Introduction

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 them with strong microwaves. AC driving also mitigates decoherence, by elevating the relevant noise frequencies to the microwave regime, where their power is suppressed.6,7 However high-power microwaves can potentially cause detrimental strong-driving effects, including Bloch–Siegert shifts of the resonant frequency8,9,10 and fast oscillations superimposed on top of Rabi oscillations.11 They can also expose the qubit to new types of decoherence such as dephasing caused by noise-induced variations of the Rabi frequency.6,7 While Bloch–Siegert shifts can be accommodated by adjusting the driving frequency or gate time, and the induced decoherence can be suppressed by employing AC sweet spots,12 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.11,13,14 However, such schemes tend to increase the complexity of the control procedure.

Here we propose an alternative control scheme for strong driving, based on rectangular pulse envelopes engineered to produce nodes in the fast oscillations at the end of a gate operation, thereby minimizing their influence. We demonstrate our method on a double-quantum-dot charge qubit, showing that high-fidelity gate operations can be achieved in charge qubits under strong driving, even while 1/f noise is applied to the double-dot detuning parameter. This noise spectrum is particularly interesting because it has both Markovian and non-Markovian components. By employing both numerical and analytical techniques, we identify specific rotations that synchronize Rabi and fast oscillations, yielding a complete set of single-qubit gates that suppress control errors. We then propose a protocol for suppressing decoherence caused by charge noise, yielding gates with fidelities higher than 99.9%, for typical charge noise magnitudes.1,3,15,16

We also develop an analytical formalism based on a cumulant expansion, to accurately describe qubit dynamics in the presence of time-averaged 1/f noise. This formalism allows us explicitly calculate and distinguish between strong driving control errors and decoherence occurring in the weak and strong driving limits.

Results

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. 1a.4,17,18 We consider ac gating of a single qubit, with the Hamiltonian \({\cal H}_{{\mathrm{sys}}}\) = \({\cal H}_q + {\cal H}_{{\mathrm{ac}}}\), where \({\cal H}_q\) = −(ε/2)σx − Δσz, the σi are Pauli matrices, ε is the detuning parameter (defined as the energy difference between the two dots), and Δ is the tunnel coupling between the dots. Here we have expressed \({\cal H}_{{\mathrm{sys}}}\) in the eigenbasis \(\left\{ {\left| 0 \right\rangle = \left( {\left| L \right\rangle - \left| R \right\rangle } \right){\mathrm{/}}\sqrt 2 } \right.\), \(\left. {\left| 1 \right\rangle = \left( {\left| L \right\rangle + \left| R \right\rangle } \right){\mathrm{/}}\sqrt {\mathrm{2}} } \right\}\), 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 \({\cal H}_{{\mathrm{ac}}} = (A_\varepsilon {\mathrm{/}}2)\sigma _x{\mathrm{cos}}(\omega _dt + \phi )\), where Aε is the driving amplitude, ωd is its angular frequency, and ϕ is the phase at time t = 0, when the drive is initiated.

Fig. 1
figure 1

Strongly driven charge qubit. a Energy level diagram of a charge qubit. The insets depict a double quantum dot in three regimes of detuning, ε. (A potential charge noise fluctuation is shown as a dashed line.) Here, filled circles indicate the position of the excess electron in the ground state, and the barrier between the dots induces a tunnel coupling, Δ. b Time evolution of the density matrix element ρ00 in the laboratory frame, including numerical simulations (dashed lines), analytical calculations obtained from Eq. (4) (solid black lines), and their differences (dotted lines). (Here, brackets 〈·〉 denote a noise average.) In all cases, we use {ε, Δ, Aε}/h = {0, 5, 4} GHz, ϕ = π/4 and initial state |0〉. We assume the charge noise follows the 1/f spectrum of Eq. (3), with frequency cutoffs ωl/2π = 0.193 MHz and ωh/2π = 80.8 GHz, and noise amplitudes cε as indicated. 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 \(N = (2\theta \tilde \omega _{{\mathrm{res}}}){\mathrm{/}}(\pi {\mathrm{\Omega }})\) 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. c Time evolution of the density matrix element \(\rho _{00}^I\) in the interaction frame, including numerical simulations (dashed gray line) and the simple asymptotic expression from Eq. (5) (solid cyan line) with corrections to Kφ up to O[γ2] and corrections to KM and KnMnφ up to O[γ], as discussed in Supplementary Sec. S3. The inset shows a short-time blow-up in the interaction frame; it further includes our full analytical calculations obtained from Eq. (4) (solid black line)

