Achieving High-Fidelity Single-Qubit Gates in a Strongly Driven Charge Qubit with $1\!/\!f$ Charge Noise

Charge qubits formed in double quantum dots represent quintessential two-level systems that enjoy both ease of control and efficient readout. Unfortunately, charge noise can cause rapid decoherence, with typical single-qubit gate fidelities falling below $90\%$. Here, we develop analytical methods to study the evolution of strongly driven charge qubits, for general and $1\!/\!f$ charge-noise spectra. We show that special pulsing techniques can simultaneously suppress errors due to strong driving and charge noise, yielding single-qubit gates with fidelities above $99.9\%$. These results demonstrate that quantum dot charge qubits provide a potential route to high-fidelity quantum computation.

Building high-quality qubits is a key objective in quantum information processing. Achieving high-fidelity gates requires both precise control and effective measures to combat decoherence arising from the environment. Semiconductor based quantum dot charge qubits, for example, suffer from strong coupling to charge noise that causes voltage fluctuations on the control electrodes [1][2][3], which has so far limited gate fidelities to below 90% [4]. To be suitable for scalable quantum computation, the fidelity must be increased to at least 99% [5].
One strategy for achieving higher fidelities is to operate the qubits as fast as possible, for example, by driving the qubits with strong microwaves. However this can potentially cause strong-driving effects, including Bloch-Siegert shifts of the resonant frequency [6,7] and fast oscillations superimposed on top of Rabi oscillations [8]. While Bloch-Siegert shifts can be accommodated by adjusting the driving frequency or gate time, fast oscillations may be difficult to control, resulting in gate errors. There are several known approaches for mitigating control errors, including pulse-shaping methods that suppress oscillations by engineering the pulse envelopes [8,9]. While such schemes can be applied to charge qubits, we explore a different method here, based on rectangular pulse envelopes engineered to produce nodes in the fast oscillations at the end of a gate. In principle, the scheme explored here can also be combined with pulseshaping methods, although this lies outside the scope of the present work.
The paper is organized as the follows. We first explain how to synchronize the fast oscillations and Rabi oscillations in a strongly driven charge qubit, in the absence of noise. We then introduce charge noise, focusing on a 1/f noise spectrum, and characterize the qubit dynamics both numerically and analytically. Finally, we demonstrate that gates with fidelities higher than 99.9% can be achieved, for typical charge noise magnitudes [1,3,10,11].
Noise-free evolution. The basis states of a double quantum dot charge qubit, |L and |R , represent the localized positions of an excess charge in the left or right dot, as indicated in Fig. 1(a) [4,13,14]. We consider ac gating of a single qubit, with the Hamiltonian H sys =H q + H ac , where H q =−(ε/2)σ x − ∆σ z , the σ i are Pauli matrices, ε is the detuning parameter (defined as the energy difference between the two dots), and ∆ is the tunnel coupling between the dots. Here we have expressed H sys in the eigenbasis {|0 = (|L − |R )/ √ 2, |1 = (|L + |R )/ √ 2}, corresponding to the charge qubit "sweet spot" ε=0, where it is first-order insensitive to electrical noise [4]. Unless otherwise noted, we assume that the nominal operating point is ε=0 throughout the remainder of this work. When a microwave signal is applied to ε, the driving Hamiltonian is given by H ac = (A ε /2)σ x cos(ω d t+φ), where A ε is the driving amplitude, ω d is its angular frequency, and φ is the phase at time t=0, when the drive is initiated.
In the rotating frame defined by H rot =U † rot H sys U rot − i U † rot (d/dt)U rot , with U rot =diag[e iω d t/2 , e −iω d t/2 ], the ideal evolution term U The insets of (b) show blow-ups of the evolution near the end of a π-rotation (t = tπ), decomposed into their Rabi (dark purple) and fast-oscillation components (red). The oscillations are synchronized at tπ when N = (2θωres)/(πΩ) is an even integer, resulting in high-fidelity gates. (The main panels also use N = 10.) Charge noise causes a slight decay of ρ00 at the end of the simulation period (2tπ 0.5 ns), which can be observed more clearly at long times in (c). Here, the inset shows a short-time blow-up of the interaction-frame simulations (solid lines), our full analytical calculations obtained from Eq. (S18) (dashed white line), and the simple asymptotic expression from Eq. (5)  , as discussed in [12].
