488 | Nature | Vol 577 | 23 January 2020
Article
For example, in silicon the execution of a controlled NOT (CNOT) gate
implemented with an on-chip stripline has been shown using microsec-
ond long pulses^6 ,^8 , and this timescale can be reduced to 0.2–0.5 μs by
incorporating nanomagnets^9. Here we demonstrate that the spin–orbit
coupling of holes in germanium together with the sizable exchange
interaction enables a CNOT within 75 ns.
A scanning electron microscope image of the germanium two-qubit
device is shown in Fig. 1a. To accumulate holes and define two quantum
dots, the circular plunger gates are set to negative potential (VP1,
VP2 ≈ −2 V). The tunnel coupling between the dots t 12 and the tunnel
couplings to the source and drain reservoirs (t1S, t2D) are controlled by
the barrier gates BC, BS and BD, respectively. Working in a virtual gate
voltage space (VVP1, VVP2, Vt1S, Vt2D and Vt 12 ), we can independently tune
these properties (see Supplementary Videos 1–3 online for video-mode
operation). We measure the transport current through the double dot
system (Fig. 1c, d), and for certain hole occupations (Extended Data
Fig. 3) we observe a suppression of the transport current for a positive
bias voltage VSD = 1 mV, caused by Pauli spin blockade (PSB) (see Fig. 1e).
We make use of the blockade as an effective method for spin-to-charge
conversion^2 ,^11 , as well as to initialize our two-qubit system in the blocked
↓↓ ground state.
Taking advantage of the strong spin–orbit coupling^17 , we are able to
implement a fast manipulation of the qubit states by electric dipole
spin resonance (EDSR). We tune the device to a readout point within
the PSB region (indicated by the label R in Fig. 1d) and apply an electric
microwave excitation to gate P1. When the frequency of the microwave
excitation matches the spin resonance frequency of either qubit, PSB
is lifted and an increase in the transport current can be observed. We
extract the resonance frequency of each qubit as a function of external
magnetic field strength B 0 (Extended Data Fig. 4) and observe two
distinct qubit resonance lines with g-factors g 1 = 0.35 and g 2 = 0.38
(Fig. 1g). The difference in g-factors between the two dots is likely to be
caused by slightly different hole fillings and thus quantum dot orbitals.
As an effect of the spin–orbit coupling, a strong orbital dependence of
the effective g-factor is typically measured in hole quantum dots^18 ,^25.
Furthermore, the effective g-factor can be tuned electrically as a direct
result of the SOC^26 (see for example Fig. 3c, d), thereby guaranteeing
independent control of the different qubits. We observe that the reso-
nance frequency of both qubits remains stable over several hours, with
discrete jumps at longer timescales as presented in Extended Data Fig. 5.
We developed a measurement technique in which we measure the
averaged transport current over N repeated pulse cycles and subtract
a reference measurement using a lock-in amplifier, to mitigate slow
variations in the transport current (see Methods), as is indicated
in Fig. 2a. After readout, the system is left in the blocking ↓↓
state, serving as the initialization of our qubits. We now operate the
device in the single-qubit transport mode in an external field of
B 0 = 0.5 T and use the second qubit (Q2) as a readout ancilla. Coherent
control over the qubit is demonstrated in a Rabi experiment, where
the spin state of qubit 1 (Q1) is measured as a function of microwave
pulse length tp and power P, as shown in Fig. 2b. By increasing the power
of the microwave pulse, we can reach Rabi frequencies of over 100 MHz,
at an elevated field B 0 = 1.65 T (Extended Data Fig. 6).
To determine the control fidelity, which describes the accuracy of
our quantum gates, we implement randomized benchmarking of the
single-qubit Clifford group^27 (Fig. 2c). The measured decay curve of the
qubit state as a function of sequence length m is shown in Fig. 2d, from
which we extract a single-qubit control fidelity of FC = 99.3%, using gate
times tπ = 20 ns and tπ/2 = 10 ns. In Fig. 2e, we show the gate fidelities for
the different π and π/2 gates as obtained by interleaved randomized
benchmarking, where each randomly drawn gate is followed by the
respective interleaved gate (see Fig. 2c). All individual gate fidelities
are FC > 99%, with the infidelity for π/2 gates being approximately twice
as low as for the π gates, on account of the difference in pulse length.
We extensively characterize the coherence in our system
at an exchange coupling of J/h ≈ 20 MHz and find T*2,Q1=8 33 ns
and T*2,Q2=4 19 ns, which can be extended by performing a Hahn echo
to TH2,Q1=1.9μs and T2,HQ2=0.8μs (data in Extended Data Fig. 7), as indi-
cated in Fig. 2f. These coherence times compare favourably to
T* 2 =1 30 ns for germanium hut wires^17 and T* 2 =2 70 ns for holes
in silicon^28. Electrons in GaAs have an even shorter dephasing time^11 ,
with T* 2 =10ns . The limited T* 2 in GaAs is due to hyperfine interactions,123
B (T)681012fres(GHz)
g 1 = 0.35
g 2 = 0.39SDB1 P1 BC P2 B2BS 200 nm BDB 0–3, 142 –3, 137 –3, 145 –3, 140–3, 140–3, 130–3, 125–3, 120ISD–3, 150–3, 115Hab–3,140–3,130–3,120–3,110VVP2(mV)VVP2(mV)VVP2(mV)VVP2(mV)VVP1 (mV) VVP1 (mV)VVP1 (mV)VVP1 (mV)–3,160 –3,140–3,140–3,130–3,120–3,110c def
Q1 Q2gP1 BCP2Vt 12 = –902 mVSDBS BD
P1 BC P2t 12Q1 Q2ESTRMtS1 t2DVt 12 = –882 mVFig. 1 | Fabrication and operation of a planar germanium double quantum
dot. a, False-coloured scanning electron microscope image of the two-qubit
device, where Ohmic contacts are indicated in yellow, the barrier gate layer is
depicted in green and the plunger gate layer in purple. Two hole quantum dots,
indicated by the blue and red arrows, are formed in a high-mobility Ge quantum
well and controlled by the electric gates. The direction of the external field B 0 is
indicated by the black arrow. b, Schematic cross-section of the system, where
quantum dots are formed below plunger gates P1 and P2, while the different
tunnelling rates can be controlled by barrier gates BS, BD and BC. c, Transport
current through the double dot as a function of plunger gate voltages for weak
(top) and strong (bottom) interdot coupling, mediated by a virtual tunnel gate.
d, Charge stability diagram of the qubit operation point, where the dashed lines
correspond to the charge transitions. The detuning axis ε is indicated by the
dotted line, with label R corresponding to the qubit readout point. To allow
coherent control of the isolated spin states, a two-level voltage pulse on gates
P1 and P2 is used to detune the dot potentials and prevent tunnelling to and
from the dots during the manipulation phase (label M). e, Transport current
through the double dot as a function of plunger gate voltage for positive (left)
and negative (right) bias. Pauli spin blockade becomes apparent from the
suppression of the transport current for the positive bias direction, up to the
singlet–triplet energy splitting of EST = 0.6 meV. f, Illustration of the energy
landscape in our double-quantum dot system. g, Resonance frequency, fres, of
the two qubits as a function of the external magnetic field, showing the
individual qubit resonances.