## Introduction

Perfecting the quality of qubits hinges on high fidelity and fast single- and two-qubit gates. Electron spin qubits in Silicon (Si) quantum dots (QD) are considered promising building blocks for scalable quantum information processing1,2,3,4,5, with long coherence times and high gate fidelities already demonstrated6,7,8,9,10,11,12,13,14. The conventional approach of using magnetic fields to operate single-qubit gates results in relatively low Rabi frequencies12, spurring the development of electrically driven spin resonance based on the spin–orbit interaction15 as an alternative. This all-electrical approach promises faster Rabi rotations and reduced power consumption, as well as paving the way towards scalability since electric fields are much easier to apply and localize than magnetic fields. In a Si QD, with relatively weak intrinsic SOI, a synthetic SOI has been introduced to provide fast and high-fidelity gates: a Rabi frequency above 10 MHz and gate fidelity of 99.9% have been reported in an isotopically enriched dot6. However, the magnetic field gradient enabling the synthetic SOI also exposes the system to charge-noise-induced spin dephasing6,16, posing a formidable technical challenge. As such, the search for a high-quality spin qubit with fast manipulation and slow decoherence remains open17.

Hole spins provide an intriguing alternative for encoding qubits as compared to conduction electrons18,19,20,21, in particular in group IV materials such as Si and Ge22,23,24,25,26,27,28,29,30,31,32,33,34. Thanks to their underlying atomic P orbitals, which carry a finite angular momentum and have odd parity, holes experience an inherently strong SOI and weak hyperfine interaction35,36. The strong spin-orbital hybridization of hole states opens the door to fast all-electrical spin control. To date, several works on hole spin qubits have been reported in a multitude of systems, such as Si metal-oxide-semiconductor (MOS)37, undoped strained Ge quantum well38,39 and GHW40 structures, with Rabi frequencies in the range of 70–140 MHz. A recent experiment in Ge/Si core/shell nanowire has reached a very fast Rabi frequency of 435 MHz, with a short spin-orbit length of 3 nm41.

Here we advance the ultrafast control of hole spin qubits by performing faster spin rotations than any reported to date. By applying microwave bursts to one gate of a GHW hole double quantum dot (DQD)31 and utilizing Pauli spin blockade (PSB) for spin-to-charge conversion and measurement, we observe multiple electric dipole spin resonance (EDSR) signals in the DQD. At one of these resonances, we achieve a Rabi frequency exceeding 540 MHz, with a dephasing time of 84 ns and a quality factor of ~ 45. This ultrafast driving is enabled by a very strong SOI, with an equivalent hole spin–orbit length of 1.5 nm. The driving speed is a strong function of the EDSR peaks we study, hence even higher quality factors are likely as qubit encoding is optimized.

## Results

### Measurement techniques and EDSR spectrum

A scanning electron microscope (SEM) image of the DQD device is shown in Fig. 1a (Supplementary Fig. 1a shows a schematic of the device). The device consists of an insulating layer and five electrodes above a GHW grown using the Stranski-Krastanow (S-K) method42,43. The charge stability diagram of the DQD is mapped out and given in Supplementary Fig. 1b, with a zoom-in to one particular triple point given in Fig. 1b. Charge occupations of the DQD are about 5 holes in the left dot and 10 holes in the right dot. When measuring the current through the DQD at a d.c. bias of 3 mV (Fig. 1b and Supplementary Fig. 1c) and −3 mV (Supplementary Fig. 1d), a clear signature of PSB is observed: The zero-detuning current drops to 1 pA in the forward biased (Vsd = 3 mV), blocked configuration (dash base line of the triangle), compared to 30 pA (Vsd = 3 mV) in the reversed biased, non-spin-blocked regime. While PSB44 is usually detected in the (0,2) or (2,0) to (1,1) charge configurations, it has been observed in other charge configurations as well45. With this in mind, we conjecture that the transition we observe occurs near the (n + 1, m + 1) to (n, m + 2) charge transition, which can be equivalently described in terms of two-hole states near the (1,1) to (0,2) transition.

In the PSB regime, with a magnetic field B perpendicular to the substrate, we generate EDSR by applying a microwave pulse to gate R. The a.c. electric field displaces the hole wave function around its equilibrium position periodically, leading to spin rotation mediated by the strong SOI (Fig. 1d). When the microwave frequency matches the resonant frequency of the spin states and causes spin flips, PSB is lifted and an increase in the transport current is observed (a pure orbital transition without spin flip cannot lift the PSB and cannot affect the current). By mapping out the current as a function of B and microwave frequency f, we find multiple spin resonances, as shown in Fig. 1c and Supplementary Fig. 2.