(cos φ, − sin φ, 0) axis, the fast oscillations may prevent the density matrix element ρ 00 from reaching 0 at the end of a gate period, t π . We see this more clearly by plotting the fast and Rabi oscillation components separately in the top inset of Fig. 1(b). On the other hand, we may adjust the pulse parameters A ε and φ to synchronize the fast oscillations with the slower Rabi oscillations, as shown in the bottom inset, to obtain an R θ (φ) gate with much higher fidelity. We characterize the infidelity arising from the fast oscillations by computing the process fidelity F θ (φ), defined by comparing the ideal evolution operator U (0) 0 , to the actual evolution U 0 , for R θ (φ) in the rotating frame. In [12], we show that specific A ε 's give rotations with perfect fidelity when θ = π, 2π, 3π, . . . for any φ. More importantly, when φ = π/4, 3π/4, 5π/4, . . . , we obtain (2) where, again, γ = A ε /(16∆). For these values of φ, the infidelity due to strong-driving errors is bounded above by ∼4γ 2 . Moreover, the oscillations are synchronized, yielding perfect fidelity (up to O[γ 4 ]), when 2θω res /Ω=N π, with N an even integer. Since this condition can be met for a continuous range of θ by adjusting Ω (i.e., A ε ), and since φ = π/4 and 3π/4 represent orthogonal rotation axes, the rotations {R θ (π/4), R θ (3π/4)} therefore generate a complete set of high-fidelity singlequbit gates. Additional phase control is provided by adjusting the waiting time between ac pulses. Unless otherwise noted, we set φ = π/4 for the remainder of our analysis.
Modeling charge noise. We introduce charge noise into our analysis through the Hamiltonian term H n = h n δε(t), where δε(t) is a random variable affecting the detuning parameter and h n = −σ x /2 is referred to as the noise matrix. The noise is characterized in terms of its time correlation function S(t 1 − t 2 ) = δε(t 1 )δε(t 2 ) , where the brackets denote an average over noise realizations, and the corresponding noise power spectrum is [15]. Although we obtain analytical solutions for generic noise spectra in [12], below we focus on 1/f noise, including in our simulations, due to its relevance for charge noise in semiconducting devices [16,17]: where c ε is related to the standard deviation of the detuning noise, σ ε , via σ ε = c ε [2 ln( √ 2πc ε / ω l )] 1/2 [18,19], and ω l (ω h ) are low (high) cutoff angular frequencies.
We now present numerical simulations of a strongly driven, noisy charge qubit by averaging the density matrix ρ(t) over noise realizations. Time sequences for , as discussed in [12]. Here, f Rabi is the Rabi frequency, Γ Rabi is the Rabi decay rate, and the 1/f noise spectrum is given in Eq. (S90), with cε = 0.5 µeV, ω l /2π = 1 Hz, and ω h /2π = 100 THz. The upper inset shows a line-cut along Aε = ∆ in the main figure (blue dashed line), revealing a FOM as high as 700. In the lower inset, we fix ∆/h = 5 GHz and Aε/h = 4 GHz (cyan star), but allow ε to vary, confirming that the FOM is maximized at the sweet spot, ε = 0, where the qubit is first-order insensitive to detuning noise. δε(t) are obtained by generating a white noise sequence, then scaling its Fourier transform by an appropriate spectral function [12,20], such as Eq. (S90). A typical result is shown in the middle panel of Fig. 1(b), where the suppression of Rabi oscillations is a direct consequence of the charge noise. To differentiate the effects of decoherence from those arising from strong driving, we present the same results in an interaction frame defined by U 0 , ρ I = U † 0 ρU 0 , in which the fast oscillations due to strong driving are not observed. Figure 1(c) shows the resulting long-time decay of the density matrix, while the inset shows the short-time behavior on an expanded scale. Note that the fast oscillations observed here do not arise directly from strong driving, but rather from non-Markovian noise terms, as discussed below.
Analytical solutions, with charge noise. Several theoretical techniques have been applied to noisy, driven two-level systems, including master equations [19,[21][22][23][24][25][26], dissipative Lander-Zener-Stückelberg interferometry [27,28], and treatments of spin-Boson systems [29][30][31]. Here, we solve the dynamical equation i dρ I /dt = δε(t)Lρ I in the interaction frame, where Lρ I ≡ [h I n , ρ I ] and h I n = U † 0 h n U 0 . We then average over the noise via a cumulant expansion [32,33], truncated at O[(δε/ Ω) 2 ]: where we have assumed that the noise is stationary, with zero mean ( δε = 0). We can rewrite Eq. (S18) in the form r I (t) = exp[K(t)]r I (0) by defining ρ I = 1/2(I 2 + r I x σ x + r I y σ y + r I z σ z ). Here, I 2 the 2 × 2 identity matrix, r I = (r I x , r I y , r I z ) is the Bloch vector, and K(t) is a 3 × 3 evolution matrix, to be determined. Since h I n (t) can be expanded into Fourier components, Eq. (S18) is solved by evaluating simple integrals of the form Details of the calculations are provided in [12], while the main results are discussed below. The accuracy of our cumulant approach is evident in Figs. 1(b) and 1(c), where the theoretical results (dashed white lines) are seen to capture all the fine structure of the simulations. Indeed, the bottom panel of Fig. 1(b) indicates that the analytical and numerical solutions differ by < 10 −3 over the entire range plotted.