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First, we follow ref. 11 and obtain exact solutions for strongly-driven qubits in the absence of noise, up to arbitrary order in the strong-driving parameter γ = Aε/(16Δ). Expanding the time-evolution operator order-by-order as U0(t) = \(\mathop {\sum}\nolimits_{n = 0}^\infty {\kern 1pt} \gamma ^nU_0^{(n)}\), we obtain

$$U_0^{(0)}(t) = \left( {\begin{array}{*{20}{c}} {e^{i\tilde \omega _{{\mathrm{res}}}t}{\mathrm{cos}}({\mathrm{\Omega }}t{\mathrm{/}}2)} & { - ie^{i(\tilde \omega _{{\mathrm{res}}}t + \phi )}{\mathrm{sin}}({\mathrm{\Omega }}t{\mathrm{/}}2)} \\ { - ie^{ - i\phi }{\mathrm{sin}}({\mathrm{\Omega }}t{\mathrm{/}}2)} & {{\mathrm{cos}}({\mathrm{\Omega }}t{\mathrm{/}}2)} \end{array}} \right).$$
(1)

[Higher-order terms are provided in Supplementary Section S2, Eq. (S5)]. Here, we consider only resonant driving, ωd = \(\tilde \omega _{{\mathrm{res}}}\), where \(\hbar \tilde \omega _{{\mathrm{res}}}\) = 2Δ(1 + 4γ2) is the renormalized resonant angular frequency, including Bloch–Siegert corrections, and ħΩ = Aε(1 + γ2)/2 is the renormalized Rabi angular frequency.

In the rotating frame defined by \({\cal H}_{{\mathrm{rot}}}\) = \(U_{{\mathrm{rot}}}^\dagger {\cal H}_{{\mathrm{sys}}}U_{{\mathrm{rot}}} - i\hbar U_{{\mathrm{rot}}}^\dagger (d{\mathrm{/}}{\rm{d}}t)U_{{\mathrm{rot}}}\), with Urot = \({\mathrm{diag}}\left[ {e^{i\omega _{\mathrm{d}}t/2},e^{ - i\omega _{\mathrm{d}}t/2}} \right]\), the ideal evolution term \(U_0^{(0)}\) generates smooth, sinusoidal, Rabi oscillations, corresponding to rotations about the (cosϕ, −sinϕ, 0) axis of the Bloch sphere. The \(U_0^{(1)}\) term represents the dominant fast oscillations associated with strong driving, with amplitude ~γ. Both drive components can be observed in the top panel of Fig. 1b, where we show that our analytical results (shown here up to O[γ2]) agree well with the results of numerical simulations of the full Hamiltonian.

Fast oscillations can cause gate infidelity. For example, if we consider an Rθ(ϕ) rotation of angle θ about the (cosϕ, −sinϕ, 0) axis, the fast oscillations may prevent the density matrix element ρ00 from reaching 0 at the end of a gate period, tπ. We see this more clearly by plotting the fast and Rabi oscillation components separately in the top inset of Fig. 1b. On the other hand, we may adjust the pulse parameters Aε and ϕ to synchronize the fast oscillations with the slower Rabi oscillations, as shown in the bottom inset, to obtain an Rθ(ϕ) gate with much higher fidelity.

We characterize the infidelity arising from the fast oscillations by computing the process fidelity Fθ(ϕ), defined by comparing the ideal evolution operator \(U_0^{(0)}\) to the actual evolution U0, for Rθ(ϕ) in the rotating frame. [see Eq. (S2) of Supplementary Section S1 for a precise definition of the process fidelity]. We find that specific Aε’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

