## Abstract

Universal quantum information processing requires the execution of single-qubit and two-qubit logic. Across all qubit realizations^{1}, spin qubits in quantum dots have great promise to become the central building block for quantum computation^{2}. Excellent quantum dot control can be achieved in gallium arsenide^{3,4,5}, and high-fidelity qubit rotations and two-qubit logic have been demonstrated in silicon^{6,7,8,9}, but universal quantum logic implemented with local control has yet to be demonstrated. Here we make this step by combining all of these desirable aspects using hole quantum dots in germanium. Good control over tunnel coupling and detuning is obtained by exploiting quantum wells with very low disorder, enabling operation at the charge symmetry point for increased qubit performance. Spin–orbit coupling obviates the need for microscopic elements close to each qubit and enables rapid qubit control with driving frequencies exceeding 100 MHz. We demonstrate a fast universal quantum gate set composed of single-qubit gates with a fidelity of 99.3 per cent and a gate time of 20 nanoseconds, and two-qubit logic operations executed within 75 nanoseconds. Planar germanium has thus matured within a year from a material that can host quantum dots to a platform enabling two-qubit logic, positioning itself as an excellent material for use in quantum information applications.

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## Data availability

All data underlying this study are available from the 4TU ResearchData repository at https://doi.org/10.4121/uuid:95bc1f2e-0218-4c55-8e5b-2b59e8fcc5e6.

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## Acknowledgements

We thank L. M. K. Vandersypen, S. Dobrovitski and J. Helsen for valuable discussions. We acknowledge support through a FOM Projectruimte of the Foundation for Fundamental Research on Matter (FOM), associated with the Netherlands Organisation for Scientific Research (NWO).

## Author information

### Affiliations

### Contributions

N.W.H. and D.P.F. performed the experiments. N.W.H. fabricated the device. A.S. and G.S. supplied the heterostructures. N.W.H., D.P.F. and M.V. wrote the manuscript with the input of all other authors. M.V. conceived and supervised the project.

### Corresponding author

Correspondence to M. Veldhorst.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Additional information

**Publisher’s note** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Instrumentation set-up for the lock-in transport measurements.

Illustration of the set-up and relevant signals for the lock-in transport measurements. The AWG is used to generate alternating pulse cycles consisting of a repeated measurement and a repeated reference. The signal is demodulated in a lock-in amplifier to give a direct measure of the difference between the two measurements and subtract slow variations in the transport signal.

### Extended Data Fig. 2 Pulse cycles used for the transport measurements.

**a** Pulse cycles used for the randomized benchmarking experiments. The measurement pulse cycle consists of *m* gates randomly drawn from the Clifford group *C*_{rand} and a final Clifford gate projecting the qubit onto the spin-up state. The reference pulse cycle consists of the same *m* Clifford gates and a different final Clifford gate projecting the qubit onto the spin-down state. Each cycle is repeated *N* times, and a series of typically *k* = 50 independent randomly drawn measurement and reference pulse cycles are alternated. These *k* = 50 different draws are thus hardware-averaged on the lock-in amplifier, and the entire experiment is repeated and averaged 30 times, yielding a total approximate 10^{5} repetitions of 1,500 different randomly drawn Clifford sequences of length *m*. An example of the qubit evolution for each pulse cycle is plotted on the Bloch sphere below. **b**, Pulse cycles used for the exchange mapping experiments. The measurement pulse cycle consists of a broad preparation and restoring pulse at frequency *f*_{3} (*f*_{1}), around a probing pulse at frequency *f*_{prb}. The reference pulse cycle consists solely of the probing pulse at *f*_{prb}. The qubit evolutions for the different resonance conditions are plotted on the Bloch sphere and illustrate the different signals measured in Fig. 4c, d.

### Extended Data Fig. 3 Demonstration of qubit operation at a second hole occupancy.

**a**, Charge stability diagram showing the (*m*, *n*) hole occupancy used during all experiments in the main text, as well as the (*m*, *n* + 1) occupancy for which we observe PSB as well. For an unpolarized filling of the quantum dots, one expects an alternating suppression of the transport current due to PSB, as spin blockade occurs only when an orbital level is fully occupied. However, the spin-filling for holes is known to be highly polarized^{34,35}, and therefore PSB can occur in sequential quantum dot fillings. **b**, Coherent Rabi oscillations measured in the (*m*, *n* + 1) occupancy. A slight linear offset is observed for Q1, which can be attributed to the microwave power. We note that, for the same microwave power, the Rabi frequency of Q2 in the (*m*, *n* + 1) occupancy is increased substantially compared to the (*m*, *n*) filling. We attribute this to the hole being in a different orbital, where the effective SOC may be different.

### Extended Data Fig. 4 Qubit resonance frequencies as a function of magnetic field.

Colour plot indicating the transport current Δ*I* through the double dot system, as a function of external magnetic field *B*_{0} and the frequency *f* of the applied microwave signal. We have numerically subtracted the mean of each row and column in each of the three individual colour plots, to account for the slow drifts in transport current, as well as the line resonances in our fridge cabling. The two bright lines indicate an increase in the transport current due to the microwave rotating either spin and thus lifting PSB.

