Universal control of a six-qubit quantum processor in silicon – Nature

Device fabrication

Devices are fabricated on an undoped ²⁸Si/SiGe heterostructure featuring an 8 nm strained ²⁸Si quantum well, with a residual ²⁹Si concentration of 0.08%, grown on a strain-relaxed Si_0.7Ge_0.3 buffer layer. The quantum well is separated from the surface by a 30-nm-thick Si_0.7Ge_0.3 spacer and a sacrificial 1 nm Si capping layer. The gate stack consists of three layers of Ti:Pd metallic gates (3:17, 3:27 and 3:27 nm) isolated from each other by 5 nm Al₂O₃ dielectrics, deposited using atomic layer deposition. A ferromagnetic Ti:Co (5:200 nm) layer on top of the gate stack creates a local magnetic field gradient for qubit addressing and manipulation. The ferromagnetic layer is isolated from the gate layers by 10 nm of Al₂O₃ dielectric. The cobalt layer is not covered with a dielectric. Further details of device fabrication methods can be found in ref. ³³.

After fabrication, all the devices are screened at 4 K. We check for current leakage, accumulation below the gates and device stability (for example, drifts in current). The best device (if it meets our requirements) is selected and cooled down in a dilution refrigerator. The fraction of devices that pass these 4 K checks varies from 0 to 50% per batch (a batch contains either 12 or 24 devices).

Microwave crosstalk and synchronization condition

In Fig. 2, the single-qubit gates are chosen to be operated at a 5 MHz Rabi frequency and all single-qubit randomized benchmarking results are taken at this frequency as well. When operating all qubits within the same sequence, we were unable to operate at a 5 MHz Rabi frequency as qubits Q2(Q3) and Q5(Q4) are too close to each other in frequency. We used the synchronization condition^53,54 to choose Rabi frequencies for the single-qubit gates for which the qubit that suffers crosstalk does not undergo a net rotation while the target qubit is rotated by 90 degrees or multiples thereof (Extended Data Fig. 5). The Rabi frequencies for the state tomography experiments are as follows (qubits Q1–Q6): 4.6 MHz, 1.9 MHz, 4.2 MHz, 3.6 MHz, 2.4 MHz and 5 MHz.

Automated calibration routines

Calibrations are a crucial part in operating a multi-qubit device. Figure 1f lists the necessary calibration types that need to be corrected periodically and Extended Data Fig. 4 shows an example calibration for each parameter type. Every calibration uses an automated script to extract the optimal value for the measured parameter, which is recorded in a database. In our framework, the operator chooses to accept this value or to re-run the calibration.

Sensing dot (5 s)

The calibrations routine starts by calibrating the sensing dots (Extended Data Fig. 4a) to the most sensitive operating point for parity mode PSB readout. We scan the (virtual) plunger voltage of the sensing dot for two different charge configurations of the corresponding double dot, corresponding to the singlet and triplet states. One configuration is in the (3,1) region and the other in the (4,0) region, in order to be insensitive to small drifts in the gate voltages. The calibration returns the plunger voltage for which the largest difference is obtained in the sensing dot signal between these two cases (Extended Data Fig. 4a). From this difference, we also set the threshold in the demodulated in-phase (I) and quadrature (Q) signals (in short, IQ signals) of the radio frequency (RF) modulated readout, to allow singlet/triplet differentiation (the IQ signal is converted to a scalar by adjusting the phase of the signal). The threshold is chosen halfway between the signals for the two charge configuration. During qubit manipulation, the sensing dot is kept in Coulomb blockade. It is only pulsed to the readout configuration when executing the readout.

Readout point (35 s)

The parity mode PSB readout is calibrated by finding the optimal voltage of the plunger gates near the anticrossing for the readout. The readout point is only calibrated along one axis (vP1 or vP5), for simplicity, and as the performance of the PSB readout is similar at any location along the anticrossing. In the calibration shown in Extended Data Fig. 4b, we initialize either a singlet ((left|uparrow downarrow rightrangle )) or a triplet ((left|downarrow downarrow rightrangle ), using a single-qubit gate) state and sweep the plunger gate to find to the optimal readout point.

