Abstract
Identifying and ameliorating dominant sources of decoherence are important steps in understanding and improving quantum systems. Here, we show that the free induction decay time (\(T_2^*\)) and the Rabi decay rate (Γ_{Rabi}) of the quantum dot hybrid qubit can be increased by more than an order of magnitude by appropriate tuning of the qubit parameters and operating points. By operating in the spinlike regime of this qubit, and choosing parameters that increase the qubit’s resilience to charge noise (which we show is presently the limiting noise source for this qubit), we achieve a Ramsey decay time \(T_2^*\) of 177 ns and a Rabi decay time 1/Γ_{Rabi} exceeding 1 μs. We find that the slowest Γ_{Rabi} is limited by fluctuations in the Rabi frequency induced by charge noise and not by fluctuations in the qubit energy itself.
Introduction
There has been much progress in the development of qubits in semiconductor quantum dots,^{1} making use of one,^{2,3,4,5,6,7,8,9,10,11} two,^{12,13,14,15,16,17,18,19,20} and three quantum dots^{21,22,23,24,25,26} to host qubits. Charge noise is often the leading source of decoherence in semiconductor qubits,^{27} and an advantage of using two or more quantum dots to host a single qubit is the ability to work at sweet spots, a technique pioneered in superconducting qubits,^{28} that make the qubit more resistant to charge noise.^{29,30,31,32,33,34,35,36}
In this work we focus on one such qubit, the quantum dot hybrid qubit (QDHQ),^{37,38,39,40,41,42,43,44,45} which is formed from three electrons in a double quantum dot, and can be viewed as a hybrid of a spin qubit and a charge qubit. Fast, full electrical control of the QDHQ was recently implemented experimentally using ac gating,^{46} demonstrating a free induction decay (FID) time of 11 ns through operation in the spinlike operating region (see Fig. 1). While QDHQ gating times are fast, substantial further improvements in QDHQ coherence times are required to achieve the highfidelity gating necessary for faulttolerant operation.^{47} True sweet spots, which are used to increase resistance to noise and thus increase coherence, are defined by a zero derivative of the qubit energy with respect to a parameter subject to noise. Sweet spots are usually found at specific points of zero extent in parameter space, so that noninfinitesimal noise amplitude temporarily moves a qubit off the sweet spot. The spinlike regime of the QDHQ has no true sweet spot; however, it has a large and extended region of small dE_{Q}/dε, where E_{Q} is the qubit energy and ε is the detuning between the two quantum dots.
Here, we show that the spinlike operating regime for the QDHQ can be made resilient to charge noise by appropriate tuning of the internal parameters of the qubit. By measuring dE_{Q}/dε, we are able to identify dot tuning parameters that increase resiliency to charge noise. These measurements show that the threeelectron QDHQ can be tuned insitu in ways that have a predictable and understandable impact on the qubit coherence: the qubit dispersion can be tuned smoothly by varying device gate voltages, and we find that the dephasing rate is proportional to dE_{Q}/dε, consistent with a charge noise dephasing mechanism. Reducing dE_{Q}/dε significantly enhances the coherence of the qubit. We have achieved an increase the coherence times by more than an order of magnitude over previous work, decreasing the Rabi decay rate Γ_{Rabi} from 67.1 to 0.98 MHz, and increasing the FID time \(T_2^*\) to as long as 177 ns. These parameters correspond to an infidelity contribution from pure dephasing of about 1%.
Results
Figure 1 shows the energy levels of the QDHQ as a function of the detuning ε. At negative detuning the energy difference between the 0〉 and 1〉 states is dominated by the Coulomb energy, while at large positive detunings, where both logical states have the same electron configuration (one electron on the left and two on the right), the energy difference is dominated by the singleparticle splitting E_{R} between the lowest two valleyorbit states in the right dot. Here, the logical states are described by their spin configuration: 0〉 = ↓〉S〉 and 1〉 = \(\sqrt {1{\rm{/}}3} \)↓〉T_{0}〉−\(\sqrt {2{\rm{/}}3} \)↑〉T_{−}〉, where ↓〉 and ↑〉 represent the spin configuration of the single electron in the left quantum dot and S〉, T_{0}〉, and T_{−}〉 represent the singlet (S) and triplet (T_{0}, T_{−}) spin configurations of the two electrons in the right quantum dot. The tunnel coupling Δ_{1(2)} describes the anticrossings between the right dot ground (first excited) state and left dot ground state.
