations for other quantum gates, namely the phase gate, $\pi/8$ gate and controlled-NOT (CNOT) gate, in the fault-tolerant universal set in terms of which any unitary operation can be expressed to arbitrary accuracy. The $J_{1}+\left\langle J_{2}\right\rangle$ values versus $\gamma_{ZZ}$ and versus $\gamma_{XX}$ obtained from the three strategies are shown in Figs. 2(a) and (b), respectively, for the phase gate and in Figs. 2(c) and (d), respectively, for the $\pi/8$ gate. Their performances are similar to those in Fig. 1(a) and (b) of the Hadamard gate. The optimization results for the two-qubit CNOT gate are presented in Sec. III.1.2.

Next, we take the optimal control parameter sets of Hadamard gate from these three strategies to show their robust performance against $Z$-noise, $X$-noise, and $Z$-&-$X$-noise at a low frequency ($\gamma_{ZZ}=\gamma_{XX}=10^{-7}\omega_{0}$) in Figs. 3(a), (b), and (c) and at a high frequency ($\gamma_{ZZ}=\gamma_{XX}=10^{-1}\omega_{0}$) in Figs. 4(a), (b), and (c). For low-frequency noises and for small noise strength ($\sigma_{XX}<10^{-1}$, $\sigma_{ZZ}<10^{-1}$), one can see from Fig. 3 that the full-order ensemble average infidelity $\left\langle\mathcal{I}\right\rangle$ scales for the IDG strategy as the second power of the noise standard deviation ($\sigma_{ZZ}$, $\sigma_{XX}$), but scales for the TVN and QSN strategies as the fourth power. This implies that $\left\langle\mathcal{I}\right\rangle\cong\left\langle J_{2}\right\rangle$ for the IDG strategy, but the TVN and QSN strategies can nullify the contribution from $\left\langle J_{2}\right\rangle$ for the low-frequency (quasi-static) noise and the dominant contribution in $\left\langle\mathcal{I}\right\rangle$ comes from the next higher-order term, i.e., $\langle\mathcal{I}\rangle\cong\langle\mathcal{O}(\tilde{\mathcal{H}}_{N}^{4})\rangle$. In this case, our method, the TVN strategy, still performs slightly better than the QSN strategy. For gate error (infidelity) less than the error threshold of $10^{-2}$ of surface codes Fowler et al. (2012) required for fault-tolerant quantum computation (FTQC), the Hadamard gate of TVN strategy can be robust to $\sigma_{ZZ}\sim 30\%$ for low-frequency $Z$-noise (i.e., against noise fluctuation with standard deviation up to about $30\%$ of $\omega_{0}/2$), robust to $\sigma_{XX}\sim 20\%$ for the $X$-noise [i.e., against noise fluctuation with standard deviation up to about $20\%$ of $\Omega_{X}(t)/2$], and robust to $\sigma_{ZZ}=\sigma_{XX}\sim 10\%$ for $Z$-$\&$-$X$-noise as shown in Figs. 3(a), (b), and (c), respectively. The corresponding optimal control pulses of the TVN strategy are shown in Figs. 3(d), (e), and (f), respectively.