Resonance frequency of qubits Q1, Q2, Q5 and Q6 (rough) (17 s)

We perform a course scan of the resonance frequencies of qubits Q1, Q2, Q5 and Q6 (Extended Data Fig. 4c) around the previously saved values. We fit the Rabi formula

to the experimental data and extract the resonance frequency, where P_s(t) is the spin probability, Ω is the Rabi frequency, and Δ is the frequency difference between the resonance frequency of the qubit and the applied microwave tone.

QND readout: CROT Q32, Q45 resonance frequency (14 s)

Subsequently, we calibrate the QND readout for qubits Q3 and Q4. To perform QND readout, we need to calibrate a CROT gate. We choose to use a controlled rotation two-qubit gate, as it requires little calibration (compared to the CPhase) given that we can ignore phase errors during readout.

We set the exchange to 10–20 MHz by barrier gate pulses and scan the CROT driving frequency (Extended Data Fig. 4d) around the previously saved values. Again, we fit the Rabi formula in equation (1) to extract the optimal resonance frequency.

QND readout: CROT Q32, Q45 pulse width (25 s)

Next, we tune the optimal microwave burst duration for the CROT gate, by driving Rabi oscillations (Extended Data Fig. 4e) in the presence of the exchange coupling. We fit the decaying sinusoid

and extract the pulse width the for CROT gate.

Resonance frequency of qubits Q3 and Q4 (rough) (28 s)

With QND readout established, we scan the driving frequency for qubits Q3 and Q4 in a similar manner as we did for Q1, Q2, Q5, Q6 (Extended Data Fig. 4f). The calibration scripts will automatically use QND readout for Q3 and Q4 calibration, in place of the PSB readout for Q1, Q2, Q5 and Q6.

Resonance frequency and amplitude (fine) (Q1, Q2, Q5, Q6→frequency 22 s, amplitude 23 s; Q3, Q4→frequency 32 s, amplitude 34 s)

We calibrate more accurately the qubit frequency and driving amplitude using an error amplification sequence (Extended Data Fig. 4g,h), where we execute an X₉₀ gate 18 times and sweep either the frequency or the amplitude of the microwave burst. We fit the data using the Rabi formula in equation (1) once again to extract the resonancy frequency. The amplitude of the microwave burst is controlled by the IQ input channels of the vector source we used. To calibrate the amplitude for an X₉₀ rotation, we vary the amplitude applied to the IQ input and fit the result to a Gaussian function,

where x is the input amplitude of the IQ signal, µ is the centre of the peak (optimal amplitude) and σ is the peak width. This functional form is not strictly correct but it does find the optimal amplitude for an X₉₀ rotation. We suspect that the longer amplification sequences gave better results, as they more closely resemble the sequence lengths used for randomized benchmarking (including some ‘heating effects’).

In these calibrations, we only calibrate the X₉₀ gate. The Y₉₀ gate is implemented similarly to the X₉₀, but phase shifted. Z gates are performed in software by shifting the reference frame. X₁₈₀ and Y₁₈₀ rotations are performed by applying two 90 degree rotations. We do not simultaneously drive two or more qubits.

X
₉₀ phase crosstalk (Q1, Q2, Q5, Q6 → 27 s; Q3, Q4 → 45 s)

Any single-qubit gate causes the Larmor frequency of the other qubits to shift slightly because of the applied microwave drive. We compensate for this by applying a virtual Z rotation to every qubit after a single-qubit gate has been performed. The Ramsey-based sequence is used to calibrate the required phase corrections (Extended Data Fig. 4i) and data is fitted with the equation

where A and B are fitting parameters that correct for the limited visibility of the spin readout, ϕ is the applied virtual Z rotations in the calibration and ϕ₀ is the fitted phase correction. A single X₉₀ pulse on one qubit will impart phase errors on qubits Q2 to Q6. Thus we need to calibrate separately 30 different phase factors, five for each qubit.

J
_ij versus vB_ij (qubit pairs Q12, Q56 → 146 s; qubit pairs Q23, Q45 → 207 s; qubit pair Q34 → 299 s)

Two-qubit gates are implemented by applying a voltage pulse that increases the tunnel coupling between the respective quantum dots. To enable two-qubit gates, we take the following elements into account:

• Exchange strength. We operate the two-qubit gates at exchange strengths J_on where the quality factor of the oscillations is maximal. This condition is found for J_on ≈ 5 MHz.

