Universal control of a six-qubit quantum processor in silicon

Abstract

Future quantum computers capable of solving relevant problems will require a large number of qubits that can be operated reliably¹. However, the requirements of having a large qubit count and operating with high fidelity are typically conflicting. Spins in semiconductor quantum dots show long-term promise^2,3 but demonstrations so far use between one and four qubits and typically optimize the fidelity of either single- or two-qubit operations, or initialization and readout^{4,5,6,7,8,9,10,11}. Here, we increase the number of qubits and simultaneously achieve respectable fidelities for universal operation, state preparation and measurement. We design, fabricate and operate a six-qubit processor with a focus on careful Hamiltonian engineering, on a high level of abstraction to program the quantum circuits, and on efficient background calibration, all of which are essential to achieve high fidelities on this extended system. State preparation combines initialization by measurement and real-time feedback with quantum-non-demolition measurements. These advances will enable testing of increasingly meaningful quantum protocols and constitute a major stepping stone towards large-scale quantum computers.

Main

On the path topractical large-scale quantum computation, electron spin qubits in semiconductor quantum dots¹² show promise because of their inherent potential for scaling through their small size^13,14, long-lived coherence⁴ and compatibility with advanced semiconductor manufacturing techniques¹⁵. Nevertheless, spin qubits currently lag behind in scale when compared to superconducting, trapped ions and photonic platforms, which have demonstrated control of several dozen qubits^16,17,18. By comparison, using semiconductor spin qubits, partial¹⁹ and universal¹¹ control of up to four qubits was achieved and entanglement of up to three qubits was quantified^9,10,20. In a six-dot linear array, two qubits encoded in the state of three spins each were operated²¹ and spin exchange oscillations in a 3 × 3 array have been reported²².

Furthermore, the experience with other qubit platforms shows that, in scaling up, maintaining the quality of the control requires substantial effort, particularly, for instance, to deal with the denser motional spectrum in trapped ions²³, to avert crosstalk in superconducting circuits²⁴ or to avoid increased losses in photonic circuits²⁵. For small semiconductor spin qubit systems, state-of-the-art single-qubit gate fidelities exceed 99.9%^5,26,27 and two-qubit gates well above 99% fidelity have been demonstrated recently^6,7,8,10. Most quantum-dot-based demonstrations suffer from low initialization or readout fidelities, with typical visibilities of no more than 60–75%, with only a few recent exceptions^8,21,28. Conversely, high-fidelity spin readout has been claimed on the basis of an analysis of the readout error mechanisms, but these claims have not been validated in combination with high-fidelity qubit control^29,30. Although high-fidelity initialization, readout, single-qubit gates and two-qubit gates have thus been demonstrated individually in small systems, almost invariably one or more of these parameters are appreciably compromised while optimizing others. A major challenge and important direction for the field is therefore to achieve high fidelities for all components while at the same time enlarging the qubit count.

Here we study a system of six spin qubits in a linear quantum dot array and test what performance can be achieved using known methods, such as multi-layer gate patterns for independent control of the two-qubit exchange interaction^31,32,33 and micromagnet gradients for electric-dipole spin resonance (EDSR) and selective qubit addressing³⁴. Furthermore, we introduce several new techniques for semiconductor qubits that, collectively, are critical for the improvement of the results and help facilitate scalability, such as initialization by measurement using real-time feedback ³⁵, qubit initialization and measurement without reservoir access, and efficient calibration routines. Initialization and readout circuits span the full six-qubit array. We characterize the quality of the control by preparing maximally entangled states of two and three spins across the array.

The six-qubit array is defined electrostatically in the ²⁸Si quantum well of a ²⁸Si/SiGe heterostructure, between two sensing quantum dots, as seen in Fig. 1a (Methods). The multi-layer gate pattern enables excellent control of the charge occupation of each quantum dot and of the tunnel couplings between neighbouring quantum dots. These parameters are controlled independently through linear combinations of gate voltages, known as virtual gates³⁶. The interdot pitch is chosen to be 90 nm, which for this 30-nm-deep quantum well yields easy access to the regime with one electron in each dot, indicated for short as the (1,1,1,1,1,1) charge occupation. Low valley splittings on Si/SiGe devices have hindered progress in the past³⁷, but in this device all valley splittings are in the range of 100–300 μeV (Supplementary Information).

Fig. 1: Device initialization, measurement and calibration.

a, A false-coloured scanning electron microscope image of a device similar to the one used in the experiments. Each colour represents a different metallization layer. Plunger (P, blue) and barrier (B, green) gates are used to define quantum dots in the channel between the screening gates (red) and sensing dots (SD1 and SD2) on the side. Two cobalt micromagnets (yellow) are placed on top of the gate stack. b,c, Buildings blocks used for readout (read) and initialization (init) in this experiment, showing the circuit used to perform a single QND measurement of qubit Q3 (b) and the circuit used for spin measurement and initialization using a parity measurement (c). The double line in the diagram indicates that X₁₈₀ rotation is conditional on the measurement outcome. d, An example of a CROT used to initialize the qubits. The sequence shown is applied repeatedly with short time intervals, with the final state of one cycle being the initial state of the next. (1) shows the even parity probability of the first measurement; (2) shows the even parity probability after the bit flip conditional on the first measurement outcome. e, Schematic showing the total scheme used for the initialization and readout of all six qubits, with U, the unitary matrix of the manipulation stage (see Extended Data Fig. 2 for an expanded view). f, Calibration graph used in the experiments. The numbers on the right show the number of parameters that are calibrated in each step.