$$\begin{array}{*{20}{l}} {1 - F_\theta (\phi )} \hfill & = \hfill & {2\gamma ^2\left[ {1 - cos\left( {2\theta \tilde \omega _{{\mathrm{res}}}{\mathrm{/\Omega }}} \right)} \right]} \hfill \\ {} \hfill & {} \hfill & { + 4\gamma ^3{\kern 1pt} {\mathrm{sin}}\theta {\mathrm{sin}}\left( {2\theta \tilde \omega _{{\mathrm{res}}}{\mathrm{/\Omega }}} \right) + O[\gamma ^4],} \hfill \end{array}$$
(2)

where, again, γ = Aε/(16Δ). For these values of ϕ, the infidelity due to strong-driving errors is bounded above by ~4γ2. Moreover, the oscillations are synchronized, yielding perfect fidelity (up to O[γ4]), when \(2\theta \tilde \omega _{{\mathrm{res}}}{\mathrm{/\Omega }}\) = Nπ, with N an even integer. Since this condition can be met for a continuous range of θ by adjusting Ω (i.e., Aε), and since ϕ = π/4 and 3π/4 represent orthogonal rotation axes, the rotations {Rθ(π/4), Rθ(3π/4)} therefore generate a complete set of high-fidelity single-qubit 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.

Charge noise

We introduce charge noise into our analysis through the Hamiltonian term \({\cal H}_n = h_n\delta \varepsilon (t)\), where δε(t) is a random variable affecting the detuning parameter and hn = −σx/2 is referred to as the noise matrix.1,3,15,16 The noise is characterized in terms of its time correlation function S(t1 − t2) = 〈δε(t1)δε(t2)〉, where the brackets denote an average over noise realizations, and the corresponding noise power spectrum is \(\tilde S(\omega ) = {\int}_{ - \infty }^\infty {\kern 1pt} {\rm{d}}t{\kern 1pt} e^{i\omega t}S(t)\).19 Although we obtain analytical solutions for generic noise spectra in Supplementary Sec. S3, below we focus on 1/f noise, including in our simulations, due to its relevance for charge noise in semiconducting devices:20,21

$$\tilde S(\omega ) = \left\{ {\begin{array}{*{20}{c}} {c_\varepsilon ^2\frac{{2\pi }}{{\left| \omega \right|}}} & {\left( {\omega _l \le \left| \omega \right| \le \omega _h} \right)} \\ 0 & {({\mathrm{otherwise}})} \end{array}} \right.,$$
(3)

where cε is related to the standard deviation of the detuning noise, σε, via \(\sigma _\varepsilon = c_\varepsilon \left[ {2\,{\mathrm{ln}}\left( {\sqrt {2\pi } c_\varepsilon {\mathrm{/}}\hbar \omega _l} \right)} \right]^{1/2}\),6,22 and ωl (ωh) are low (high) cutoff angular frequencies. We note that all frequencies relevant for qubit operation occur between these two cutoffs, so that the decoherence includes both Markovian and non-Markovain contributions.

We now present numerical simulations of a strongly driven, noisy charge qubit. A typical result is shown in the middle panel of Fig. 1b, 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 U0, \(\rho ^I = U_0^\dagger \rho U_0\), in which the fast oscillations due to strong driving are not observed. Figure 1c 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,6,7,23,24,25,26,27 dissipative Lander-Zener-Stückelberg interferometry,28,29 and treatments of spin-Boson systems.30,31,32 In Supplementary Sec. S3, we solve the dynamical equation in the interaction frame via a cumulant expansion,33,34 truncated at O[(δε/ħΩ)2]. The time evolution can be written in the form rI(t) = exp[K(t)]rI(0) by expressing \(\left\langle {\rho ^I} \right\rangle = 1{\mathrm{/}}2\left( {I_2 + r_x^I\sigma _x + r_y^I\sigma _y + r_z^I\sigma _z} \right)\). Here, I2 is the 2 × 2 identity matrix, \({\bf{r}}^I = \left( {r_x^I,r_y^I,r_z^I} \right)\) is the Bloch vector, and K(t) is a 3 × 3 evolution matrix, given by

$$[K(t)]_{ij} = - \frac{4}{{\hbar ^2}}\mathop {\sum}\limits_{\omega _1,\omega _2} \left[ { - \alpha _{j,\omega _1}\alpha _{i,\omega _2} + \delta _{ij}\mathop {\sum}\limits_{k = 1}^3 {\kern 1pt} \alpha _{k,\omega _1}\alpha _{k,\omega _2}} \right]I(t,\omega _1,\omega _2),$$
(4)