• Adiabacity condition. When the Zeeman energy difference (ΔE_z) and the exchange (J(t)) are of the same order of magnitude, care has to be taken to maintain adiabaticity throughout the CPhase gate. We do this by applying a Tukey-based pulse, where the ramp time is chosen as ({tau }_{{rm{ramp}}}=frac{3}{sqrt{Delta {E}_{{rm{z}}}^{2}+{J}_{{rm{on}}}^{2}}}) (ref. ⁴³).

• Single-qubit phase shifts. As we apply the exchange pulse, the qubits will physically be slightly displaced. This causes a frequency shift and hence phase accumulation, which needs to be corrected for.

In order to satisfy these conditions, we need to know the relationship between the barrier voltage and the exchange strength. We construct this relation by measuring the exchange strength (Fig. 3a) for the last 25% of the virtual barrier pulsing range (J > 1 MHz regime). We fit the exchange to an exponential and extrapolate this to any exchange value (Extended Data Fig. 4j). This allows us to generate the adiabatic pulse as described in the main text and choose the target exchange value.

CZ duration (qubit pairs Q12, Q56 → 29 s; qubit pairs Q23, Q45 → 34 s; qubit pair Q34 → 45 s)

The gate voltage pulse to implement a CZ operation uses a Tukey shape in J by inverting the relationship J(vB_ij). The maximum value of J is capped at J_on. The actual largest value of J used and the length of the pulse then determine the phase acquired under ZZ evolution. We first analytically evaluate the accumulated ZZ evolution as a function of these parameters around the target of π evolution under ZZ, and then experimentally fine tune the actual accumulated ZZ evolution by executing a Ramsey circuit with a decoupled CPhase evolution in between the two π/2 rotations. An example of such a calibration measurement is shown in Extended Data Fig. 4k.

CZ phase crosstalk (Q1, Q2, Q5, Q6 → <30 s; Q3, Q4 → <50 s)

After the exchange pulse is executed, single-qubit phases have to be corrected. We correct these phases on all the qubits, whether participating or not in the two-qubit gate. We calibrate the required phase corrections in a very similar way as done for the single-qubit gate phase corrections. An example of the circuit and measurement is given in Extended Data Fig. 4l,m. The exact calibration run-time depends on the CZ pulse width and can vary by a couple of seconds depending on the target qubit.

Heating effects

We observed several effects that bear a signature of heating in our experiments. When microwaves are applied to the EDSR line of the sample, several qubit properties change by an amount that depends on the applied driving power and the duty cycle of applying power versus no power. This effect has also been observed in other works⁵⁵. We report our findings in Extended Data Fig. 6 and will discuss adjustments made to the sequences of the experiments to reduce their effects. The main heating effects are a reduction of the signal-to-noise ratio (SNR) of the sensing dot and a change of the qubit resonance frequency and T₂^*.

In Extended Data Fig. 6a–d, we investigate the effect of a microwave burst applied to the EDSR driving gate, after which the signal of the sensing dot is measured. We observe changes in the background signal and in the peak signal (the electrochemical potential of the sensing dot is not affected, as the peak does not shift in gate voltage). As the background signal rises more than the peak signal, the net signal is reduced. This reduction depends on the magnitude and duration of the applied microwave pulse (Extended Data Fig. 6b). The original SNR can be recovered by introducing a waiting time after the microwave pulse. The typical timescale needed to restore the SNR is of the order of 100 μs (Extended Data Fig. 6c,d). We added for all (randomized benchmarking) data taken in this paper a waiting of 100 μs (500 μs) after the manipulation stage to achieve a good balance between SNR and experiment duration. Spin relaxation between manipulation and readout is negligible, given that no T₁ decay was observed on a timescale of 1 ms within the measurement accuracy. We did not introduce extra waiting times after feedback/CROT pulses in the initialization/readout cycle, as the power to perform these pulses did not limit the SNR.