Results. The physics of noise-averaged qubit dynamics is encoded in K(t), which can be decomposed into a sum of Markovian terms K M , and non-Markovian terms. The latter may be further divided into pure-dephasing terms K ϕ [25,34,35], and non-Markovian-non-dephasing terms K nMnϕ . Pure-dephasing terms are conventionally associated with the integral I(t, ω 1 =0, ω 2 =0) ∼ t 2 ln(1/ω l t) [25,34,35]. However, since K is defined in a rotating frame, "pure dephasing" has a different meaning than in the laboratory frame [23,25]: here, the leading order contributions to K ϕ are proportional to γ 2 , and are therefore attributed to strong driving. Markovian terms are associated with the integral Re[I(t, ω, −ω)], corresponding to short correlation times [12], and exponential decay (∼e −Γt ). The dominant non-Markoviannon-dephasing terms are associated with the integral Im[I(t, ω, −ω)]], yielding slow oscillations in the rotating frame, as well as the fast oscillations in the inset of Fig. 1(c). Since it is common in the literature to treat the dephasing and depolarizing channels separately [25], it is significant that our method encompasses both phenomena (and other behavior, including K nMnϕ ) within a common framework, allowing us to compare and contrast their effects.
We can compute the asymptotic behavior of r I (t) analytically, using the cumulant expansion. In the limit t 1/ω, where ω is any characteristic qubit frequency, many terms drop out, yielding the leading-order solution in γ: for the initial state |0 .
Here, the decoherence is dominated by the integral The real part describes exponential decay, giving the Markovian decoherence rate for driven evolution, as observed in Fig. 1(c), which can also be derived from Bloch-Redfield theory [25,26].
We can define the figure of merit (FOM), f Rabi /Γ Rabi , which corresponds to the number of coherent R 2π (π/4) rotations (not including strong-driving control errors), where the Rabi decay rate Γ Rabi is determined from Eq. (5) as ρ I By exploring a range of control parameters in Fig. 2, we first confirm that the FOM is strongly enhanced at the sweet spot ε = 0 (lower inset). Increasing the tunnel coupling ∆ and the driving amplitude A ε both enhance the FOM, as shown in the main panel. By increasing ∆ and A ε simultaneously, as shown in the upper inset, we find that the FOM can exceed 700 for a physically realistic charge noise amplitude of c ε = 0.5 µeV (σ ε = 3.12 µeV) [1,3,10,11].
R π (π/4) gate fidelity. We now compute process fidelities for R π (π/4) gates, using the χ-matrix method [12]. Control errors due to strong driving are investigated by considering U (0) 0 as the ideal evolution. The results of both numerical and analytical calculations are shown in Fig. 3. For no noise (c ε =0), the simulations are essentially identical to Eq. (2), revealing "dips" of low infi-delity, enabled by synchronized oscillations. For φ = π/4, the dip minima are proportional to γ 4 , while their widths are proportional to γ 2 [12], suggesting potential benefits of working at large A ε ∝γ. When c ε >0, the infidelity grows. Initially, the envelope of the infidelity oscillations decreases with A ε , because fast gates have less time to be affected by noise; then it increases, due to strongdriving effects. For smaller A ε , the simulations deviate slightly from the analytical results when the high-order noise terms become non-negligible. In all cases, the infidelity is locally minimized when A ε is positioned at a dip.
For the noise levels considered in Fig. 3(a), which are consistent with recent experiments [1, 3, 10, 11], we obtain F max 99%, which is insufficient for achieving highfidelity gates. However, the following procedure can be used to suppress both control errors and decoherence. First, A ε is tuned to a dip. Then, ∆ and A ε are simultaneously increased while holding γ = A ε /16∆ (and thus N ) fixed. In this way, we remain in a dip, while increasing the gate speed to suppress noise effects. The results are shown in Fig. 3(b). Here, when c ε = 1 µeV (σ ε = 6.36 µeV), we obtain fidelities >99% when ∆>40 µeV, and >99.9% when ∆>120 µeV. The corresponding qubit frequencies, 2∆/h=29.3 GHz and 58.0 GHz, are comparable to the qubit frequency of the quantum dot spin qubit in Ref. [36], and the Rabi frequencies 4∆/hN are generally lower.
Conclusions. Based on numerical and analytical calculations with 1/f detuning noise, we demonstrated that high-fidelity gate operations can be achieved in charge qubits employing strong-driving methods. We identified rotations that synchronize Rabi and fast oscillations, yielding a complete set of single-qubit gates that suppress control errors. We also outlined a protocol for suppressing decoherence caused by charge noise. Our protocol, and our analytical formalism, are both applicable to other solid-state systems, including superconducting flux qubits [37,38] and quantum-dot singlettriplet qubits [10,39], and can be extended to systems with multiple levels, including quantum-dot hybrid qubits [3,8,[40][41][42][43][44] and charge-quadrupole qubits [45,46].