Figure 2a–g shows results of FID measurements for four different values of the measured dE_{Q}/dε, performed using the pulse sequence of diagram IV of Fig. 1, in order to determine \(\Gamma _2^*{\rm{ = }}1{\rm{/}}T_2^*\). For short times (panels a, c, e), Ramsey fringes are visible for all dE_{Q}/dε; in contrast, for t_{Free} = 22 ns, Ramsey fringes are attenuated in Fig. 2b (large dE_{Q}/dε), yet are still clearly visible in Fig. 2f (small dE_{Q}/dε). As shown in Fig. 2g, by tuning the qubit to achieve dE_{Q}/dε = 0.0025, Ramsey fringes are still visible at t_{Free} = 120 ns, and at this tuning a Gaussian fit to the Ramsey fringe amplitude (shown in Fig. 2h) yields \(T_2^*{\rm{ = }}177 \pm 9\,{\rm{ns}}\). Fits to the Ramsey fringe amplitude of the other three detunings are shown in Fig. 2i, demonstrating a strong correlation between small dE_{Q}/dε and long \(T_2^*\). Although we have shown Gaussian fits in Fig. 2, consistent with quasistatic charge noise, we note that the FID decay also can be fit by an exponential decay, which would be consistent with noise that is dominated by only a few twolevel fluctuators,^{48} and therefore we cannot distinguish between these two limiting cases (see Supplemental Material for fit parameters extracted from exponential decays).
Figure 2j shows \(\Gamma _2^*{\rm{ = }}1{\rm{/}}T_2^*\) for a wide range of dE_{Q}/dε, demonstrating a significant improvement in coherence for reduced values of dE_{Q}/dε. For a Gaussian distribution of quasistatic fluctuations of the detuning parameter, with a standard deviation of σ_{ε}, one expects that^{15, 27}
In Fig. 2j, we observe such a linear relation between \(\Gamma _2^*\) and dE_{Q}/dε, with a fitting constant σ_{ε} = 4.39 ± 0.32 μeV.
We now turn to a discussion of the Rabi decay time, 1/Γ_{Rabi}, and its dependence on the qubit dispersion dE_{Q}/dε. Figure 3a shows both E_{Q} and dE_{Q}/dε as a function of detuning, calculated using the measured tuning parameters for Fig. 3b–e (see Supplementary Section 1 and 4), showing the decrease in the slope dE_{Q}/dε with increasing ε. Figure 3b–e shows Rabi oscillation measurements, performed with a microwave burst of duration t_{RF} and acquired at the detunings labeled b–e in Fig. 3a, showing that with increasing ε (and therefore decreasing dE_{Q}/dε) the Rabi decay rate Γ_{Rabi} decreases by more than an order of magnitude for the data reported here.
For quantum gates, the contribution to infidelity arising from qubit decoherence is minimized when the ratio of the gate duration to the Rabi decay time is minimized. The data in Fig. 3f, acquired at a different dot tuning, show that this ratio can be made small enough that an X_{π/2} gate can be performed over 100 times within one Rabi decay time. In the absence of any other nonideality in the experiment, this would limit the fidelity of an X_{π/2} rotation on the Bloch sphere to 99.0% and would represent a sevenfold improvement over previous results.^{46}
It is also interesting to consider how long the Rabi decay time, 1/Γ_{Rabi}, itself can be. Figure 3g shows Rabi oscillations acquired at a different dot tuning and a very small dE_{Q}/dε = 0.005. Here, Γ_{Rabi} = 0.98 MHz, representing a decrease by more than a factor of 30 from previously reported Rabi decay rates.^{46}
The decay of Rabi oscillations is caused by at least two different mechanisms,^{49} both of which are observed in these experiments. First, for relatively large values of dE_{Q}/dε, fluctuations in E_{Q} from charge noise dominate the decoherence. This is similar to FID measurements, with the important difference that the microwave drive effectively reduces the range of frequencies decohering the qubit. This results in Rabi decoherence rates Γ_{Rabi} that are slower than the FID rates \(\Gamma _2^*\) at the same dE_{Q}/dε. For this mechanism, the Rabi decay is expected to be exponential and depend quadratically on dE_{Q}/dε.^{48, 50} Figure 3h shows Γ_{Rabi} vs. dE_{Q}/dε and a quadratic fit to the data; the data are well described by this functional form, and decreasing dE_{Q}/dε yields nearly two orders of magnitude decrease in Γ_{Rabi}.