For high-frequency noise shown in Fig. 4, the full-order ensemble average infidelity $\left\langle\mathcal{I}\right\rangle$ scales as the second power of noise standard deviation ($\sigma_{ZZ}$, $\sigma_{XX}$) for all three strategies and noises. This indicates that for high frequency noise $\left\langle J_{2}\right\rangle$ is not nullified completely, and is only minimized. Even in this case, the TVN strategy still has over two orders of magnitude improvement in $\left\langle\mathcal{I}\right\rangle$ compared with the IDG strategy, and over one order of magnitude improvement compared with the QSN strategy for the $Z$-noise at small noise strengths as shown in Fig. 4(a). For $\left\langle\mathcal{I}\right\rangle\lesssim 10^{-2}$ less than the FTQC error threshold of the surface codes, the Hadamard gate implemented by our optimal control pulse shown in Fig. 4(d) can be robust to $\sigma_{ZZ}\sim 20\%$ for the $Z$-noise. On the other hand, for the high-frequency $X$-noise, $\left\langle\mathcal{I}\right\rangle$ obtained by the QSN strategy has even slightly higher values than those by the IDG strategy. The improvement in $\left\langle\mathcal{I}\right\rangle$ by the TVN strategy over the other two strategies is less than one order of magnitude. To improve the gate performance, we increase the degrees of freedom for optimization by adding a control term $\Omega_{Y}(t)Y/2$ and its accompanying $Y$-noise term $\beta_{Y}(t)\Omega_{Y}(t)Y/2$ in the Hamiltonian. We choose, for simplicity, $\gamma_{YY}=\gamma_{XX}$ and $\sigma_{YY}=\sigma_{XX}$, and use the same optimal procedure as the TVN strategy. The improvement in $\left\langle\mathcal{I}\right\rangle$ of the TVN strategy with an additional $Y$ control as compared with the TVN strategy is over a half order of magnitude. As a result, the Hadamard gate with the optimal control pulses of the TVN strategy with an additional $Y$ control shown in Fig. 4(e) can be robust to $\sigma_{XX}=\sigma_{YY}\sim 20\%$ for $\left\langle\mathcal{I}\right\rangle\lesssim 10^{-2}$. Note that the optimization algorithm seems to find control pulses with stronger strengths to suppress the $Z$-noise, but searches weaker control pulses to minimize the $X$-noise cost function since the system coupling operator term of the $X$-noise is proportional to the control field $\Omega_{X}(t)$ in our noise model. So for the case with the $Z$-noise and $X$-noise simultaneously present, there is a trade-off in the control pulse strength for the cost function optimization between the $Z$-noise and the $X$-noise. Consequently, the ensemble infidelity of the $Z$-$\&$-$X$ noise does not reach a low value as the case with only the $Z$-noise or the $X$-noise. Thus one can see from Fig. 4(c) that the improvement in $\left\langle\mathcal{I}\right\rangle$ of the TVN strategy over the IDG strategy is just near one order of magnitude, and only a half order as compared with the QSN strategy. Similar trade-off also takes place for the TVN strategy with additional $Y$ control although it performs slightly better than the TVN strategy with only the $\Omega_{X}(t)$ control field. Nevertheless, the Hadamard gate implemented with the optimal pulse obtained by the TVN strategy with additional $Y$ control shown in Fig. 4(f) can be still robust to $\sigma_{ZZ}=\sigma_{XX}=\sigma_{YY}\sim 6\%$ for $\left\langle\mathcal{I}\right\rangle\lesssim 10^{-2}$.

III.1.2 Two-qubit gates

Next, we demonstrate that our method can find control pulses for high-fidelity two-qubit CNOT gate operations in the presence of multi-sources of high-frequency noise. The two-qubit Hamiltonian is chosen ass

[TABLE]

where $Z_{j}$ and $X_{j}$ denote the Pauli’s matrix operators for qubit $j$, $\Omega_{X_{j}}(t)$ is the control field applied to qubit $j$ and $J(t)$ is the two-qubit coupling strength. We assume OU noise can be present in each of the five terms, and $\sigma_{ZZ1}$, $\sigma_{ZZ2}$, $\sigma_{XX1}$, $\sigma_{XX2}$, and $\sigma_{JJ}$ are respectively the corresponding standard deviation $\sigma_{OU}$, and $\gamma_{ZZ1}$, $\gamma_{ZZ2}$, $\gamma_{XX1}$, $\gamma_{XX2}$, and $\gamma_{JJ}$ are respectively the corresponding $\gamma_{OU}$. We choose the control fields $\Omega_{X_{1}}(t)$ and $\Omega_{X_{2}}(t)$ as composite sine pulses, and the two-qubit control $J(t)$ as a composite sine pulse with a constant shift.

