Abstract
We implement resonant single qubit operations on a semiconductor hybrid qubit hosted in a threeelectron Si/SiGe double quantum dot structure. By resonantly modulating the double dot energy detuning and employing electron tunnellingbased readout, we achieve fast (>100 MHz) Rabi oscillations and purely electrical manipulations of the threeelectron spin states. We demonstrate universal single qubit gates using a Ramsey pulse sequence as well as microwave phase control, the latter of which shows control of an arbitrary rotation axis on the X–Y plane of the Bloch sphere. Quantum process tomography yields π rotation gate fidelities higher than 93 (96)% around the X (Z) axis of the Bloch sphere. We further show that the implementation of dynamic decoupling sequences on the hybrid qubit enables coherence times longer than 150 ns.
Introduction
Isolated spins in semiconductors provide a promising platform to explore quantum mechanical coherence and develop engineered quantum systems.^{1–13} Silicon has attracted great interest as a host material for developing spin qubits because of its weak spinorbit coupling and hyperfine interaction, and several architectures based on gate defined quantum dots have been proposed and demonstrated experimentally.^{14,15} Recently, a quantum dot hybrid qubit formed by three electrons in a double quantum dot was proposed,^{16,17} and nonadiabatic pulsedgate operation was implemented experimentally,^{18} demonstrating simple and fast electrical manipulations of spin states with a promising ratio of coherence time to manipulation time. However, the overall gate fidelity of the pulsegated hybrid qubit is limited by relatively fast dephasing due to charge noise during one of the two required gate operations. Here we perform the first microwavedriven gate operations of a quantum dot hybrid qubit, avoiding entirely the regime in which it is most sensitive to charge noise. Resonant detuning modulation along with phase control of the microwaves enables a π rotation time of <5 ns (50 ps) around X (Z) axis with high fidelities >93 (96)%. We also implement Hahn echo^{19–21} and Carr–Purcell (CP)^{22} dynamic decoupling sequences with which we demonstrate a coherence time of over 150 ns. We further discuss a pathway to improve gate fidelity to above 99%, exceeding the threshold for surface code based quantum error correction.^{23}
The quantum dot hybrid qubit combines desirable features of charge (fast manipulation) and spin (long coherence time) qubits. The qubit states can be written as 0〉=↓〉S〉, where S denotes a singlet state in the right dot, and \mathrm{1}\u3009\mathrm{=1/}\sqrt{3}\downarrow \u3009{T}_{0}\u3009\sqrt{\mathrm{2/3}}\uparrow \u3009{T}_{}\u3009, where T_{0} and T_{_} are two of the triplet states in the right dot. The states 0〉 and 1〉 have the nearly same dependence on ε in the range of detuning at which the qubit is operated (Figures 1c and d), enabling quantum control that is largely insensitive to charge fluctuations. Moreover, electric fields couple to the qubit states and enable highspeed manipulation.^{16,17,24–27} Previously, we experimentally demonstrated nonadiabatic quantum control (direct current (DC)pulsed gating) of the hybrid qubit, where the manipulation and measurement scheme required the use of a detuning regime that is sensitive to charge noise (with ε near but not equal to zero—see Figure 1d).^{18} Moreover, DC gating requires abrupt changes in detuning. With a given bandwidth in the transmission line, pulse imperfections arising, e.g., from frequency dependent attenuation, lead to inaccurate control of rotation axes. In this work, we demonstrate resonant microwavedriven control and statedependent tunnelling readout of the qubit, which together overcome this limitation of DCpulsed gating and enable full manipulation on the Bloch sphere at a single operating point in detuning that is wellprotected from charge noise.
The experiments here are performed in a double dot with a gate design as shown in Figure 1a and with electron occupations as shown on the stability diagram of Figure 1b. The electron occupations and energy level alignments used for qubit initialisation, readout and microwave spectroscopy of the qubit states are shown schematically in Figure 1c. All the experiments reported here start with an initial dot occupation of (1,2) and with the system in state 0〉, prepared at a detuning ε≈230 μeV; this detuning is also used for measurement and corresponds to point M in Figure 1b. After initialisation, we apply a microwave burst pattern at point O, which either coincides with point M or is reached through an adiabatic ramp in detuning (the latter case is illustrated in Figure 1b). The tunnel coupling between the two sides of the double quantum dot mediates an exchange interaction that enables transitions between the qubit states and can be driven by modulating the detuning.^{16} Qubit rotation occurs when the frequency of the applied microwave electric field is resonant with the qubit energy level difference. The measurement point M is chosen so that the Fermi level of the right reservior is in between the energies of 0〉 and 1〉, and we use qubit statedependent tunnelling to project states 0〉 and 1〉 to the (1,2) and (1,1) charge states, respectively. Waiting at point M for ∼10 μs also resets the qubit to state 0〉, by tunnelling an electron from the reservoir, if needed. Thus, the qubit state population following the microwave burst is measured by monitoring the current I_{QPC} (QPC, quantum point contact) through the chargesensing quantum point contact (Figure 1a). Details of the measurement procedure and probability normalisation are in Supplementary Information S1.