The major observed resonances in Fig. 1c are well described by a two-hole model built upon a single singlet in the (0,2) charge configuration (S02) and two-spin states |↓↑〉, |↑↓〉), |↑↑〉 and |↓↓〉 in the (1,1) charge configuration (Supplementary Note 2), as evidenced by Fig. 1e, f. The large number of resonances and different slopes in Fig. 1c are clear hints of different g-factors for the two dots. Indeed, to generate the theoretical spectrum in Fig. 1f, we use g-factors of 7 and 3.95 for the two dots. With such different g-factors30,31,46, the two-spin states in the (1,1) regime should be spin product states for any magnetic field above 0.1 T. In the following, we focus on two of these resonances, denoted as mode A and mode B in Fig. 1c. Within our model, the corresponding transitions involve single-spin-flip in the left (A, between |↓↓〉 and |↑↓〉) and the right (B, between |↓↓〉 and |↓↑〉) dot.

### Rabi oscillations

To demonstrate coherent control of a hole spin qubit, we apply a three-step pulse sequence on gate R (Fig. 2a) to generate Rabi oscillations for mode A at B=100 mT. The probability of a parallel spin state (spin blocked) or anti-parallel state (unblocked) is measured by the averaged current through the DQD as a function of the microwave burst duration τburst and microwave frequency f (Fig. 2b). We can resolve up to seven oscillations within 180 ns, and the standard chevron pattern helps us to pinpoint the qubit Larmor frequency at 7.92 GHz. To investigate how fast the qubit can be driven coherently, we vary the microwave power P of the driving field from 0 dBm to 9 dBm and measure the Rabi frequency of mode A (Fig. 2d). Rabi oscillations at f = 7.92 GHz with a fit to $${{{\rm{A}}}}\cdot\cos(2{{{{{\rm{\pi }}}}}}{\it{f}}_{{{{{{\rm{Rabi}}}}}}}{\tau }_{{{{{{\rm{burst}}}}}}}+{\it{\varphi}} )\cdot \exp (-{({\tau }_{{{{{{\rm{burst}}}}}}}/{\it{T}}_{2}^{{{{{{\rm{R}}}}}}})}^{2})+{\it{I}}_{0}$$ (An offset of 0.5 pA is set between two oscillations for clarity) are shown in Fig. 2c at three different microwave power P = −5, 0, 6 dBm. At the strongest driving with P = 9 dBm (Fig. 4b), we achieve a Rabi frequency of fRabi = 542 ± 2 MHz for mode A and 291 ± 1 MHz for mode B.

### Free evolution and decoherence

Decoherence determines the quality of the hole spin qubit. To evaluate the dephasing time $${T}_{2}^{\ast }$$ for mode A, we perform a Ramsey fringe experiment6,7,37,38,39,40, with the pulse sequence shown in the top panel of Fig. 3f. A pattern of Ramsey fringes is shown in Fig. 3a when we vary the waiting time τ and microwave frequency detuning ∆f = f − f0 (f0 = 7.92 GHz is the Larmor frequency). The fringes remain visible up to τ ~ 60 ns, giving a qualitative indication of the dephasing time. We perform a Fast Fourier Transformation (FFT) of the Ramsey pattern in Fig. 3b, where the Ramsey oscillation frequency (fRamsey) equals ∆f. Alternatively, two-axis control can be achieved by varying the relative phase ∆φ of the microwave modulation between the two pulses6,15,37. The results of relative phase (∆φ) in cycles identify the control of the rotation axis in addition to ∆f (Fig. 3c). From the Ramsey experiment of mode A, dephasing times of $${T}_{2}^{\ast }=84\pm 9$$ ns and $${T}_{2}^{\ast }=42\pm 4$$ ns are extracted at P = −10 dBm (Fig. 3d) and P = 9 dBm (Fig. 3e), respectively. The former is a better representation of hole spin dephasing, while the latter reflects coherence degradation from the onset of microwave-induced heating. The measured coherence times can be extended by performing a Hahn echo pulse sequence. For instance, the coherence time is extracted to be $${T}_{2}^{\ast }=66\pm 6$$ ns in mode B at P = 0 dBm (Fig. 3f), while an enhanced coherence time of 523 ns is obtained using Hahn echo (Fig. 3g).

### Spin–orbit coupling strength

Unlike z-rotation speed, which is simply determined by the control microwave, i.e., frequency detuning, the x-rotation speed for the hole spins is determined by SOI strength together with the strength of the driving electric field22. With a three dimensional confinement (with z-direction confinement much stronger than the other two dimensions, Ly = 40 nm >> Lx >> Lz) and an out-of-plane magnetic field, the lowest states in the subspace spanned by the spin-3/2 hole states can be calculated, yielding states that are 95% HH (Supplementary Note 9). With the relatively small number of holes per dot (5–10), we attribute the manipulated spin states to be HH throughout the manuscript. The EDSR signals we observe are thus mediated by the strong SOI of 2D heavy-holes.