Extended Data Figure 6f gives more insight in what makes the background and peak signal of the sensing dots change. The impedance of the sensing dot is measured using RF reflectometry. The background of the measured signal depends on the inductance of the surface-mount inductor, the capacitance to ground^29,56,57 and the resistance to ground of the RF readout circuit. Extended Data Figure 6f shows the response of the readout circuit under different microwave powers (the RF power is kept fixed). A frequency shift (0.5 MHz) and a reduction in quality factor is observed. This can be indicative of an increase in capacitance and dissipation in the readout circuit. Currently the microscopic mechanisms that cause this behaviour are unknown.

The second effect is observed when looking at the qubit properties themselves. Extended Data Figure 6e shows that both the dephasing time T₂^* measured in a Ramsey experiment and the qubit frequency are altered by the microwave radiation. In the actual experiments, we apply a microwave pre-pulse of 1–4 μs before the manipulation stage to make the qubit frequency more predictable, although this comes at the cost of a reduced T₂^*. The pre-pulse can be applied either at the start or at the end of the pulse sequence, with similar effects. This indicates that heating effects on the qubit frequency persist for longer than the total time of a single-shot experiment (approximately  600 μs), which is different from the effect on the sensing dot signal. Also the microscopic mechanisms behind the qubit frequency shift and T₂^* reduction remain to be understood.

Parity mode PSB readout

PSB readout is a method used to convert a spin state to a more easily detectable charge state⁵⁸. Several factors need to be taken into account for this conversion, to enable good readout visibilities. Extended Data Figure 1a,b shows the energy level diagrams for PSB readout performed for (1,1) and (3,1) charge occupation. The diagrams use valley energies E_v of 65 μeV to illustrate where problems can occur. When looking at Extended Data Fig. 1, we can observe two potential issues:

• The excited valley state with (left|downarrow downarrow rightrangle ) is located below the ground valley state with (left|uparrow downarrow rightrangle ). We assume in the diagram that the (2,0) singlet state ((left|{rm{S,; 0}}rightrangle )) is coupled to both the (1,1) ground valley state and the (1,1) excited valley state. In this case, during the initialization/readout pulses, a population can be moved into the excited valley state. This problem can be solved by working at a lower magnetic field, such that E_v > E_z (Extended Data Fig. 1b).

• When operating in the (1,1) charge occupation, the readout window is quite small, as the size is determined by the difference between the valley energy and the Zeeman energy. A common way to prevent this problem is by operating in the (3,1) electron occupation.

With both measures in place, we consistently obtain high visibilities of Rabi oscillations (≥94%) on every device tested.

In the following we describe the procedure used to tune up the parity mode PSB.

• Find an appropriate tunnelling rate at the (3,1) anticrossing. An initial guess of a good tunnelling rate can be found using video mode tuning. We use the arbitrary waveform generator to record at high speed the frames of the charge stability diagram (5 μs averaging per point, a full image is acquired in t_image = 200 ms). During the measurement of the frames, we vary the tunnel coupling, while looking at the (3,1) ↔ (4,0) anticrossing until the pattern shown in Extended Data Fig. 1c is observed. This figure shows that, depending on the (random) initial state, the transition from (3,1) to (4,0) occurs at either location (i) or location (ii). This is exactly what needs needs to happen when the readout is performed.

• Find the readout point. We hold point (1) fixed in the centre of the (3,1) charge occupation (Extended Data Fig. 1c). Point (2) is scanned with the AWG along the detuning axis as shown in Extended Data Fig. 1c. We pulse from point (1) to point (2) and measure the state (with ramp time of  around 2 μs), and then we pulse back to point (1). When plotting the measured singlet probability, a gap is seen between the case where a singlet is prepared and the case where a random spin state is prepared (Extended Data Fig. 1d). The centre of this region is a good readout point.

• Optimizing the readout parameters. The main optimization parameters are the detuning (ϵ), tunnel coupling (t_c) and ramp time to ramp towards the PSB region. We also independently calibrate the ramp time and tunnel coupling from the readout zone towards the operation point of the qubits. When ramping in towards the readout point, it is important to be adiabatic with respect to the tunnel coupling. We do not need to be adiabatic with respect to spin, as both (left|uparrow downarrow rightrangle ) and (left|downarrow uparrow rightrangle ) relax quickly to the singlet state (faster than we can measure,   in less than 1 ns). When pulsing from the readout to the operation point, more care has to be taken. When using the ramp time that performs well for the readout, we notice that we initialize a mixed state, as we are not adiabatic with respect to spin. This can be solved by pulsing the tunnel coupling to a larger value before initiating the initialization ramp (Extended Data Fig. 1g).