Materials and methods
The details of the Si/SiGe double quantum dot device are presented in refs 28 and 29. We work in the region of the charge stability diagram where the valence electron occupation of the double dot is (1,1) or (1,2), as confirmed by magnetospectroscopy measurements.^{29,30} All manipulation sequences, including the microwave bursts, are generated by a Tektronix 70002A arbitrary waveform generator and are added to the dotdefining DC voltage through a bias tee (Picosecond Pulselabs 5546107) before being applied to gate R. We map the states 0〉 and 1〉 to the (1,2) and (1,1) charge occupation states, respectively, leading to conductance changes through the quantum point contact. We measure with a lockin amplifier (EG&G model 7265, Oak Ridge, TN, USA) the difference in conductance with and without the applied microwave burst. When converting time averaged conductance differences to the reported probabilities, tunnelling between the (1,2) and (1,1) charge states during the measurement phase is taken into account using the measured times for tunnelling out of (T_{o}≃200 ns) and into (T_{i}≃2.1 μs) the dot. Supplementary Information S1 presents the details of the measurement technique and the probability normalisation.
Results
We perform microwave spectroscopy of the qubit intrinsic frequency—the energy difference δE in Figure 1d—by applying the voltage pulse shown in the inset to Figure 1e. The colour plot in that figure shows the resulting probability of measuring state 1〉 after applying this pulse to initial state 0〉. The measured resonance and qubit energy dispersion agrees well with the green dashed curve, which is the calculated energy level diagram with Hamiltonian parameters measured in our previous study.^{18} As is clear from the colour plot in Figure 1e, the linewidth of the resonant peak narrows significantly at ε>200 μeV, becoming much narrower than the resonance in the charge qubit regime (ε≈0).^{31} This linewidth narrowing corresponds to an increase in the inhomogeneous dephasing time, and it is this range in detuning that corresponds to the hybrid qubit regime. The two states in the right quantum dot that are separated by δE most likely correspond to two combinations of the zvalleys, which are weakly mixed by the step in potential at the quantum well interface.^{32}
Applying microwave bursts to gate R in the hybrid qubit regime yields Rabi oscillations, as shown in Figures 1f–i. The Rabi frequency increases as a function of increasing microwave amplitude V_{ac} (measured at the arbitrary waveform generator), resulting in Rabi frequencies as high as 100 MHz. Figure 1j shows the power dependence of the qubit oscillations, revealing an oscillation frequency that is linear in the applied amplitude, as expected for Rabi oscillations. The speed of the X axis rotation demonstrated here is comparable to electrically manipulated spin rotations in InSb and InAs, which rely on strong spinorbit coupling of the host material;^{33,34} here we achieve fast rotations solely through electric field coupling to the qubit states. This coupling is also highly tunable, since it is determined by the ground and excited state interdot tunnel couplings.^{35} Below, we characterize gates with the qubit frequency chosen to be ≈11.52 GHz.
We characterize the inhomogeneous dephasing time by performing a Ramsey fringe experiment, which also demonstrates Z axis rotations on the qubit Bloch sphere. The microwave pulse sequence is shown schematically in Figure 2a. We first prepare the state Y\u3009=\sqrt{\mathrm{1/2}}\mathrm{(0}\u3009+i\mathrm{1}\u3009) by performing an X_{π/2} rotation. Z axis rotation results from the evolution of a relative phase between states 0〉 and 1〉, given by \mathit{\phi}={t}_{\mathrm{e}}\delta E/\hslash, where t_{e} is the time spent between the state preparation and measurement X_{π/2} pulses, the latter of which is used to project the Y axis component onto the Z axis. The final probability P_{1} is measured as described above. Figure 2b shows the resulting quantum oscillations as a function of V_{L}, which controls the detuning energy, and t_{e}. In Figure 2b, the frequency of the oscillations increases slightly as V_{L} becomes more negative, as the qubit energy levels are not quite perfectly independent of detuning. Figure 2c shows a line cut taken near the optimal resonant condition (V_{L}≈−392 mV), showing clear oscillations in P_{1} consistent with the qubit frequency of ≈11.5 GHz. By fitting the oscillations to an exponentially damped sine wave (red solid curve), we extract an inhomogeneous dephasing time {T}_{2}^{*}\mathrm{=11}\mathrm{ns}, consistent with the value measured previously with nonadiabatic pulsed gating on the same device with similar intrinsic qubit frequency.^{18} We estimate the typical tunnellingout time T_{o}≈200 ns (Supplementary Information S1), so that the inhomogeneous coherence time is not likely limited by electron tunnelling to the reservoir during the measurement phase.