The robust performance of the CNOT gate using the three strategies for high frequency ($\gamma_{ZZ1}=\gamma_{ZZ2}=\gamma_{XX1}=\gamma_{XX2}=\gamma_{JJ}=10^{-1}\omega_{0})$ $Z$-noise, $X$-&-$J$-noise, and $Z$-&-$X$-&-$J$-noise are shown in Figs. 5(a), (b), and (c), respectively. The corresponding optimal control pulses of the TVN strategy for operation time $t_{f}=100/\omega_{0}$ are shown in Figs. 5(d), (e), and (f), respectively. For $\omega_{0}t_{f}=100$, our method (the TVN strategy) for the case of the $Z$-noise and the case of the $X$-&-$J$-noise has one-order of magnitude improvement in $\left\langle\mathcal{I}\right\rangle$ values as compared with the QSN strategy for small noise strength, but only half-order improvement for the case of the $Z$-&-$X$-&-$J$-noise. This is because for the case of the $Z$-&-$X$-&-$J$-noise, there is a trade-off in the control pulse strength for the cost function optimization between the $Z$-noise and the $X$-&-$J$-noise, similar to that in the single-qubit case. The robust performance can be improved by reducing gate operation time $t_{f}$, for example, from $t_{f}=100/\omega_{0}$ to $t_{f}=20/\omega_{0}$, to decrease the duration of the influence of the noises. This can be seen from the purple pentagrams in Figs. 5(a) and (c). For the case of the $X$-&-$J$-noise in Fig. 5(b), only slight improvement is observed for the $t_{f}=20/\omega_{0}$ case because when the operation time decreases, it is hard to make the strengths of the control fields $\Omega_{X}^{j}(t)$ and $J(t)$ all small as in the $t_{f}=100/\omega_{0}$ case. For high-frequency noise and for FTQC error threshold $\left\langle\mathcal{I}\right\rangle\leq 10^{-2}$ of the surface codes, the CNOT gate with operation time $t_{f}=20/\omega_{0}$ can be robust to $\sigma_{ZZ1}=\sigma_{ZZ2}\sim 10\%$ for the $Z$-noise, robust to $\sigma_{XX1}=\sigma_{XX2}=\sigma_{JJ}\sim 10\%$ for the $X$-&-$J$-noise, and robust to $\sigma_{ZZ1}=\sigma_{ZZ2}=\sigma_{XX1}=\sigma_{XX2}=\sigma_{JJ}\sim 3\%$ for the $Z$-&-$X$-&-$J$-noise by our method.

We describe briefly about the computational resources and computation time in our calculations. For the case of the $Z$-&-$X$-&-$J$-noise, we use 40 control parameters in a parameter set to run the two-step optimization for the two-qubit CNOT gate, and choose 100 initial random guesses of the parameter sets for the first-step optimization and 10 parameter sets obtained in the first-step optimization as initial guesses for the second step optimization. We use a total of 60 2GHz-CPU cores and it takes about 2 days to obtain the control pulses and robust performance calculations of Fig. 5(c). These resources and time spent to construct the robust high-fidelity CNOT gates against five sources of high-frequency noise are quite acceptable.

III.2 Comparison with filter-transfer-function method

In this subsection, we compare our method with the FTF method Soare et al. (2014); Ball and Biercuk (2015); Green et al. (2013). The cost function $\left\langle J_{2}\right\rangle$ in Eq. (6) can be transformed to frequency domain as

[TABLE]

where $S_{j}(\omega)$ is the noise PSD for the $j$-th noise, and $F_{j}(\omega)$ is the corresponding filter-transfer function. The cost function of the $j$-th noise for optimization in the FTF method is defined as Green et al. (2013); Soare et al. (2014); Ball and Biercuk (2015) $A_{j}\equiv\int_{\omega_{L}}^{\omega_{c}}F_{j}(\omega)d\omega$. The region $\left[\omega_{L},\omega_{c}\right]$ of the integration of the cost function $A_{j}$ is determined by the non-negligible region of the noise PSD. In order to compare with our method, we use the same form of control pulse, the same number of control parameters, and the same optimal procedure except changing for the FTF method the cost function from $J_{1}+\left\langle J_{2}\right\rangle$ to $J_{1}+A_{j}$ in the second step of the two-step optimization. We call this procedure the FTF strategy. Then we apply the IDG strategy, FTF strategy, and TVN strategy to find high-fidelity Hadamard gate for one-qubit system with single $Z$-noise. To demonstrate the advantage of our method over the FTF method, we choose the noise PSD for the $Z$-noise to contain a high-frequency distribution as

[TABLE]

