The sample was bonded to a printed circuit board (PCB) mounted onto the mixing chamber of a dilution refrigerator. All measurements were performed at the base temperature of the fridge,T_base ≈ 20 mK. DC voltages were applied to all of the gate electrodes using room-temperature digital-to-analogue converters via filtered lines (not shown). Voltage pulses were applied to plunger gates P1 and P2 using a Tektronix 5014C arbitrary waveform generator (AWG) with 1-GHz clock rate. The signals from the AWGs passed through a room-temperature low-pass filter and attenuators at different stages of the fridge and were added to the DC signals via bias tees mounted on the PCB. Two Keysight E8267D vector microwave sources, MW1 and MW2, were used to apply microwaves (18–20 GHz) to perform EDSR on Q1 and Q2, respectively. The signals passed through room-temperature DC blocks and custom-built 15-GHz high-pass filters and attenuators at different stages of the fridge and were added to the DC signals via bias tees mounted on the PCB. The output of the microwave source (phase, frequency, amplitude and duration) was controlled with I/Q vector modulation. The I/Q signals were generated with another Tektronix 5041C, which was the master device for the entire set-up and provided trigger signals for the other devices. In addition to the vector modulation, we used pulse modulation to give an on/off microwave power output ratio of 120 dB. Although I/Q modulation can be used to output multiple frequencies, the bandwidth of the AWG was not sufficient to control both qubits with one microwave source owing to their large separation in frequency (1.3 GHz). The sensor currentI was converted to a voltage signal with a custom-built preamplifier, and an isolation amplifier was used to separate the signal ground, with the measurement equipment grounded to reduce interference. Following this, a 20-kHz Bessel low-pass filter was applied to the signal using a SIM965 analogue filter. A field-programmable gate array (FPGA) analysed the voltage signal during the readout and assigned the trace to be spin-up if the voltage fell below a certain threshold. The voltage signal could also be measured with a digitizer card in the computer. The shapes of the pulses generated by the AWGs and microwave sources during qubit manipulation, along with the typical timescales, are shown in the lower right. Square pulses were used to perform the CZ gate and as the input for the I/Q modulation to generate microwave pulses. The pulse modulation was turned on 40 ns before turning on the I/Q signal, owing to the time needed for the modulation to switch on.

a, Stability diagram of the double quantum dot, showing the positions in gate space used to perform single-qubit gates (red circle) and two-qubit gates (yellow circle). The map shows the numerically obtained differential current dI/dV_P1 through the quantum-dot charge sensor as a function of the voltages on P1 and P2. The white dashed line is the (1, 1)–(0,2) inter-dot transition line. The white arrow indicates the detuning axisε used in the experiments. Although the detuning pulse for the two-qubit gate crosses the charge-addition lines of D1 and D2, the quantum dots remain in the (1, 1) charge state because the pulse time is much shorter than the electron tunnel times to the reservoirs.b, Plot of the voltage pulses applied to plunger gates P1 and P2 and the response of the quantum-dot charge sensor over one measurement cycle. D2 is unloaded by pulsing into the (1, 0) charge region for 1.5 ms (purple circle). The electron on D1 is initialized to spin-down by pulsing to a spin-relaxation hotspot at the (1, 0) and (0, 1) charge degeneracies (orange circle) for 50 μs (seeExtended Data Fig. 5). D2 is loaded with a spin-down electron by pulsing to the readout position for 4 ms (blue circle). During manipulation, the voltages on the plunger gates are pulsed to the red circle for single-qubit gates and to the yellow circle for two-qubit gates, where the exchange is about 6 MHz. After manipulation, the spin of the electron on D2 is measured by pulsing to the readout position (blue circle) for 0.7 ms, where the Fermi level of the reservoir is between the spin-up and spin-down electrochemical potentials of D2. If the electron is spin-up then it can tunnel out, after which a spin-down electron tunnels back in. These two tunnel events are detected by the quantum-dot sensor as a single blip in the current signal. An additional 1.3 ms is spent at the readout position so that D2 is initialized to spin-down with high fidelity. Following this, Q1 is measured by first performing a CROT gate at the yellow circle so that, where CROT12 indicates a CROT gate with Q1 as the control and Q2 as the target. A projective measurement of Q1 is then performed by measuring Q2 at the readout position for 0.7 ms (blue circle). Finally, we add a compensation pulse to VP1 and VP2 so that over the measurement cycleV_DC = 0 to mitigate charging effects in the bias tees.c, Close-up of the stability diagram ina showing the positions in gate space used for initialization and readout.

The purple (top) and orange (bottom) data points correspond to measurements performed on Q1 and Q2, respectively, in the (1, 1) regime (red circle inExtended Data Fig. 2).a, Spin-up fraction as a function of the microwave frequency of an applied π pulse, showing a resonant frequency of 18.387 GHz (19.670 GHz) for Q1 (Q2).b, The spin-relaxation time is measured by preparing the qubit to spin-up and varying the wait time before readout. From the exponential decay in the spin-up probability we measureT₁ > 50 ms (T₁ = 3.7 ± 0.5 ms) for Q1 (Q2).c, Spin-up probability as a function of microwave duration, showing Rabi oscillations of 2.0 MHz for Q1 and Q2.d, The dephasing time is measured by applying a Ramsey pulse sequence and varying the free evolution timeτ. Oscillations were added artificially to improve the fit of the decay by making the phase of the last microwave pulse dependent on the free evolution time,ϕ = sin(ωτ), whereω = 4 MHz. By fitting the data with a Gaussian decay,, we extract () for Q1 (Q2). In the measurement for Q1, the first π/2 microwave pulse is aY gate. The Ramsey measurement was performed over about 20 min with the frequency calibrated every approximately 1 min.e, The coherence time of Q1 (Q2) can be extended toT_2Hahn = 19 ± 3 μs (T_2Hahn = 7 ± 1 μs) by a Hahn echo sequence. The coherence time is extracted from an exponential fit to the spin-up probability as a function of the free evolution time in the Hahn echo sequence.f, Full two-axis control is demonstrated by applying two π/2 pulses and varying the phase of the second one. All error bars are 1σ from the mean.

