Extending the coherence of a quantum dot hybrid qubit

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 (T2*\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_2^*$$\end{document}) and the Rabi decay rate (ΓRabi) 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 spin-like 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 T2*\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_2^*$$\end{document} of 177 ns and a Rabi decay time 1/ΓRabi exceeding 1 μs. We find that the slowest ΓRabi is limited by fluctuations in the Rabi frequency induced by charge noise and not by fluctuations in the qubit energy itself. Researchers in the United States demonstrate high tunability of spin qubits in silicon-based quantum dots. Mark Eriksson at the University of Wisconsin-Madison and colleagues have achieved more than a tenfold improvement in the performance of these three-electron double dot qubits by tuning the electric fields used to confine electrons to quantum dots to a regime where the qubit was predicted to be much less susceptible to the effects of charge noise. Since charge noise limits the performance of many such qubits, these findings provide a path toward the fabrication of electrically gated qubits in silicon quantum dots with very high fidelities.


INTRODUCTION
There has been much progress in the development of qubits in semiconductor quantum dots, 1 making use of one, 2-11 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 spin-like operating region (see Fig. 1). While QDHQ gating times are fast, substantial further improvements in QDHQ coherence times are required to achieve the high-fidelity gating necessary for fault-tolerant 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 non-infinitesimal noise amplitude temporarily moves a qubit off the sweet spot. The spin-like 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 spin-like 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 three-electron QDHQ can be tuned in-situ 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 Ã 2 to as long as 177 ns. These parameters correspond to an infidelity contribution from pure dephasing of about 1%. 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 valley-orbit states in the right dot. Here, the logical states are described by their spin configuration: |0〉 = |↓〉|S〉 and |1〉 = ffiffiffiffiffiffiffi ffi 1=3 p |↓〉|T 0 〉− ffiffiffiffiffiffiffi ffi 2=3 p |↑〉|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.  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 Ã 2 ¼177 ± 9 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 Ã 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 two-level 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 Γ Ã 2 ¼1=T Ã 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
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 Γ Ã 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  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 high-fidelity 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 Ã 2 ¼1=Γ Ã 2 as long as 177 ns. These coherence times exhibit the utility of the extended near-sweet 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 frequency-dependent 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. Fig. 3 Rabi decay rate is limited by charge noise and applied microwave power, A ε . a Plots of E Q = hf Q (black) and dE Q /dε (red) vs. ε for the tuning used in b-e. Here, f Q is the qubit frequency, and the spectroscopy methods used are described in Supplementary Section 4. b-e Rabi oscillations of the probability P 1 of being in state |1〉, all obtained at the same tuning but at different ε, ranging from~210 to 340 μeV. Γ Rabi clearly decreases as dE Q /dε decreases. The decrease in f Rabi between b-e is caused by the decreased coupling to the left dot as ε is increased (see Eq. (2)). A ε is nominally the same but changes slightly between b-e due to changes in f Q as discussed in Supplementary Section 5. f Rabi oscillations, taken at a different device tuning, demonstrating over 100 coherent X π/2 rotations within a Rabi decay time. g Rabi oscillations demonstrating a Rabi decay time longer than 1 μs, taken at a device tuning differing from those in b-f. h Γ Rabi , obtained by fitting to an exponential decay, plotted as a function of dE Q /dε for measurements at multiple tunings and operating points, with A ε within 10% of 25 μeV. Here, the black line has slope 2, indicating that Γ Rabi depends quadratically on dE Q /dε, consistent with Γ Rabi being limited by fluctuations of f Q . 48,50 Here, the different tunings are labeled with different colors (red, green, and blue), as specified in Supplementary Section 1. i Rabi oscillations taken at ε = 323 μeV (dE Q /dε = 0.005), as a function of the microwave amplitude A ε , showing that the Rabi decay rate Γ Rabi ∝ A ε , consistent with Γ Rabi being limited by fluctuations of f Rabi for small values of dE Q /dε Extending the coherence of a quantum dot hybrid qubit B Thorgrimsson et al.
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