that has two peaks at $\omega=0$ and $\omega=5\gamma$. As the value of $\gamma$ increases, the dominant distribution associated with the second peak of the PSD $S(\omega)$ moves to a high frequency region in which the FTF method may not work very effectively. We demonstrate that our method including the detailed noise PSD distribution in the cost function can still in this case suppress the gate error coming from $S(\omega)$, a different PSD from that of the OU noise model used previously. The lower limit $\omega_{L}$ of the integral of the cost function $A_{z}$ for the FTF strategy is chosen to be zero, and the upper limit $\omega_{c}$ is chosen to be $1\omega_{0}$, $2\omega_{0}$, and $3\omega_{0}$ to enclose the dominant distribution of $S(\omega)$ [see Figs. 6(a), (b), and (c)] for $\gamma=0.1\omega_{0}$, $0.3\omega_{0}$, and $0.5\omega_{0}$, respectively. For the single $Z$-noise considered here, the infidelity from Eq. (13) is $\left\langle J_{2}\right\rangle=\int_{-\infty}^{\infty}d\omega S(\omega)\left[F_{z}(\omega)/(2\pi\omega^{2})\right]$. The improvement of $\left\langle J_{2}\right\rangle$ can be analyzed through the overlap of $S(\omega)$ with $\left[F_{z}(\omega)/(2\pi\omega^{2})\right]$ Gordon et al. (2008). If the control pulses can make $\left[F_{z}(\omega)/(2\pi\omega^{2})\right]$ small in the dominant distribution region of $S(\omega)$, then $\left\langle J_{2}\right\rangle$ can be significantly improved (reduced). We plot $\left[F_{z}(\omega)/(2\pi\omega^{2})\right]$ evaluated by the optimal control parameter sets obtained from the above three strategies for three different values of $\gamma=0.1\omega_{0}$, $0.3\omega_{0}$, and $0.5\omega_{0}$ of $S(\omega)$ in Figs. 6(d), (e), and (f), respectively. The corresponding $\left\langle J_{2}\right\rangle$ values are shown in Fig. 6(g). By taking the case of $\gamma=0.3\omega_{0}$ as an example, the function $\left[F_{z}(\omega)/(2\pi\omega^{2})\right]$ of the TVN strategy shows apparent drops near the two peaks of the noise PSD at $\omega=0$ and $\omega=1.5\omega_{0}$, but the function for the FTF strategy and the IDG strategy does not. Thus, about one order of magnitude improvement in $\left\langle J_{2}\right\rangle$ of the TVN strategy over the other two strategies is observed. In short, as the range of dominant distribution of the PSD enlarges [e.g., from Fig. 6(a) to Fig. 6(c)], the TVN strategy, including the detailed noise information (CF) in the optimization cost function Gordon et al. (2008), can suppress the dominant infidelity contribution more effectively than the FTF strategy. Furthermore, the concatenation method is used to construct control pulses against two different non-commuting noises in the FTF method Soare et al. (2014); Ball and Biercuk (2015). But using the concatenation method to deal with the case of multi-controls, multi-sources of noise, and multi-qubits may be very complicated. On the other hand, our method can find robust control pulses for high-fidelity CNOT gates that involve three control knobs and up to five sources of high-frequency noise as demonstrated in Fig. 5.

IV Conclusion

To conclude, our two-step optimization method can provide robust control pulses of high-fidelity quantum gates for stochastic time-varying noise. Besides, our method is quite general, and can be applied to different system models, noise models, and noise CF’s (PSD’s). If the system is very clean, i.e., quantum noise is very weak, our method presented in this paper can be used directly to find control pulses for high-fidelity gates in the presence of time-varying classical noises. For example, decoherence (dephasing) time is rather long in quantum-dot electron spin qubits in purified silicon Veldhorst et al. (2014, 2015) as compared to those in GaAs Petta et al. (2005); Koppens et al. (2006), that is, quantum noise coming from the coupling to the environmental spins is very weak, and the dominant gate error is due to the electric classical control noise Veldhorst et al. (2015). We will present the results for quantum-dot electron spin qubits in purified silicon elsewhere. Our method will make essential steps toward constructing high-fidelity and robust quantum gates for FTQC in realistic quantum computing systems.

Acknowledgements.

We acknowledge support from the the Ministry of Science and Technology of Taiwan under Grant No. 103-2112-M-002-003-MY3, from the National Taiwan University under Grant Nos. 105R891402 and 105R104021, and from the thematic group program of the National Center for Theoretical Sciences, Taiwan.

Appendix A Derivation of Eqs. (2)-(4) in the main text

We present the derivation of Eqs. (2)-(4) in the main text and discuss the role of the extra term $\epsilon$ in Eq. (2). Substituting the total system propagator in Dyson expansion $U(t_{f})=U_{I}(t_{f})\cdot(I+\Psi_{1}+\Psi_{2}+\cdots)$ into the infidelity definition $\mathcal{I}$ of Eq. (1) in the main text, we obtain

[TABLE]

The first term $J_{1}$ on the right hand side of the equal sign of Eq. (15) is the gate infidelity for the ideal system defined in Eq. (3). Then we define the error shift matrix $U_{\epsilon}$ of the ideal propagator $U_{I}(t_{f})$ at time $t_{f}$ from the target gate $U_{T}$ up to a global phase $\phi$ as

[TABLE]