The leading SOI for a 2D HH gas is the Rashba term47

$${H}_{{{{{{\rm{so}}}}}}}=i{\alpha }_{2}({k}_{+}^{3}{\sigma }_{-}-{k}_{-}^{3}{\sigma }_{+})$$
(1)

where σ± = (σx± iσy)/2, k± = kx± iky and the Rashba constant α2 arises from the spherical component of the Luttinger Hamiltonian48. We have determined that the cubic-symmetry term α3 is negligible in a nanowire, being an order of magnitude smaller (Supplementary Note 9). We have performed a single-hole model calculation with in-plane confinement of an asymmetric harmonic potential $$V(x,y)=\frac{1}{2}m{\omega }_{x}^{2}{x}^{2}+\frac{1}{2}m{\omega }_{y}^{2}{y}^{2}$$. By projecting Hso onto the eigenstates of the 2D harmonic oscillator we obtain the transition matrix element for Rabi oscillations:

$$h{f}_{{{{{{\rm{Rabi}}}}}}}=g{\mu }_{{{\rm{B}}}}B\cdot \frac{{a}_{x}}{{l}_{{{{{{\rm{so}}}}}}}}\cdot \frac{e{E}_{{{\rm{ac}}}}{a}_{x}}{\hslash {\omega }_{y}}$$
(2)

where $${a}_{x}=\sqrt{\hslash /({m}^{\ast }{\omega }_{x})}$$ is the transverse QD size, g the Lande g-factor for a single spin, μB the Bohr magneton, B the static magnetic field, m* the Ge HH effective mass, Eac the effective electric field at the QD generated by the microwave pulses applied to gate R, and lso  1/α2 the spin-orbit length defined in the Supplementary Note 10.

We can obtain the spin–orbit coupling strength according to Eq. (2) from observations of Rabi oscillations of both mode B (Fig. 4a and Supplementary Fig. 7.1) and mode A (Fig. 4a and Supplementary Fig. 6) in the range of P ≤ 9 dBm. In mode A, nine oscillation periods are observed within 16 ns at P = 9 dBm (Fig. 4b), with a Rabi frequency of fRabi = 542 ± 2 MHz at f = 7.92 GHz. When the microwave power is further increased (Supplementary Fig. 6b, c), photon-assisted tunneling (PAT) limits the detection of coherent control at higher Rabi frequencies6,7,15,19,20. Replacing Eac (Supplementary Fig. 8) with P in Eq. (2), spin-orbit lengths of 1.5 nm and 1.4 nm are obtained by fitting the linear dependence of fRabi on Eac in mode A and B, respectively, corresponding to a strong spin-orbit coupling strength of α2 ~ 680 meV·nm3. Due to the smaller a.c. the field in the right dot compared to the left dot (Supplementary Note 6) and the absence of a low-energy intermediate state (Supplementary Note 2), mode B shows a slower speed of Rabi rotation compared to mode A at the same microwave power even though the spin–orbit coupling strengths are similar. Notice that while spin–orbit length is a concept more appropriate for free carriers, it is nonetheless useful for comparing different systems. For example, for strongly spin-orbit coupled conduction electrons confined in InAs quantum dots of similar size to our dots, typical spin-orbit length ranges from 100 to 200 nm49. In comparison, our hole system has, inherently, a much stronger spin-orbit coupling (thus a much smaller lso), which is the determining factor for the ultrafast operation of our qubit.

## Discussion

Ultrafast control of hole spins has also been achieved in a Ge/Si core/shell nanowire41, though that hole system is quite different from ours. The key difference stems from the respective geometries. Our GHW has a confinement potential that is much stronger in one direction, akin to a quantum well, for which theory predicts a spin-3/2 (i.e. ‘heavy hole’) ground state. Ge/Si core/shell nanowires have cylindrical symmetry, where theory predicts a spin-1/2 (‘light-hole’) ground state5. Due to possible mass reversal, the magnitude of the effective mass is not a reliable indicator of spin character. On the other hand, the two systems do share the feature of a strong spin–orbit coupling, which enables fast control.