One challenge for our proposal is the requirement of large tunnel couplings, or fast qubits. However, by employing high-order synchronized oscillations (N ∼10), we can reduce gate speeds to be compatible with current experiments. Improvements in ac control technology and materials with lower charge noise can also mitigate the technical challenges. On the other hand, large tunnel couplings determine the phonon-induced relaxation rate [47][48][49], which sets an upper bound on qubit coherence. Moving forward, we note that the phase, φ, represents an important control knob in our proposal, and can be viewed as a simple pulse-shaping tool. In future work, we therefore expect more sophisticated pulseshaping techniques to improve the gate fidelities [8,50].
We thank M. Eriksson for helpful discussions. Y.-C. Y. was supported by a Jeff and Lily Chen Distinguished Graduate Fellowship. This work was also supported in part by ARO (W911NF-12-1-0607, W911NF-17-1-0274) and the Vannevar Bush Faculty Fellowship program sponsored by the Basic Re-search Office of the Assistant Secretary of Defense for Research and Engineering and funded by the Office of Naval Research through grant N00014-15-1-0029.

SUPPLEMENTARY INFORMATION
In these Supplemental Materials, in Sec. , we provide the definition of the process fidelity, and in Sec. we present a brief derivation of the fidelity formula, Eq. (2) in the main text. In Sec , we sketch the derivation of the analytic formula for the qubit time evolution for general noise spectra, and derive the asymptotic formulas for Markovian noise, quasi-static noise, and 1/f noise. Sec. provides the details of the simulations shown in the main text, including the method used to generate the noise realizations and a method used to speed up the simulations. In Sec , we show that high-fidelity rotations about an axis tilted away from X-axis by φ = π/4 by an arbitrary angle θ, R θ (φ = π/4), can be achieved using a method similar to the one discussed in the main text.

PROCESS FIDELITY
Following Ref. [51], a generic quantum process E on a two dimensional Hilbert space can be expressed as where ρ 0 is the initial-state density matrix, {E m } = {I, σ x , −iσ y , σ z } is a basis for the vector space of 2 × 2 matrices, and χ mn is a 4 × 4 process matrix, commonly referred to as the chi matrix. The process fidelity is then defined as where χ sys is the process matrix for the actual physical evolution, including strong-driving effects, and χ ideal is the process matrix for the ideal rotation.

DERIVATION OF FIDELITY FORMULA IN THE ABSENCE OF CHARGE NOISE
We now evaluate Eq. (S2) in the absence of charge noise, including only strong-driving effects. Additional effects due to charge noise are presented in Sec. .
For a resonantly driven charge qubit operated at the charge anti-crossing (ε = 0) with only detuning drive (A ∆ = 0), the Hamiltonian in the eigenbasis of the qubit is given by whereω res is the resonant angular frequency. The evolution operation of the system in the absence of noise, E sys (ρ 0 ) = U 0 ρ 0 U † 0 , can be obtained analytically up to arbitrary order in the strong-driving parameter γ ≡ A ε /(16∆) 0 . The ideal unitary evolution of an R θ (φ) rotation corresponds to the O[γ 0 ] terms in this expansion, which can also be obtained from the rotating wave approximation, giving where t θ = θ/Ω is the duration of the rotation. The leading order contribution to the fast oscillations which we treat here as infidelity, occurred at O[γ 1 ], and are given by cos(ω res t θ + 2φ) sin(θ/2) e −iφ sin(ω res t θ ) cos(θ/2) e i(ωrest θ +φ) sin(ω res t θ ) cos(θ/2) −e iωrest θ cos(ω res t θ + 2φ) sin(θ/2) .
We now calculate the χ matrix of E sys order by order, sys . Since E sys describes the unitary evolution (without decoherence), we must have 1 = Tr(χ sys χ sys ) = 1 + 2γTr(χ ideal χ sys )] + · · · . Equating terms order by order in γ, we arrive at Using Eq. (S2) and Eq. (S6)-(S9) and applying our analytical results for U 0 , i.e. Eqs. (S4) and (S5) and high-order contributions, the fidelity F up to O[γ 3 ] is given by If φ = (2m + 1)π/4 for any integer m, this expression can be simplified to yield Eq. (2) in the main text: where N ≡ (2θω res )/(πΩ). Note here that which can be simplified as F (4) θ (φ) = −16 sin 2 (θ/2) when N is an even integer. This represents an upper bound on the gate fidelity in any dip, as discussed in Sec. below, and in the main text.