### Extended Data Fig. 5 Temporal dependence of the resonance frequency.

We track the resonance frequency of both Q1 and Q2 over the time of approximately 110 h. We observe that the qubit frequency remains remarkably stable over this period, but do observe discrete, uncorrelated steps in the resonance frequencies of both qubits. The resonance frequency of Q1 only shows steps of Δ*f* ≈ 2 MHz between two distinct levels, whereas for Q2 we observe steps of both Δ*f* ≈ 1 MHz and Δ*f* ≈ 2 MHz, between three different levels, as also becomes apparent from the histogram. The origin of these steps could be, for example, the slow loading and unloading of charge traps, which manipulates the qubit resonance frequency through the change in electric field, or hyperfine coupling to a nearby nuclear spin^{36}.

### Extended Data Fig. 6 Magnetic field dependence of the driving speed of Q1.

**a**, **b**, Rabi frequency dependence on the applied microwave power *P* in arbitrary units, for *B*_{0} = 0.5 T (**a**) and *B*_{0} = 1.65 T (**b**). Multiple mechanisms can be at play for the EDSR driving of the spins^{37} and these are typically all linearly dependent on *B*_{0}. As a result of this, considerably higher driving frequencies can be reached at higher magnetic fields. We note that the exact microwave power cannot be compared between the two measurements, owing to the strong frequency dependence of the attenuation of our fridge lines.

### Extended Data Fig. 7 Relaxation, dephasing and coherence times.

We perform a Ramsey experiment, in which two π/2 pulses are separated by time *τ*, during which the qubit will evolve as a result of the implemented detuning. We fit the decay of the observed oscillations to \(\Delta {I}_{{\rm{S}}{\rm{D}}}=a\cos (2{\rm{\pi }}\Delta f\tau +\varphi )\exp [-{(\tau /{T}_{2}^{* })}^{\alpha * }]\), with *a* a scaling factor, Δ*f* the detuning and *ϕ* a phase offset, and find a spin coherence time of \({T}_{2,{\rm{Q}}1}^{* }=833\,{\rm{n}}{\rm{s}}\) and \({T}_{2,{\rm{Q}}2}^{* }=419\,{\rm{n}}{\rm{s}}\) and decay coefficients of \({\alpha }_{{\rm{Q}}1}^{* }=1.2\pm 0.2\) and \({\alpha }_{{\rm{Q}}2}^{* }=1.5\pm 0.2\), for Q1 and Q2, respectively. The spin coherence can be extended by performing a Hahn echoing sequence, consisting of π/2, π and π/2 pulses separated by waiting times *τ*. Fitting the observed decay as a function of the total waiting time 2*τ* to a power law \(\Delta {I}_{{\rm{S}}{\rm{D}}}=a\exp [-{(2\tau /{T}_{2}^{* })}^{{\alpha }^{{\rm{H}}}}]\), we find extended coherence times of \({T}_{2,{\rm{Q}}1}^{{\rm{H}}}=1.9\,\mu {\rm{s}}\) and \({T}_{2,{\rm{Q}}2}^{{\rm{H}}}=0.8\,\mu {\rm{s}}\) and decay coefficients of \({\alpha }_{{\rm{Q1}}}^{{\rm{H}}}=1.5\pm 0.1\) and \({\alpha }_{{\rm{Q2}}}^{{\rm{H}}}=2.5\pm 0.3\), for Q1 and Q2, respectively. Finally, we perform a measurement of the spin lifetime by applying a single π pulse, after which we wait for a time *τ*. We fit the decay to \(\Delta {I}_{{\rm{S}}{\rm{D}}}=\exp [-(\tau /{T}_{1})]\) and find lifetimes of *T*_{1,Q1} = 9 μs and *T*_{1,Q2} = 3 μs.

### Extended Data Fig. 8 Relaxation time *T*_{1} as a function of gate voltage on the tunnel barriers between dot and reservoir.

**a**, **b**, The relaxation time *T*_{1} of the dots increases approximately exponentially as a function of the respective dot–reservoir gate voltage, for Q1 (**a**) as well as for Q2 (**b**). The relaxation time of Q1 increases exponentially from *T*_{1} < 1 μs to *T*_{1} > 10 μs, and a similar scaling is observed for Q2. For even smaller dot–reservoir couplings, the transport signal drops below our measurement limit, but switching to charge sensing could allow a further increase in *T*_{1}.

### Extended Data Fig. 9 Phase corrections on the qubits.

**a**, Extended Ramsey experiment on each of the four resonance line, using five different test gates between the π/2 pulses to observe the effect on the resonance frequency. A linear phase shift as a function of test gate pulse length *τ* can be observed for some lines, indicating a frequency shift during the pulsing. **b**, We compensate for this effect by performing a software update of δ*ϕ* = δ*ντ* to each additional pulse, with δ*ν* the frequency shift of the qubit as a result of the microwave signal.

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Hendrickx, N., Franke, D., Sammak, A. *et al.* Fast two-qubit logic with holes in germanium.
*Nature* **577, **487–491 (2020). https://doi.org/10.1038/s41586-019-1919-3

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