The large Rabi frequencies in our system are achieved with a small driving electrical field ~104 V/m, compared to ~106 V/m static electrical field used for QD confinement (Supplementary Fig. 8). We thus attribute the large Rabi frequencies of both mode A and B to a large value of α2. The particularly large Rabi frequency in mode A may have been enhanced by the low-energy excited state included in our model (Supplementary Note 2 and 10), a fact that has previously been exploited to enhance spin-electric-field coupling50,51. To obtain a deeper understanding of such strong spin–orbit coupling, anisotropy spectroscopy would be a viable method to reveal the underlying mechanisms46,52. Even though the Rabi frequency of 540 MHz is limited by unwanted PAT and heating effects at high microwave power, we believe it is still not the upper bound for the Rabi frequency of a GHW hole spin qubit. For instance, faster operation is possible if another branch of EDSR with a larger Larmor frequency can be identified in the DQD. A particular example is the transition between |↓↓〉 and |↑↑〉, which can be observed in our system (brown in Supplementary Fig. 2a). This spin-flip transition corresponds to a larger energy splitting compared to mode A and mode B at the same magnetic field and could result in a larger Rabi frequency as EDSR frequency is proportional to Zeeman splitting. Moreover, faster Rabi oscillation can be achieved by changing the manipulation position. Our test results show that Rabi frequency can be increased from 63 MHz to 111 MHz by switching the manipulation position from M2 to M1 (Supplementary Fig. 4). We are thus optimistic that even faster Rabi operation is achievable after optimization.

A high-quality qubit requires both fast manipulation and slow decoherence. We have thus investigated dephasing for both modes A and B (Supplementary Fig. 6d and Fig. 3f). The dephasing rate appears approximately uniform across the two modes, so that the qubit quality factor $$Q=2{f}_{{{\rm{Rabi}}}}{T}_{2}^{{{\rm{Rabi}}}}$$ is roughly determined by the Rabi frequency of the different modes. We thus obtain a lower bound estimate of the quality factor ~ 45 using fRabi = 542 MHz and $${T}_{2}^{\ast }=42$$ ns at P = 9 dBm. This value predicts a fidelity of the π gate to be e−1/Q= 97.8%. A benchmark for Rabi frequency of QD spin qubits is set in Supplementary Fig. 10. Our hole spin qubit has one of the fastest Rabi frequency, with a quality factor around 45 that we believe can be further improved.

In conclusion, we have achieved ultrafast spin manipulation in a Ge HW. We report a Rabi frequency of up to 540 MHz at a small magnetic field of 100 mT, and obtain a dephasing time of 84 ns from a Ramsey fringe experiment. A hole spin qubit with a quality-factor of 45 is thus realized in our experiment. As dephasing appears to change little across different modes, higher-quality qubits could be achievable for state combinations with stronger spin–orbit coupling. We report a small spin-orbit length in a smaller Ge double quantum dot compared to existing work on GHW in the literature40 with narrower electrodes. We attribute the ultrafast control of a hole spin qubit that we have observed to an overall strong spin-orbit coupling, possibly assisted by a nearby excited state, even though the relative smaller longitude dot size (along y) may have reduced the Rabi frequency. In other words, our results demonstrate that hole spins in GHW QDs are intriguing candidates for semiconductor quantum computing, providing the ability of all-electrical ultrafast control without the need for a micromagnet13 or a co-planar stripline14.

## Methods

### Device fabrication

Our hut wire was grown on Si (001) by means of a catalyst-free method based on molecular beam epitaxy. A Ge layer (1.5 nm) was deposited by S-K growth mode on a Si buffer layer (100 nm). A 3.5-nm-thick Si cap was then grown on top of the Ge layer to protect the nanowire with a width of 20 nm (Details in Supplementary Fig. 1a). The dot size along x direction of ax= 5 nm is obtained based on the ground state calculation. The length of our wire can be longer than 1 μm, although a 500 nm wire is sufficient for our DQD device. On both ends of the wire, two 30-nm-thick palladium electrodes were defined as the ohmic contacts. A layer of aluminum oxide (25 nm) was then deposited on top of the nanowire and ohmic contacts as an insulator. Three 30-nm-wide Ti/Pd electrodes are deposited to generate the DQD potential in the wire.

### Experimental setup

The experiments were performed in an Oxford Triton dilution refrigerator at a base temperature of 10 mK. The sinusoidal waves from the output of the arbitrary waveform generator Keysight M8190A are inputted to I/Q ports of the vector source Keysight E8267D in order to generate the modulated sequences for spin control. Combined with pulses from M8190A, these sequences are then transmitted by a semi-rigid coaxial line connecting to gate R, where the total attenuation is 36 dB. As shown in Fig. 2a, we apply two-stage pulses to gate R for spin initialization, control and readout. The length of one cycle is fixed at 640 ns and 320 ns of it is for spin control. The average transport current is measured by a digital multimeter after a low-noise current preamplifier SR570.