We also note that, for arbitrary φ and any integer k, the fidelity can be written as which equals unity, corresponding to perfect fidelity, whenever (θω res /Ω) is an integer multiple of π for θ = 2kπ, or whenever (θω res /Ω + 4φ) is an integer multiple of π for θ = (2k + 1)π.

ANALYTIC FORMULA FOR DYNAMICS IN THE PRESENCE OF DETUNING NOISE WITH DIFFERENT SPECTRA
In this section we analytically solve the dynamics of a strongly driven two-level system in the presence of classical detuning noise with different spectra. We first outline the derivation, and then present the results for noise with a generic spectrum. We then discuss in detail the asymptotic formulas that describe the large-time behavior for three different spectra: Markovian noise, quasi-static noise, and 1/f noise.
The dynamics of a strongly driven two-level system coupled to classical noise affecting the detuning ε can be described using the Hamiltonian where ω q is the qubit angular frequency, ω d is the angular frequency of the drive, A t (A ) is the transverse (longitudinal) amplitude of the drive, and h n,x (h n,z ) is the transverse (longitudinal) coupling to the detuning noise δε. Here we start with a more general situation to include, taking a charge qubit as an example, the case of driving tunnel coupling at the sweet spot (A = 0 when ε = 0), and also the case of working away from the sweet spot (ε, A t , A , h n,x , h n,z = 0). In the main text, we only consider driving detuning while working at the sweet spot, the special case where ω q = 2∆, A = 0, h n,x = −1/2, and h n,z = 0. In the absence of noise, the evolution operator U 0 satisfies the equation where U 0 (t = 0) = I. In Ref. [8], we showed how to obtain U 0 order-by-order in the perturbation parameter γ ∼ A/ ω d , where A = A t , A , using a dressed-state formalism. Transforming into the interaction picture, the equation of motion in the presence of the detuning noise can be written as where ρ I = U † 0 ρU 0 is the density matrix in the interaction picture, and Lρ I ≡ [h I n , ρ I ] with h I n = U † 0 h n U 0 and h n = h n,x σ x + h n,z σ z . The evolution can be expressed as a cumulant expansion [32,33] where · · · is the ensemble average over δε(t) and · · · c is the cumulant average. To O[(δε/ Ω) 2 ], this can be written as where · · · describes the ensemble average over δε(t), and S(t 1 − t 2 ) ≡ δε(t 1 )δε(t 2 ) is the time autocorrelation function of δε(t). We note that the power spectrum density of the noise,S(ω), is related to S(t) by [15] S(ω) = ∞ −∞ dt e iωt S(t). (S19) To simplify the calculation, we express the qubit state as a Bloch vector r I in the interaction frame, [ρ I = 1/2(I 2 + r I x σ x +r I y σ y +r I z σ z )], and the super-operator L as a matrix by using {σ x , σ y , σ z } as the basis of its domain (two-by-two Hermitian operator with vanishing trace). Eq. (S18) can then be written as r I (t) = exp[K(t)]r I (0). (S20) By expressing h I n (t) in terms of the Pauli matrices, h I n (t) = h I n,x (t)σ x + h I n,y (t)σ y + h I n,z (t)σ z , and further expanding the coefficients in Fourier series, h I n,i (t) = i,ω α i,ω e iωt , the matrix K(t) in Eq. (S20) can be written as where In general, K(t) = K M +K ϕ +K nMnϕ where K M is a Markovian term and K ϕ is a pure-dephasing term [25,34,35]. We denote the remaining term, K nMnϕ , as the non-Markovian-non-dephasing term; it is the correction to the Markovian approximation. The Markovian term is defined as the decoherence within the Markovian approximation (i.e., when the correlation time is much smaller than the characteristic time scale of the system dynamics). If only the Markovian term is present, the decoherence yields an exponential decay and K(t) is linear in time t. For further discussion, please see Sec. and the main text. The pure-dephasing term describes pure dephasing in the rotating frame defined by U rot =diag[e iω d t/2 , e −iω d t/2 ], and is associated with I(t, ω 1 = 0, ω 2 = 0) [25,34,35]. The non-Markovian-non-dephasing term is the correction to the Markovian approximation, describing, for example, the decoherence due to low-frequency noise, and oscillatory terms typically ignored in the asymptotic limit t 2π/ω, where ω represents any relevant angular frequency in the system.
In the following subsections, we evaluate the three terms of K(t), {K M , K ϕ , K nMnϕ }, for general S(t 1 − t 2 ), and then describe the asymptotic form of K(t) for three cases: Markovian noise, quasistatic noise, and 1/f noise. To be concise, we assume the qubit is driven resonantly (ω d =ω res ).