Resonant microwave drive also enables arbitrary twoaxis control on the X–Y plane of the Bloch sphere by varying the relative phase ϕ of the X_{π/2} pulses. Figure 2d shows a measurement of P_{1}, demonstrating both twoaxis control and phase control. Starting from a maximum (minimum) P_{1} at ϕ=0, when we apply X_{π/2}Ω_{π/2} (X_{π/2}Z_{π}Ω_{π/2}) on the state 0〉, P_{1} oscillates as a function of the relative phase ϕ that determines the axis of the Ω_{π/2},_{ϕ} rotation. The deviation from an ideal sinusoidal oscillation stems from limited phase resolution of our method of waveform generation (Supplementary Information S1).
We now turn to echo and dynamic decoupling pulse sequences. Figure 3a shows Hahn echo^{19–21} and CP dynamic decoupling^{22} pulse sequences. Provided that the source of dephasing fluctuates slowly on the timescale of the electron spin dynamics, inserting an X_{π} pulse between state initialisation and measurement, which is performed with X_{π/2} gates, corrects for noise that arises during the time evolution. Figure 3b shows a typical echo measurement. While keeping the total free evolution time τ fixed at 10 ns, we sweep the position of the decoupling X_{π} pulse to reveal echoed oscillations.^{21,28} In Figure 3b, the oscillations of P_{1} as a function of δt are at twice the Ramsey frequency (2F_{Ramsey}≈23 GHz), as expected for an echo measurement, and the visibility of the oscillations is ~0.35, because the data were acquired at the relatively long free evolution time of 10 ns.
Improvement in coherence times can be obtained by implementing CP sequences (Figure 3a), which use multiple X_{π} pulses inserted during the free evolution. Since the timescale for the CP sequence is typically longer than T_{o}≈200 ns, an adiabatic detuning offset of amplitude ≈60 μeV was applied to shift point O (Figure 1b) during free evolution in order to prevent electron tunnelling to the reservoir. In the absence of dephasing, the CP sequence with an even number of X_{π} pulses applied on the state 0〉 yields P_{1}=1. The measured P_{1} as a function of τ, shown in Figure 3c, decays exponentially due to dephasing: {P}_{1}\left(\tau \right)=0.5+A{e}^{{(\tau /{T}_{2})}^{\alpha}}, where α depends on the frequency spectrum of the dominant noise sources.^{36} Figure 3d shows the results of fits as a function of the number n of decoupling X_{π} pulses with fixed α=2, 3 and 4. The resulting coherence time T_{2} shows more than an order of magnitude improvement (>150 ns) with n=8, and the resulting times are approximately independent of the α used in the fit. Beyond n=8 we typically observe a decrease in T_{2}, which is not completely understood at this time. We expect that optimisation of microwave pulses can increase T_{2} further.
Characterisation of the fidelity
We now present tomographic characterisation of the microwavedriven hybrid qubit. Because this device does not allow singleshot measurement of this qubit, the short number of pulses required for quantum process tomography make that procedure more reliable for the present work than randomized benchmarking, even though at high values of gate fidelity quantum process tomography is typically less reliable than randomised benchmarking.^{37} Here, to reconstruct the time evolution of the single qubit density matrix, we use the microwave pulse sequences shown schematically in Figure 4a to perform repeated state evolution under an X(Z) gate and perform independent X, Y and Z axes projective measurements. For state tomography under X(Z) axis rotation, we prepare initial states near 0〉 and near Y〉. After time evolution under the gate operation, we measure X, Y and Z axes projections of the timeevolved Bloch vector using −Y(π/2), X(π/2), and identity operations, respectively, and measure the resulting P_{1}. Note that the pulse sequences shown in Figure 4a represent state tomography in the rotating (laboratory) frame for Rabi (Ramsey) oscillations, because the phase of the second π/2 pulses for the Rabi oscillation tomography evolves as the length of Rabi manipulation t_{b} is increased, whereas the second microwave pulse in the state tomography of the Ramsey fringes has fixed relative phase with respect to the first microwave pulse. Figure 4b and c show X (black circles), Y (green triangles) and Z (orange squares) axes projections of the timeevolved Bloch vector under continuous X (b) and Z (c) axes rotation gates.