Note that when the gate infidelity $J_{1}$ for the ideal system is made small, the matrix elements of $U_{\epsilon}$ also become small. Substituting the expression of $U_{I}(t_{f})$ of Eq. (16) back to Eq. (15), we obtain

[TABLE]

where $J_{2}$ is defined in Eq. (4),

[TABLE]

and $\mathcal{O}(\tilde{\mathcal{H}}_{N}^{m},m\geq 3)$ denotes other higher-order terms without containing $U_{\epsilon}$. The first-order noise term, $-{\rm Re}[{\rm Tr}(\Psi_{1})]/2^{n-1}$, in Eq. (17) actually vanishes. Because the noise Hamiltonian $\mathcal{H}_{N}$ is Hermitian [with $\beta_{j}(t)$ being real], ${\rm Tr}[\mathcal{H}_{N}(t^{\prime})]$ is a real number. Thus the first-order term proportional to the real part of ${\rm Re}[{\rm Tr}(\Psi_{1})]$, with ${\rm Tr}(\Psi_{1})=-i\int_{0}^{t_{f}}{\rm Tr}[\mathcal{H}_{N}(t^{\prime})]dt^{\prime}$, vanishes. This result of no first-order noise contribution in $\mathcal{I}$ is similar to that in Ref.Daems et al. (2013). This is also the reason why there is no first-order noise contribution in ensemble average $\langle\mathcal{I}\rangle$ of Eq. (5). Equations (2)-(4) in the main text can then be easily obtained from Eq. (17) with the identification of $\epsilon=\epsilon(U_{\epsilon},\Psi_{j})$.

We discuss below the property and the role of $\epsilon=\epsilon(U_{\epsilon},\Psi_{j})$ in Eq. (2) or in Eq. (17). The extra contribution $\epsilon\cong\epsilon(U_{\epsilon},\Psi_{j})$ to the gate infidelity with detailed form shown in Eq. (18) is related to the error shift matrix $U_{\epsilon}$ and all Dyson expansion terms $\Psi_{j}$. As noted earlier, if $J_{1}$ is small, then the matrix elements of $U_{\epsilon}$ are also small. Moreover, if the noise strength is not too strong such that $|\Psi_{j+1}|\ll|\Psi_{j}|$, then the extra contribution $\epsilon\cong\epsilon(U_{\epsilon},\Psi_{j})$ is also small. Therefore when running optimization for small noise strength for which the higher-order terms $\mathcal{O}(\tilde{\mathcal{H}}_{N}^{m},m\geq 3)$ becomes negligible (see Appendix B), the extra contribution $\epsilon$ can be omitted as $J_{1}$ is minimized to a small number. Consequently, one can focus on the optimization of only $J_{1}+\left\langle J_{2}\right\rangle$.

The advantage of introducing $J_{1}$ and $\epsilon$ in our method is to enable more degrees of freedom in control parameters for optimization. There are actually no $J_{1}$ and the extra controbution $\epsilon$ in the gate infidelity expression of the robust control method of SUPCODE Wang et al. (2012); Kestner et al. (2013) and the filter-transfer-function method Soare et al. (2014); Ball and Biercuk (2015). In these methods, $J_{1}$ or equivalently the error shift matrix $U_{\epsilon}$ is set exactly to zero by imposing some constraints on the control parameters. In contrast, our method can tolerate some error of $U_{\epsilon}$ and thus have more degrees of freedom in control parameters as long as $J_{1}$ and the extra contribution $\langle\epsilon\rangle$ in gate infidelity $\langle{\cal I}\rangle$ are made just smaller than $\langle J_{2}\rangle$. This advantage of having more degrees of freedom for optimization plays an important role in finding better control pulses as the number of qubits, the number of controls, and the number of noise sources increase.

Appendix B Estimation of

higher-order contributions

Here we estimate the contributions of higher order terms $\mathcal{O}(\tilde{\mathcal{H}}_{N}^{m},m\geq 3)$ and discuss when they can be neglected. We express the higher-order terms as $\mathcal{O}(\tilde{\mathcal{H}}_{N}^{m},m\geq 3)=\sum_{p\geq 3}J_{p}$, where $J_{p}$ denotes the $p$-th order noise term of the gate infidelity. Detailed forms of the first two lowest-order terms in $\mathcal{O}(\tilde{\mathcal{H}}_{N}^{m},m\geq 3)$ are