We note that this technique is easy to use and is applicable to many other kinds of noise that are present in condensed matter devices such as Lorentzian noise and power-law noise, and can also be generalized to multi-level systems such as quantum-dot hybrid qubits [3,8,[40][41][42][43][44].

Analytic formula for general noise spectra
For a generic noise spectrumS(ω), we define the symmetric and antisymmetric versions of Eq. (S22) as follows: such that I(t, ω 1 , ω 2 ) = I S (t, ω 1 , ω 2 ) + I A (t, ω 1 , ω 2 ). The partially evaluated integrals for different cases of {ω 1 , ω 2 } are given by in the main text compare the analytical results and the simulation results of the gate fidelity F π (φ = π/4). The analytical results match the simulation results in the regimes of greatest interest, for example when the driving amplitude is large. In Fig. 3(a), deviations between theory and simulations begin to appear when the driving amplitude is small because the higher order effects of the noise become visible when the gate time is long. In Fig. 3(b), deviations also arise from the higher order terms in the noise δε when the tunnel coupling is small. When the tunnel coupling is large, deviation appears again because of the higher order terms in γ. However, we note that the deviations are quite small, on the order of 10 −4 , and are mainly visible here because the data are plotted on a log scale. In order to describe the asymptotic behavior of the dynamics at long times, we calculate the asymptotic form of K(t) by taking the limit t 2π/ω, where ω represents any relevant angular frequency in the system, and neglecting the oscillatory terms. The asymptotic forms of K M (t), K ϕ (t), and K nMnϕ (t) typically can be expressed as We note that in some cases, I(t, ω 1 = 0, ω 2 = 0) also contributes to the asymptotic form of K nMnϕ . In the following, we focus on three typical types of noise: Markovian noise, quasi-static noise, and 1/f noise. We provide the results for the functions defined in Eqs. (S28)-(S33), the corresponding integrals I(t, ω 1 , ω 2 ), and the asymptotic formulas for these quantities. We express K(t) as a perturbation series in γ, using K (i) x to label the O[γ i ] contribution to K x , where x can be M, ϕ, or nMnϕ.

Markovian Noise
In the Markovian limit, we assume that the correlation time τ c is much smaller than the time scale we are interested in, t, and 1/ω, where ω is the characteristic frequency of the system (i.e., for a resonantly driven charge qubit, the resonant angular frequency ω res , the Rabi angular frequency Ω or their linear combinations), such that t/(2π) 1/ω τ c /(2π). By setting S(t ) ≈ 0 when t > τ c and neglecting any term that scales with τ c , Eqs. (S28)-(S33) can be evaluated to yield Inserting Eq. (S37) to Eq. (S42) in Eq. (S26) and Eq. (S27) and neglecting O[1/ω] terms, we obtain I = I S + I A , with: Applying this formula to Eq. (S14) and only keeping the terms up to O[γ 0 ] (consistent with the rotating wave approximation) yields where Note that, if φ = 0, the expressions for Γ x and Γ z = Γ y are equivalent to those obtained using Bloch-Redfield theory [15,25,26]. For this special case, the elements of the Bloch vector exhibit an exponential decay, If one now restore the O(1/Ω) terms, previously neglected while deriving Eqs. (S43) and (S44), one obtains an equation of motion for the Bloch vector within the rotating wave approximation, where η = 1 2 h 2 n,x [S(ω res + Ω) −S(ω res − Ω)], λ = − 1 4 h 2 n,x [S(ω res + Ω) +S(ω res − Ω)] + h 2 n,zS (Ω), and sgn(x) = +1 (−1) when x > 0 (x < 0). We note that Equation (S51) is equivalent to the master equation results derived in [21]. However, the current method also provides a systematic approach for including higher-order terms in γ, beyond the rotating-wave approximation.

Quasistatic Noise
For quasistatic noise with standard deviation σ ε , the time correlation function is and the corresponding noise spectral density isS Using these forms in Eqs. (S28)-(S33) yields Taking the limit t/(2π) 1/ω for any relevant angular frequencies of the system ω, and keeping only the nonoscillatory terms to O[1/ω], the integrals can be approximated as ε t/ω 2 (ω 1 = 0 and ω 2 = 0), 0 (Otherwise). (S60) For quasistatic noise, K(t) = K ϕ + K nMnϕ (t), and K M = 0. The leading order term in K nMnϕ (t) is O[γ 0 ]: where n,zSimag (Ω) + h 2 n,x (S imag (−ω res + Ω) +S imag (ω res + Ω))], (S62) The leading order contribution to K ϕ (t) is O[γ 2 ]: where We note that, for the charge qubit operated at the sweet spot as discussed in the main text, this term is non-vanishing only if the tunnel coupling driving amplitude is not zero. This indicates that one should only drive detuning in order to suppress this decoherence effect, as in the main text.