We show in Extended Data Fig. 1e,f that the histograms for parallel and antiparallel spin states are well separated, which enbales a spin readout fidelity exceeding 99.97% for both qubits Q1–Q2 and for qubits Q5–Q6. This number could be further increased by integrating the signal for longer, but is not the limiting process. This method of quantifying the spin readout fidelity is commonly used in the literature but it leaves out errors occurring during the ramp time (the mapping of qubit states to the readout basis states). This can be a pronounced effect, as seen from the measured visibility of the Rabi oscillations.

Postselection of data

When using parity readout on a single qubit pair, around 5% of the runs are discarded on average as part of the initialization procedure (Fig. 1d). In the case when two outer qubit pairs are used, about 10% of the data are discarded (1 − 0.95²). When performing experiments on all six qubits, additional initialization steps with postselection are needed (in Extended Data Fig. 2, runs are postselected on the basis of 18 measurement outcomes in total), and we discard around 65% of the dataset. When we do not discard any runs, the initialization fidelity reduces by around 5–9% for a single qubit pair.

Setup and the real-time feedback using FPGA

Setup

A detailed schematic of the experimental setup is presented in Extended Data Fig. 7, listing all the key components used in the experiment.

Programming quantum circuits

The quantum circuits are implemented in the form of microwave bursts for single-qubit operations, gate voltage pulses for two-qubit gates and gate voltage pulses combined with RF bursts for readout. The gate voltage pulses are generated by an arbitrary wave generator (AWG). The microwave bursts are generated through IQ modulation of a microwave vector source carrier frequency. The input signals for the IQ modulation are generated by the same AWG as used for the voltage pulses. The IQ modulation defines the amplitude envelope of the microwave bursts, the output frequency and the phase shifts. Virtual Z gates are implemented by incrementing the reference phase of the numerically controlled oscillators (NCOs) (see below) and are used to, for example, correct phase errors introduced by crosstalk. The generated control signals are stored in memory with a resolution of 1 ns.

Microwave bursts applied to the six-qubit sample are supplied by a single microwave source with a carrier frequency set at 16.3 GHz. We address the six different qubits using single side-band IQ modulation of the carrier to displace the frequency of the microwave output signal to the frequency of the target qubit. As each qubit has a different resonance frequency (which is different from the carrier frequency), it is necessary to track the phase evolution at the qubit Larmor precession frequency to ensure phase coherent microwave bursts for successive single-qubit operations. To realize that, we define in the AWG six continuously running NCOs, one for each qubit. These NCOs keep track of the phase evolution of the qubits with respect to the carrier frequency. We choose this approach instead of precalculating the phase factors for every pulse in a sequence, which is a not a scalable approach with the growing complexity of the quantum circuits.

The digitizer is synchronized with the AWG to acquire qubit readout data. In a single shot we can include multiple readout segments, each defined in a digitizer instruction list. A step in this list specifies a measurement time window, a wait time and the threshold for the qubit state. The input signal is integrated during the measurement window and the result is compared with a threshold to determine the qubit state. This outcome, 0 or 1, can be passed directly to the AWG by a trigger line within a Keysight PCI eXtensions for Instrumentation (PXI) chassis, shared by the digitizer and the AWGs, to realize real-time feedback on the measurement output.

Real-time feedback

In the initialization and readout sequences the execution of selected gates depends on the outcomes of intermediate measurements, which enables real-time qubit state corrections. The total time from the end of the measurement until the start of the conditional gate (burst) on the device should be much shorter than the qubit relaxation time T₁, and ideally also shorter than  around 1 μs, which is the time needed for the adiabatic passage back to the manipulation point after the parity measurement, such that no unnecessary idling time is spent. This fast control loop is realized with a custom FPGA (field-programmable gate array) program in the AWG and digitizer as shown in Extended Data Fig. 8. The total latency for the closed loop feedback is 660 ns, which fits the design requirements.