Second, charge noise can also cause fluctuations in the rotation rate f_{Rabi} itself,^{49} and as dE_{Q}/dε becomes small, these fluctuations become the dominant source of decoherence. This second decay process is expected to yield a decay rate proportional to the drive amplitude A_{ε}, and as shown in Fig. 3i we observe this proportionality in the experiment for small dE_{Q}/dε. Thus, for small dE_{Q}/dε, fluctuations in f_{Rabi} dominate the Rabi decay rate. In contrast to the Rabi decay process discussed above, in which the applied microwave pulse narrows the frequency range of charge fluctuations contributing to the decay, charge fluctuations over a wide bandwidth are expected to contribute to this decay process. This contribution can be seen by applying the rotating wave approximation to Eq. (S1) in Supplementary Section 1, which yields an approximate form for f_{Rabi} that is valid at large detunings:
σ_{ε} can then be related to σ_{Rabi}, the standard deviation of fluctuations in f_{Rabi}, by
We therefore expect the decay rate from this mechanism to be proportional to dE_{Q}/dε rather than to the square of dE_{Q}/dε, explaining its dominance at small dE_{Q}/dε.
Discussion
In this work we have shown that the internal parameters of the QDHQ can alter the qubit dispersion dE_{Q}/dε over a wide range, resulting in large tunability in both the decoherence rates and the Rabi frequencies achievable. The dominant dephasing mechanism for Rabi oscillations switches from fluctuations in the qubit energy E_{Q} to fluctuations in the Rabi frequency f_{Rabi} at the smallest values of dE_{Q}/dε. By decreasing dE_{Q}/dε we have reduced both the Rabi and the Ramsey decoherence rates, important metrics for achieving highfidelity quantum gate operations, by more than an order of magnitude compared with previous work, demonstrating Γ_{Rabi} as small as 0.98 MHz and \(T_2^*{\rm{ = }}1{\rm{/}}\Gamma _2^*\) as long as 177 ns. These coherence times exhibit the utility of the extended nearsweet spot in the QDHQ for improving qubit performance in the presence of charge noise.
Methods
The Si/SiGe device is operated in a region where magnetospectroscopy measurements^{3, 51} have indicated that the valence electron occupation of the double dot is (1,2) for the qubit states studied here. Manipulation pulse sequences were generated using Tektronix 70001 A arbitrary waveform generators and added to DC gate voltages on gates L and R using bias tees (PSPL5546). Because of the frequencydependent attenuation of the bias tees, corrections were made to the applied pulses during the adiabatic detuning pulses, as described in Supplementary Section 5. The qubit states were mapped to the (1,1) and (1,2) charge occupation states as described in ref. 46. A description of the methods used to measure the qubit dispersion and lever arm can be found in Supplementary Section 4.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work was supported in part by ARO (W911NF1710274, W911NF120607, W911NF0810482), NSF (DMR1206915, PHY1104660, DGE1256259), and the Vannevar Bush Faculty Fellowship program sponsored by the Basic Research Office of the Assistant Secretary of Defense for Research and Engineering and funded by the Office of Naval Research through grant N000141510029. Development and maintenance of the growth facilities used for fabricating samples is supported by DOE (DEFG0203ER46028). We acknowledge the use of facilities supported by NSF through the UWMadison MRSEC (DMR1121288). The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office (ARO), or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
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B.T. performed electrical measurements with L.W.S. and J.C., developed measurement techniques with D.K. and R.H.F., and analyzed the data with Y.Y., M.A.E., M.F., and S.N.C. D.W.R. developed hardware and software for the measurements. C.B.S. fabricated the quantum dot device. D.E.S. and M.G.L. prepared the Si/SiGe heterostructure. All authors contributed to the preparation of the manuscript.
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Correspondence to M. A. Eriksson.
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Thorgrimsson, B., Kim, D., Yang, Y. et al. Extending the coherence of a quantum dot hybrid qubit. npj Quantum Inf 3, 32 (2017). https://doi.org/10.1038/s4153401700342
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