These two terms cause the elements of the Bloch vector to decay as a Gaussian with a sinusoidal oscillation: The 1/f power spectrum was presented in Eq. (3) of the main text where ω l (ω h ) is the low (high) angular frequency cut-off. The corresponding time correlation function is where Ci(x) ≡ − ∞ x dx cos(x )/x is the cosine integral function, and Si(x) ≡ x 0 dx sin(x )/x is the sine integral function. We further define functions For a resonantly driven two-level system, only K M (t) and K nMnϕ (t) have contributions of O[(A/ ω d ) 0 ], corresponding to the rotating-wave approximation. The lowest-order contributions to K ϕ (t) is O[(A/ ω d ) 2 ]. Below, we report the leading order contributions of K M (t), K nMnϕ (t), and K ϕ (t), respectively. The explicit expressions of O[γ] contributions of K M (t), K nMnϕ (t) are, however, omitted for brevity.
It is unsurprising that 1/f noise causes both Markovian and non-Markovian types of decoherence. At O[γ 0 ], where Note that Γ nMnϕ is the rotating frequency induced by the low-frequency part of the 1/f noise. For a driven charge qubit operated at a sweet spot ε = 0, as considered in the main text, h n,z = 0 so that terms proportional toS(Ω) or S imag (Ω) vanish. Since the remaining terms in Γ nMnϕ now have opposite signs and similar magnitudes, we see that K nMnϕ is small compared to K M . If we further take φ = π/4 as in the main text, the expressions are simplified as Γ x = Γ y and Γ x + Γ xy = Γ z , yielding Eq. (5) of the main text when the initial state is |0 . The leading order contribution to K ϕ (t) is O[γ 2 ]: As dΩ/dε = 2h nx A / ω res , we again interpret K ϕ (t) as the dephasing in the rotating frame, in which the eigenenergies of the dressed states are ± Ω. For the data of the asymptotic formula shown in the main text [ Fig. 1(c) and Fig. 2], we also include contributions to K ϕ up to O[γ 2 ], and contributions to K M and K nMnϕ up to O[γ]. To capture the essence of the dynamics in a simple and illustrative way, we did not include the O[γ 2 ] contributions to K M and K nMnϕ . This is justified by the fact that the lower-order contributions already capture the features of the simulation as shown in the Fig. 1(c) in the main text. In fact, the O[γ 2 ] contributions to K M and K nMnϕ are expected to have small effects. This is contrary to K ϕ where the leading-order terms are O[γ 2 ] and the dephasing function I(t, ω 1 = 0, ω 2 = 0) ∼ t 2 log(t) t when t is large, suggesting appreciable effects on the dynamics, as shown in the lower inset of Fig. 2.

DETAILS OF NUMERICAL SIMULATIONS
We now describe the method used for the numerical simulations. The objective is to simulate the evolution of the ensemble average of the density matrix, ρ , in the presence of detuning noise δε(t), where · · · denotes the ensemble average over δε(t).
The dynamics is governed by the Schrödinger equation Here, H sys = H q + H ac is the Hamiltonian of the system, where H q describes the qubit and H ac describes the ac driving, and H n = h n δε(t) describes the effect of charge noise on the system, where h n is the noise matrix. Using the states {|0 = (|L − |R )/ √ 2, |1 = (|L + |R )/ √ 2} as the basis, we can express H q = −( /2)σ x − ∆σ z , H ac = (A ε /2)σ x cos(ωt), and h n = −σ x /2, where the σ i are Pauli matrices.
The most obvious way to proceed is to first generate a set of noise realizations, {δε α (t)} α=1,...,αmax , having the desired spectrum,S(ω), then solve the Schrödinger equation for a given noise realization δε α (t) to obtained the final state |ψ α and the corresponding density matrix ρ α = |ψ α ψ α |. The ensemble average of the density matrix is then given by Performing numerical simulation in this manner, however, is very time-consuming, especially when α max is large. We therefore develop a more efficient method to simulate the Schrödinger equation over a large number of noise realizations. We now first describe the generation of the noise realizations, and then the simulation method.

Generation Of Noise Realizations for the Numerical Simulations
In the simulation, a noise realization δε(t) is discretized into the noise sequences {δε 1 , δε 2 , · · · , δε N }, such that δε(t) = δε k is constant over the time interval t k−1 ≤ t < t k where t k ≡ k t tot /N , t tot is the total time of consideration, and N is the total number of time segments. We follow [20] to generate the noise sequences for the numerical simulation. The method is: 1. Generate a Gaussian white noise sequence {u 1 , u 2 , · · · , u N } with zero mean and unit standard deviation.
