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. A study shows that pulsing techniques can suppress an important source of errors in qubits formed in semiconductor quantum dots. Quantum bits (qubits) are the building block of quantum technologies, and building high-quality qubits is thus a key objective. Semiconductor quantum dot-based charge qubits represent the quintessential two-level system, which enjoys both ease of control and efficient readout. Although a promising platform, such a system suffers from charge noise, which limits it potential use in applications. A team of researchers from University of Wisconsin-Madison use analytical methods to show that special pulsing techniques could be used to suppress the errors due to charge noise, as well as those from strong driving fields. This study reaffirms quantum dot charge qubits as a promising platform quantum computation.


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, 1-3 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 frequency [8][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 singlequbit 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.

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 H sys = H q þ H ac , where H q = −(ε/2)σ x − Δσ z , the σ i 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 H sys in the eigenbasis 0 , 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 ¼ ðA ε =2Þσ x cosðω d t þ ϕÞ, 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.
First, we follow ref. 11 and obtain exact solutions for stronglydriven qubits in the absence of noise, up to arbitrary order in the strong-driving parameter γ = A ε /(16Δ). Expanding the timeevolution operator order-by-order as U 0 (t) =  Fast oscillations can cause gate infidelity. For example, if we consider an R θ (ϕ) rotation of angle θ about the (cosϕ, −sinϕ, 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. 1b. 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 ð0Þ 0 to the actual evolution U 0 , for R θ (ϕ) in the rotating frame. [see Eq. (S2) of Supplementary Section S1 for a precise definition of the process fidelity]. We find that specific A ε 's give rotations with perfect fidelity when θ = π, 2π, 3π, … for any ϕ.

Charge noise
We introduce charge noise into our analysis through the Hamiltonian term H n ¼ h n δεðtÞ, where δε(t) is a random variable affecting the detuning parameter and h n = −σ x /2 is referred to as the noise matrix. 1,3,15,16 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 isSðωÞ ¼ R 1 À1 dt e iωt SðtÞ. 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 where c ε is related to the standard deviation of the detuning 6,22 and ω l (ω h ) 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 U 0 , ρ I ¼ U y 0 ρU 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, 6,7,23-27 dissipative Lander-Zener-Stückelberg interferometry, 28,29 and treatments of spin-Boson systems. 30 Here, I 2 is the 2 × 2 identity matrix, r I ¼ r I x ; r I y ; r I z is the Bloch vector, and K(t) is a 3 × 3 evolution matrix, given by (4) where we have expand the noise matrix in the interaction frame into Fourier components h I n ðtÞ U y 0 h n U 0 ¼ P i;ω α i;ω e iωt , and defined Iðt; ω 1 ; ω 2 Þ R t 0 dt 1 R t1 0 dt 2 e iω1t1 e iω2t2 S t 1 À t 2 ð Þ. Since U 0 can be expressed order-by-order in γ, the same is also true of α i,ω , allowing us to distinguish the effects arising in the weak-drive limit, O[γ 0 ], from the strong-driving limit, O[γ n ], 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 K M , and non-Markovian terms. The latter may be further divided into pure-dephasing terms K φ , 26,35,36 and non-Markovian-nondephasing 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). 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 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 [see Supplementary Eq. (S34)], and exponential decay (~e −Γt ). The dominant non-Markovian-nondephasing 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 K nMnφ ) within a common framework, allowing us to compare and contrast their effects.

Asymptotic solutions
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 Iðt; ω; ÀωÞ %SðωÞt=2 þS imag ðωÞt, whose imaginary part is given byS imag ðωÞ c 2 ε 2i=ω ð Þln ω=ω l j j. The real part describes exponential decay, giving the Markovian decoherence rate for driven evolution, 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 fre- , originating from the integrated low-frequency ("quasistatic") portion of the noise spectrum, 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), f Rabi /Γ Rabi , corresponding to the number of coherent R 2π (π/4) rotations within a Rabi decay period T Rabi = 1/Γ Rabi (not including strong-driving control errors), where the Rabi decay rate Γ Rabi is determined from Eq. (5) such that ρ I 00 t ¼ 1=Γ Rabi À Á ð1 þ e À1 Þ=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 A ε both enhance the FOM, as shown in the main panel. By increasing Δ 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). In the lower inset, we fix Δ/h = 5 GHz and A ε /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 R π (π/4) gate fidelity We now compute process fidelities for R π (π/4) gates, using the χmatrix method described in Supplementary Sec. S1. Control errors due to strong driving are investigated by considering U ð0Þ 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 infidelity, enabled by synchronized oscillations. For ϕ = π/4, the dip minima are proportional to γ 4 , while their widths are proportional to γ 2 [see Supplementary Eq. (S13)], suggesting potential benefits of working at large A ε ∝ γ. As c ε increases, the infidelity also grows, including both Markovian (K M ), and non-Markovian contributions (K φ and K nMnφ ). Initially, the envelope of the infidelity oscillations decreases with A ε , 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 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. 3a, which are consistent with recent experiments, 1,3,15,16 we obtain F max ≲ 99%, which is insufficient for achieving high-fidelity gates. However, the following procedure can be used to suppress both control errors and decoherence. First, A ε is tuned to a dip. Then, Δ and A ε are simultaneously increased while holding γ = A ε /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 c ε = 1 μeV (σ ε = 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 doubledot 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 qubits 38,39 and quantum-dot singlet-triplet qubits, 15,40,41 and can be extended to systems with multiple levels, including quantum-dot hybrid qubits 3,11,[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 highorder 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, 52-54 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

Numerical Simulation
We numerically simulate the Schrodinger equation of a strongly driven, noisy charge qubit, i hdρ=dt ¼ ½H sys þ H n ; ρ, 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 hdρ I =dt ¼ δεðtÞLρ I in the interaction frame, where Lρ I ½h I n ; ρ I and h I n ¼ U y 0 h n U 0 . We then average over the noise via a cumulant expansion, 33

DATA AVAILABILITY
The data and numerical codes that support the findings of this study are available from the corresponding author upon reasonable request. Sim.