where we have expand the noise matrix in the interaction frame into Fourier components \(h_n^I(t) \equiv U_0^\dagger h_nU_0 = \mathop {\sum}\nolimits_{i,\omega } {\kern 1pt} \alpha _{i,\omega }e^{i\omega t}\), and defined \(I(t,\omega _1,\omega _2) \equiv {\int}_0^t {\rm{d}}t_1{\int}_0^{t_1} {\kern 1pt} {\rm{d}}t_2e^{i\omega _1t_1}e^{i\omega _2t_2}S\left( {t_1 - t_2} \right)\). Since U0 can be expressed order-by-order in γ, the same is also true of αi,ω, allowing us to distinguish the effects arising in the weak-drive limit, O[γ0], from the strong-driving limit, O[γn], for n ≥ 1. The accuracy of this cumulant approach is evident in Fig. 1b, c, where the theoretical results (solid black line) are seen to capture all the fine structure of the simulations. Indeed, the bottom panel of Fig. 1b indicates that the analytical and numerical solutions differ by <10−3 over the entire range plotted.

The physics of noise-averaged qubit dynamics is encoded in K(t), which can be decomposed into a sum of Markovian terms KM, and non-Markovian terms. The latter may be further divided into pure-dephasing terms Kφ,26,35,36 and non-Markovian-non-dephasing terms KnMnφ. Pure-dephasing terms are conventionally associated with the integral I(t, ω1 = 0, ω2 = 0) ~ t2 ln(1/ωlt).26,35,36 However, since K is defined in a rotating frame, “pure dephasing” has a different meaning than in the laboratory frame:7,26 here, the leading order contributions to Kφ are proportional to γ2, and are therefore attributed to strong driving. Markovian terms are associated with the integral Re[I(t, ω, −ω)], corresponding to short correlation times [see Supplementary Eq. (S34)], and exponential decay (~e−Γt). The dominant non-Markovian-non-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. 1c. Since it is common in the literature to treat the dephasing and depolarizing channels separately,26 it is significant that our method encompasses both phenomena (and other behavior, including KnMnφ) within a common framework, allowing us to compare and contrast their effects.

Asymptotic solutions

We can compute the asymptotic behavior of rI(t) analytically, using the cumulant expansion. In the limit \(t \gg 1{\mathrm{/}}\omega\), where ω is any characteristic qubit frequency, many terms drop out, yielding the leading-order solution in γ:

$${\bf{r}}^I(t) = \frac{{e^{ - {\mathrm{\Gamma }}_zt}}}{{\sqrt 2 }}\left[ {{\mathrm{sin}}\left( {{\mathrm{\Gamma }}_{{\mathrm{nMn}}\varphi }t} \right),{\mathrm{sin}}\left( {{\mathrm{\Gamma }}_{{\mathrm{nMn}}\varphi }t} \right),\sqrt 2 {\mathrm{cos}}\left( {{\mathrm{\Gamma }}_{{\mathrm{nMn}}\varphi }t} \right)} \right],$$
(5)