On the basis of the density matrices obtained from the state tomography, we implement quantum process tomography (QPT) to extract fidelities of single qubit gates on the alternating current (AC)driven hybrid qubit through the relation,^{18,38,39}
where ε(ρ) is the density matrix specifying the output for a given input density matrix ρ, the {\tilde{E}}_{m} are the basis operators in the space of 2×2 matrices, and χ is the process matrix. Experimentally, four linearly independent input and output states are chosen from continuous evolution of the state under X and Z axes rotations available from the state tomography data set, and the maximum likelihood method^{18,39} is used to determine χ. Figures 4d–g show the results of QPT (symbols) performed on the π/2 and π rotations around the X and Z axes and comparison to corresponding ideal rotation process matrices (bars). The error bars of length ≈0.01–0.02 represent the standard deviation of the experimental result obtained by 10 distinct input and output density matrices chosen from the state tomography data. The process matrices χ obtained from QPT in the Pauli basis {I, σ_{x}, σ_{y}, σ_{z}} yield process fidelities {F}_{\mathrm{p}}=Tr({\mathit{\chi}}_{\mathrm{ideal}}\mathit{\chi}) of 93% and 96% for π rotations around the X and Z axes, respectively. Comparing these results to the process fidelities of 85% and 94% for X and Z axis rotations reported previously for the nonadiabatic DCpulsegated hybrid qubit,^{18} we find more than a factor of two reduction in the X axis rotation infidelity. The pulses we applied in this work were turned on abruptly, and pulse sequences for consecutive gates were concatenated without gaps, both of which can decrease fidelity; optimisation of the pulse sequences, like that performed in ref. 40, offers opportunities for improvement.
Discussion
The improvement in overall fidelity of the ACgated quantum dot hybrid qubit demonstrated here compared with DCpulsed gating stems mainly from (1) elimination of the need to enter the regime in which the qubit is sensitive to charge noise by using resonant manipulation and tunnellingbased readout, and (2) reduced rotation axis and angle errors because resonant driving with fixed frequency enables more accurate control of these quantities. The AC driving in this work was performed by resonantly modulating the energy detuning between the dots. For this type of modulation, the ratio of manipulation time (Rabi period) to coherence time depends strongly on the strength of ground and excited state tunnel couplings,^{35} and thus we expect that further improvement in fidelity can be achieved by increasing these tunnel couplings, making the energy level dispersion flatter, while maintaining high gate speeds. Moreover, recent theoretical work suggests that dynamically modulating tunnel coupling instead of detuning can enable Rabi frequencies exceeding 1 GHz while keeping long coherence times, enabling achievement of gate fidelites exceeding 99%.^{41} ACgating also enables much greater flexibility in the design and operation of quantum gates, as recently demonstrated for quantum control of spins on phosphorous in Si,^{42} and similar approaches should be possible for both one and twoqubit gates for the hybrid quantum dot qubit.
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Acknowledgements
We thank Xuedong Hu, John King Gamble and Brandur Thorgrimsson for useful discussions. This work was supported in part by ARO (W911NF120607) and NSF (PHY1104660). Development and maintenance of the growth facilities used for fabricating samples is supported by DOE (DEFG0203ER46028). This research utilised NSFsupported shared facilities at the University of WisconsinMadison.
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DK performed electrical measurements, state and process tomography, and analysed the data with MAE, MF and SNC. DRW developed hardware and software for the measurements. CBS fabricated the quantum dot device. DES and MGL prepared the Si/SiGe heterostructure. All authors contributed to the preparation of the manuscript.
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Supplementary Information accompanies the paper on the npj Quantum Information website (http://www.nature.com/npjqi)
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Kim, D., Ward, D., Simmons, C. et al. Highfidelity resonant gating of a siliconbased quantum dot hybrid qubit. npj Quantum Inf 1, 15004 (2015). https://doi.org/10.1038/npjqi.2015.4
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DOI: https://doi.org/10.1038/npjqi.2015.4
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