Randomized benchmarking of the single-qubit gates for each qubit is performed by applying a randomized sequence of a varying number of Clifford gatesm to either the |1〉 or |0〉 state and measuring the final spin-up probability orP_|1〉, respectively. All gates in the Clifford group are decomposed into gates from the set {I, ±X, ±X², ±Y, ±Y²}. The purple (orange) data points show the difference in the spin-up probabilities for Q1 (Q2) as a function of sequence length. For each sequence lengthm we average over 32 different randomized sequences. From an exponential fit (solid lines) of the data,, we estimate average Clifford-gate fidelitiesF_C = 1 − (1 −p)/2 of 98.8% and 98.0% for Q1 and Q2, respectively. The last three data points from both datasets were omitted from the fits because they deviate from a single exponential²⁰. All error bars are 1σ from the mean.

a, Close-up stability diagram of the (1, 0)–(0, 1) charge transition. The white arrow defines the detuning axis between D1 and D2 controlled with P1.b, Schematic of the energy-level diagram as a function of detuning for one electron spin in a double quantum dot.c, Spin-relaxation hotspots are measured by preparing the electron on D1 to spin-up using EDSR, applying a voltage pulse along the detuning axis (white arrow ina) for a wait time of 200 ns and performing readout of the electron spin. We observe three dips in the spin-up probability, which correspond to spin-relaxation hot spots. The first and third hotspot are due to anticrossings between the (0, ↓) and (↑, 0) states and the (↓, 0) and (0, ↑) states²⁴. The second hotspot occurs at zero detuning. The voltage separation between the first and third hot spot corresponds to the sum of the Zeeman energy of D1 and D2 divided by the gate lever armα along the detuning axis. Knowing precisely the Zeeman energies from EDSR spectroscopy, we can accurately extract the gate lever arm to beα = 0.09e.d, The spin-relaxation time at zero detuning (orange circle ina) is found to beT₁ = 220 ns by measuring the exponential decay of the spin-up probability as a function of wait timeτ at zero detuning. All error bars are 1σ from the mean.

a, Microwave spectroscopy of Q2 close to zero detuning between the (1, 1) and (0, 2) states (yellow dot inExtended Data Fig. 2a) with the exchange coupling on. The blue and red curves show the resonance of Q2 after preparing Q1 into spin-down and spin-up, respectively. The resonance frequency of Q2 shifts by the exchange coupling, and by applying a π pulse at one of these frequencies we can perform a CROT gate, which is equivalent to a CNOT gate up to a rotation. As discussed in the main text, this CROT gate is used to perform the projective measurement of Q1. All error bars are 1σ from the mean.

The exchange couplingJ_off during single-qubit gates is measured using a two-qubit Hahn echo sequence, which cancels out any unconditional rotations during the free evolution timeτ. Fitting the spin-up probability as a function of free evolution timeτ using the functional form sin(2πJ_offτ), we extractJ_off = 270 kHz. All error bars are 1σ from the mean.

a,b, Spectroscopy of Q1 (a) and Q2 (b) versus detuning energyε after initializing the other qubit to. Towardsε = 0 there are two resonances each for Q1 and Q2, which are separated by the exchange energyJ(ε)/h. As discussed, the Zeeman energyE_Z(ε) of Q1 and Q2 also depends on detuning because changes to the applied voltages shift the position of the electron in the magnetic-field gradient. The four resonance frequencies are fitted (green, blue, red and yellow lines) withf_jk = E_Zj(ε) + (−1)^k+1J(ε), wherej denotes the qubit andk denotes the state of the other qubit. The data are fitted well using,, andE_Z2(ε)∝ε. The fitted Zeeman energies of Q1 and Q2 are shown by the black lines. We observe that the Zeeman energy of Q1 has an exponential dependence towards the (0, 2) charge regime (ε = 0). This observation can be explained by the electron delocalizing from D1 towards D2, which has a much higher Zeeman energy.c, Schematic of the colour-coded transitions that correspond to the resonances ina andb.

The normalized spin-up probability of Q1 as a function of the total duration time 2τ of the two CZ gates in the DCZ sequence. The data are fitted using a sinusoid,P_|1〉 = 0.5sin(2πJτ) + 0.5, with either a Gaussian (black line;) or exponential (red line;) decay. From these fits we find a decay time ofT₂ = 1.6 μs. All error bars are 1σ from the mean.

a,b, Two-spin probabilities as a function of the sequence time during the Deutsch–Josza algorithm (a) and the Grover search algorithm (b) for each function, using the decoupled version of the two-qubit CZ gate (the DCZ gate). The solid lines show the outcome of the simulations, which include decoherence due to quasi-static charge noise and nuclear-spin noise. All error bars are 1σ from the mean.

This file contains supplementary notes S1-S3 and supplementary figures S1-S5. (PDF 1317 kb)