for the initial state |0〉. Here, the decoherence is dominated by the integral \(I(t,\omega , - \omega ) \approx \tilde S(\omega )t{\mathrm{/}}2 + \tilde S_{{\mathrm{imag}}}(\omega )t\), whose imaginary part is given by \(\tilde S_{{\mathrm{imag}}}(\omega ) \equiv c_\varepsilon ^2\left( {2i{\mathrm{/}}\omega } \right){\mathrm{ln}}\left| {\omega {\mathrm{/}}\omega _l} \right|\). The real part describes exponential decay, giving the Markovian decoherence rate for driven evolution, Γz = \(\left( {1{\mathrm{/}}16\hbar ^2} \right)\left[ {4\tilde S\left( {\tilde \omega _{{\mathrm{res}}}} \right)} \right.\) + \(\tilde S\left( {\tilde \omega _{{\mathrm{res}}} + {\mathrm{\Omega }}} \right)\) + \(\left. {\tilde S\left( {\tilde \omega _{{\mathrm{res}}} - {\mathrm{\Omega }}} \right)} \right]\) as observed in Fig. 1c, which can also be derived from Bloch-Redfield theory.26,27 The imaginary part corresponds to a non-Markovian-non-dephasing noise-induced rotation with frequency \({\mathrm{\Gamma }}_{{\mathrm{nMn}}\varphi } = \left( { - i{\mathrm{/}}8\hbar ^2} \right)\left[ {\left( {\tilde S_{{\mathrm{imag}}}\left( { - \tilde \omega _{{\mathrm{res}}} + {\mathrm{\Omega }}} \right) + \tilde S_{{\mathrm{imag}}}\left( {\tilde \omega _{{\mathrm{res}}} + {\mathrm{\Omega }}} \right)} \right)} \right]\), originating from the integrated low-frequency (“quasistatic”) portion of the noise spectrum, \(\left[ {{\int}_{\omega _l}^\omega {\kern 1pt} d\omega^\prime S(\omega^\prime ){\mathrm{/}}\pi } \right]^{1/2} = c_\varepsilon \left[ {2ln(\omega {\mathrm{/}}\omega _l)} \right]^{1/2}\). Besides these lowest-order results, which are the only important terms under weak driving, we can also compute higher-order corrections to these terms that become important under strong driving. Such high-order results are presented in Supplementary Section S3.

We can define a figure of merit (FOM), fRabi/ΓRabi, corresponding to the number of coherent R2π(π/4) rotations within a Rabi decay period TRabi = 1/ΓRabi (not including strong-driving control errors), where the Rabi decay rate ΓRabi is determined from Eq. (5) such that \(\left\langle {\rho _{00}^I\left( {t = 1{\mathrm{/\Gamma }}_{{\mathrm{Rabi}}}} \right)} \right\rangle \equiv (1 + e^{ - 1}){\mathrm{/}}2\). 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 Δ and the driving amplitude Aε both enhance the FOM, as shown in the main panel. By increasing Δ and Aε simultaneously, as shown in the upper inset, we find that the FOM can exceed 700 for a physically realistic charge noise amplitude of cε = 0.5 μeV (σε = 3.12 μeV).1,3,15,16

Fig. 2
figure 2

Figure of merit (FOM) of gates of a strongly driven charge qubit. Analytical calculations of the FOM fRabi/ΓRabi of a strongly driven charge qubit, as a function of the tunnel coupling Δ and driving amplitude Aε, at ε = 0 and ϕ = π/4, based on the asymptotic formula in Eq. (5), with corrections to Kφ up to O[γ2] and corrections to KM and KnMnφ up to O[γ], as discussed in Supplementary Sec. S3. Here, fRabi is the Rabi frequency, ΓRabi is the Rabi decay rate, and the 1/f noise spectrum is given in Eq. (3), with cε = 0.5 μeV, ωl/2π = 1 Hz, and ωh/2π = 100 THz. The upper inset shows a line-cut along Aε = Δ in the main figure (blue dashed line), revealing a FOM as high as 700. In the lower inset, we fix Δ/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

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R π(π/4) gate fidelity

We now compute process fidelities for Rπ(π/4) gates, using the χ-matrix method described in Supplementary Sec. S1. 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 (cε = 0), the simulations are essentially identical to Eq. (2), revealing “dips” of low infidelity, enabled by synchronized oscillations. For ϕ = π/4, the dip minima are proportional to γ4, while their widths are proportional to γ2 [see Supplementary Eq. (S13)], suggesting potential benefits of working at large Aε ∝ γ. As cε increases, the infidelity also grows, including both Markovian (KM), and non-Markovian contributions (Kφ and KnMnφ). Initially, the envelope of the infidelity oscillations decreases with Aε, because fast gates have less time to be affected by noise; it then increases, due to the combinantion of strong-driving effects and the decoherence induced by strong driving. For smaller Aε, 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 Aε is positioned at a dip.