[TABLE]

where

[TABLE]

is the $q$-th order Dyson expansion term. To make an estimation of the magnitude of $\Psi_{q}$, we take the Z-noise model for the single-qubit gate operations in Sec. III.1.1 as an example. Substituting the noise Hamiltonian $\tilde{\mathcal{H}}_{N}(t)=\beta_{Z}(t)\omega_{0}{R}_{Z}(t)$ with $R_{Z}(t)=U_{I}^{\dagger}(t)[\omega_{0}Z/2]U_{I}(t)$ in the interaction picture into $\Psi_{q}$, we obtain

[TABLE]

where $\bar{R}_{Z}(t)=U_{I}^{\dagger}(t)[Z/2]U_{I}(t)$. Since $U_{I}(t)$ is unitary, its matrix elements $\left|U_{I,jk}(t)\right|\leq 1$ for all $j$ and $k$. Consequently, $\left|\bar{R}_{Z,jk}(t)\right|<1$ for all $j$ and $k$, so is $\left|\{\bar{R}_{Z}(t_{1})\bar{R}_{Z}(t_{2})\cdots\bar{R}_{Z}(t_{q})\}_{jk}\right|<1$ for all $j$ and $k$. Taking the strength of $\beta_{Z}(t)$ to be about its standard deviation $\sigma_{ZZ}$, we estimate the noise strength contribution to be $\left|\{\beta_{Z}(t_{1})\beta_{Z}(t_{2})\cdots\beta_{Z}(t_{q})\}\right|\approx(\sigma_{ZZ})^{q}$. The time integral contribution $\{\int_{0}^{t_{f}}\omega_{0}dt_{1}\int_{0}^{t_{1}}\omega_{0}dt_{2}\cdots\int_{0}^{t_{q-1}}\omega_{0}dt_{q}\}$ can be estimated to be about $\sim(\omega_{0}t_{f})^{q}/q!$. By combining the above estimations, the magnitude of $\left|\Psi_{q,jk}\right|$ is in the order of $\sim(\omega_{0}t_{f}\sigma_{ZZ})^{q}/q!$. Then substituting the estimated value of $\left|\Psi_{q,jk}\right|$ into $J_{2}$ in Eq. (4), $J_{3}$ in Eq. (19), and $J_{4}$ in Eq. (20), we obtain the magnitude ratio $J_{3}/J_{2}\sim(\omega_{0}t_{f}\sigma_{ZZ})/3$ and $J_{4}/J_{2}\sim(\omega_{0}t_{f}\sigma_{ZZ})^{2}/12$. The single-qubit gate operation time in Sec. III.1.1 is $\omega_{0}t_{f}=20$. If we choose the noise fluctuation $\sigma_{ZZ}=10^{-3}$, then the ratio $J_{3}/J_{2}\sim(6\times 10^{-3})$ and $J_{4}/J_{2}\sim(3\times 10^{-5})$, and thus the higher-order terms $\mathcal{O}(\tilde{\mathcal{H}}_{N}^{m},m\geq 3)$ can be safely neglected. If, however, $\sigma_{ZZ}\sim 10^{-1}$, then $\omega_{0}t_{f}\sigma_{ZZ}\sim 2$. In this case, $J_{3}/J_{2}\sim 2/3$ and $J_{4}/J_{2}\sim 1/3$, so the higher-order terms $\mathcal{O}(\tilde{\mathcal{H}}_{N}^{m},m\geq 3)$ can not be neglected. Comparing our estimation with the results of the full-Hamiltonian simulation, one finds that the ensemble average of the gate infidelity $\left\langle\mathcal{I}\right\rangle$ of the IDG strategy scales as the second power of $\sigma_{ZZ}$ (because $\left\langle J_{2}\right\rangle$ dominates) for small $\sigma_{ZZ}$ until $\sigma_{ZZ}\sim 10^{-1}$ in Fig. 3(a) for low-frequency noise $\gamma_{ZZ}=10^{-7}\omega_{0}$ and in Fig. 4(a) for high-frequency noise $\gamma_{ZZ}=10^{-1}\omega_{0}$. This is consistent with our estimation. In other words, if $\sigma_{ZZ}$ is considerably smaller than $10^{-1}$, $\mathcal{O}(\tilde{\mathcal{H}}_{N}^{m},m\geq 3)$ can be ignored. Therefore, even for the case when the full-Hamiltonian simulation is not available, we can use this estimation method to determine the criterion for neglecting the higher-order terms $\mathcal{O}(\tilde{\mathcal{H}}_{N}^{m},m\geq 3)$.