2. Compute the Fourier transform of the sequence, {ũ 1 ,ũ 2 , · · · ,ũ N }, wherẽ 3. LetS(ω) represent the desired noise power spectrum. Scale the white noise in the Fourier space by taking the productũ m S m , where S m =S (2π(m − 1)/t tot ) /∆t, and ∆t = t tot /N . Then perform the inverse Fourier transformation to obtain the noise realization in the time domain: Here, we assume 1/f noise such that corresponding to the continuous noise spectrum where the low angular frequency cutoff is ω l = 2π(m l − 1)/t tot and the high angular frequency cutoff is ω h = πN/t tot .

Method of Simulation
We wish to solve the Schrödinger equation, Eq. (S86), numerically for a given noise realization δε(t) defined as in Eq. (S89). Let U 0 (t) be the evolution operation satisfying with initial condition U 0 (t = 0) = I. We transform the Schrödinger equation into the interaction picture, defining |ψ I (t) = U 0 (t) † |ψ(t) and h I n (t) = U 0 (t) † h n (t)U 0 (t), so that To include the cases where gate times are in between t k and t k−1 , we define the evolution operator U I (t, t k−1 ) as |ψ I (t) = U I (t, t k−1 )|ψ I (t k−1 ) for t k ≥ t ≥ t k−1 , which can be expanded perturbatively as where W r (t, t k−1 )'s are time-ordered integrals, Truncating the series after r = r max , one can approximate the evolution operator as where {W r (t, t k−1 )} can be calculated either analytically or numerically. The evolution operator U I (t, 0) then can be expressed as The solution of the Schrödinger equation in the lab frame can then be expressed as |ψ(t) = U (t)|ψ(0) , where U (t) = U 0 (t)U I (t, 0). The strength of this method is that W r (t, t k−1 ) does not depend on delta ε(t), so we only need to perform simulations once, to determine {W r (t, t k−1 )} r=1,...,rmax and {W r (t l , t l−1 )} r=1,...,rmax for l = 1, . . . , k − 1, numerically. After computing these matrices, the calculation of |ψ(t) for all the generated noise sequences directly follows Eq. (S95) and Eq. (S96), with no further simulations required. This allows efficient simulation of the system, even with a large number of noise realizations.
In this work, we take r max = 6, α max = 100, 000, and compute {W r } numerically. We have checked the validity of this simplified method by comparing our results to those of full simulations with parameters the same as the simulation shown in Fig. 1(b) and Fig. 1(c) for several different noise sequences, and find the deviations between the resulting density matrices are smaller than 10 −7 for the time period considered. We also study the statistics of the simulated fidelities by dividing the simulation into 10 trials, each with α max = 10, 000, and compute the ratio of the standard deviation of these 10 trials, σ F , to the average infidelity σ F /(1 − F ). For all the parameters considered in the main text, this ratio is around 10 −2 , showing that the reported infidelity is accurate to at least two significant figures.
HIGH-FIDELITY R θ (φ = π/4) GATES According to Eq. (S10), to O[γ 3 ], the intrinsic infidelity of a resonantly driven R θ (φ = π/4) gate performed on a charge qubit in the absence of noise can be expressed as 1 − F θ (φ = π/4) = 2γ 2 [1 − cos(2θω res /Ω)] + 4γ 3 sin(θ) sin(2θω res /Ω). (S97) For any rotation angle θ, the intrinsic infidelity vanishes whenever N ≡ (2ω res θ)/(Ωπ) is an even integer, producing dips like those observed in Fig. 3 of the main text. However, in the presence of the noise, the dips are suppressed and the infidelity increases. This effect can be reduced by increasing the tunnel coupling ∆ and the driving amplitude A ε simultaneously while keeping the ratio A ε /∆ fixed, so that the system remains in a dip. This is because the strong-driving effects only depend on the ratio of the driving amplitude and the tunnel coupling A ε /∆; however, by reducing the gate time (t g ∝ A −1 ), we can reduce the effects of charge noise, thus improving the fidelity. This method was applied to the special case of R π (φ = π/4) gates in the main text. To demonstrate that the method works for any θ, we now apply it to R θ (φ = π/4) gates for 0 < θ < π. . The simulation shows that as ∆ and Aε increase, the gate time decreases and the infidelities approach the intrinsic infidelity limit, corresponding to cε = 0 µeV. Note that when ∆/h 35 GHz, these gate fidelities are all greater then 99.9%.
In Fig. S1, we compare numerical results for the intrinsic infidelities (c ε = 0 µeV) of R θ (φ = π/4) gates with infidelities in the presence of 1/f detuning charge noise with c ε = 1 µeV (σ ε = 6.36 µeV), operated in the dip corresponding to N = 10. As usual, the system is operated at the sweet spot (ε = 0), and we consider the tunnel couplings ∆/h = 20, 35, 50 GHz. Since the intrinsic infidelity vanishes up to O[γ 3 ] when operating in a dip, its leading order contribution, from Eq. (S12), is given by