Fig. 3
figure 3

Characterization of the fidelity. Dependence of the infidelity 1 − Fπ(π/4) of a strongly driven charge qubit on a the driving amplitude Aε, with Δ/h = 5 GHz, and on b the tunnel coupling Δ, with N = \(\left( {2\theta \tilde \omega _{{\mathrm{res}}}} \right){\mathrm{/}}(\pi {\mathrm{\Omega }})\) = 10, 12, 14, 16 held constant. Here, ε = 0, and the charge noise is given by Eq. (3), with ωl/2π = 1 Hz, ωh/2π = 256 GHz, and the values of cε are indicated. Note that low frequencies (<0.3 MHz) are approximated as quasistatic in these simulations, while the analytical results are obtained from Eq. (4), up to O[γ3] [or O[γ4] for cε = 0, see Supplementary Eq. (S12)]. In b, the infidelities are computed at the “dips” indicated in a

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For the noise levels considered in Fig. 3a, which are consistent with recent experiments,1,3,15,16 we obtain Fmax ≲ 99%, which is insufficient for achieving high-fidelity gates. However, the following procedure can be used to suppress both control errors and decoherence. First, Aε is tuned to a dip. Then, Δ and Aε are simultaneously increased while holding γ = Aε/16Δ (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. 3b. Here, when cε = 1 μeV (σε = 6.36 μeV), we obtain fidelities >99% when Δ > 40 μeV, and >99.9% when Δ > 120 μeV. The corresponding qubit frequencies, 2Δ/h = 29.3 GHz and 58.0 GHz, are comparable to the qubit frequency of the quantum dot spin qubit in ref. 37, and the Rabi frequencies ≃4Δ/hN are generally lower. We note that this protocol is applicable for any rotation angle θ, as shown in Supplementary Section S5.

Discussion

We have developed a new scheme for effectively harnessing strong driving to perform high-fidelity gates in quantum double-dot charge qubits, even in the presence of realistic 1/f noise. Our protocol, and our analytical formalism, are both applicable to other solid-state systems, including superconducting flux qubits38,39 and quantum-dot singlet-triplet qubits,15,40,41 and can be extended to systems with multiple levels, including quantum-dot hybrid qubits3,11,42,43,44,45,46 and charge-quadrupole qubits.47,48 Phonon-induced decoherence can be also analyzed in this formalism, after first averaging the phonons over the corresponding thermal distribution.34,49,50,51 However, the effectiveness of the protocol may be reduced compared to case of charge noise, since the power spectral density of phonons typically increases with the frequency.

A possible challenge for implementing this proposal is the requirement of large tunnel couplings, which could results in fast qubits that are difficult to control. However, by employing high-order synchronized oscillations (e.g. 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, the phonon-induced relaxation rate increases strongly with tunnel coupling,52,53,54 which will set an upper bound on the 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, it should be possible to combine the methods described here with conventional pulse shaping techniques, which would be expected to further improve the gate fidelities.11,13,14

Methods

Numerical simulation

We numerically simulate the Schrodinger equation of a strongly driven, noisy charge qubit, \(i\hbar {\kern 1pt} d\rho {\mathrm{/}}dt = [{\cal H}_{{\mathrm{sys}}} + {\cal H}_{\mathrm{n}},\rho ]\), where the time sequences for δε(t) are obtained by generating a white noise sequence, then scaling its Fourier transform by an appropriate spectral function,55 such as Eq. (3). We then average the density matrix 〈ρ(t)〉 over noise realizations. Details of these procedures are provided in Supplementary Sec. S4.

Analytical formalism

We analytically solve the dynamical equation \(i\hbar {\kern 1pt} d\rho ^I{\mathrm{/}}dt = \delta \varepsilon (t){\cal L}\rho ^I\) in the interaction frame, where \({\cal L}\rho ^I \equiv [h_n^I,\rho ^I]\) and \(h_n^I = U_0^\dagger h_nU_0\). We then average over the noise via a cumulant expansion,33,34 truncated at O[(δε/ħΩ)2], yielding \(\left\langle {\rho ^I(t)} \right\rangle\) = \({\mathrm{exp}}\left[ { - \frac{1}{{\hbar ^2}}{\int}_0^t {\kern 1pt} dt_1{\int}_0^{t_1} {\kern 1pt} dt_2{\cal L}(t_1){\cal L}(t_2)S(t_1 - t_2)} \right]\rho ^I(0)\), where we have assumed that the noise is stationary, with zero mean (〈δε〉 = 0). Details of the calculations are provided in Supplementary Sec. S3.