In designing the qubit measurement scheme, we focused on achieving short measurement cycles in combination with high-fidelity readout, as this accelerates testing of all other aspects of the experiment. To measure the outer qubit pairs, we use Pauli spin blockade (PSB) to probe the parity of the two spins (rather than differentiating between singlet and triplet states), exploiting the fact that the T₀ triplet relaxes to the singlet well before the end of the 10 μs readout window. We tune the outer dot pairs of the array to the (3,1) electron occupation, where the readout window is larger than in the (1,1) regime (Extended Data Fig. 1). As the sensing dots are less sensitive to the charge transition between the centre dots, the middle qubits are measured by quantum-non-demolition (QND) measurements that map the state of qubit Q3(Q4) on qubit Q2(Q5) through a conditional rotation (CROT) (Fig. 1b)^5,38. In this way, for every iteration of the experiment, 4 bits of information are retrieved that depend on the state of all six physical qubits. Iterative operation permits full readout of the six-qubit system.

Qubit initialization is based on measurements of the spin state across the array followed by real-time feedback to place all qubits in the target initial state. This scheme has the benefit of not relying on slow thermalization and that no access to electron reservoirs is needed to bring in fresh electrons, which is helpful for scaling to larger arrays. In fact, we had experimental runs of more than one month in which the electrons stayed within the array continuously. For qubits Q3 and Q4, real-time feedback simply consists of flipping the qubit if the measurement returned $\left|\uparrow \right\rangle$. Initialization of qubits Q1 and Q2 (or Q5 and Q6) using parity measurements and real-time feedback is illustrated in Fig. 1c. First, assuming that the qubits start from a random state, we perform a parity measurement that will cause the state to either collapse to an even ($\left|\downarrow \downarrow \right\rangle$, $\left|\uparrow \uparrow \right\rangle$) or odd ($\left|\uparrow \downarrow \right\rangle$/$\left|\downarrow \uparrow \right\rangle$) parity (Methods). After the measurement, a π pulse is applied to qubit Q1 in case of even parity, which converts the state to odd parity (feedback latency 660 ns). Subsequently, we perform a second measurement, which converts either of the odd parity states to $\left|\uparrow \downarrow \right\rangle$. Specifically, when pulsing towards the readout operating point, both $\left|\uparrow \downarrow \right\rangle$ and $\left|\downarrow \uparrow \right\rangle$ relax into the singlet state ((4,0) charge occupation). When pulsing adiabatically from the (4,0) back to the (3,1) charge configuration, the singlet is mapped onto the $\left|\uparrow \downarrow \right\rangle$ state. If the qubit initialization is successful, the second measurement should return an odd parity (with typically around 95% success rate). To further boost the initialization fidelity we use the outcome of the second measurement to postselect successful experiment runs (Extended Data Fig. 1d). Figure 1d shows initialization by measurement of the first two qubits. The first readout outcome (blue) shows Rabi oscillations controlled by a microwave burst of variable duration applied near the end of the previous cycle (see Methods for more details). The second readout outcome (green) shows the state after the real-time classical feedback step. The oscillation has largely vanished, indicating successful initialization by measurement and feedback.

The sequence to initialize and measure all qubits is shown in Fig. 1e (see Extended Data Fig. 2 for the unfolded quantum circuit). We sequentially initialize qubit pair Q5 and Q6, then qubit Q4, then qubits Q1 and Q2, and finally qubit Q3, using the steps described above (for compactness, the steps appear as being simultaneous in the diagram). In order to further enhance the measurement and initialization fidelities, we repeat the QND measurement three times, alternating the order of the qubit Q3 and Q4 measurements. We postselect runs with three identical QND readout outcomes in both the initialization and measurement steps (except for Fig. 5 below, where readout simply uses majority voting). After performing the full initialization procedure depicted in Fig. 1e, the six-qubit array is initialized in the state $\left|\uparrow \downarrow \downarrow \downarrow \downarrow \uparrow \right\rangle$. In all measurements below, we initialize either two, three or all six qubits, depending on the requirement of the specific quantum circuit we intend to run. We leave the unused qubits randomly initialized, as the visibilities decrease when initializing all six qubits within a single shot sequence (Extended Data Fig. 3). When operating on individual qubits, the initialization and measurement procedures yield visibilities of 93.5–98.0% (Fig. 2e). To put these numbers in perspective, if the readout error for both $\left|0\right\rangle$ and $\left|1\right\rangle$ were 1% alongside an initialization error of 1%, the visibility would be 96%.

Fig. 2: Single-qubit gate characterization.

a, Rabi oscillations for every qubit, taken sequentially. The spin fraction refers to the spin-up fraction for qubits Q2–Q5 and to the spin-down fraction for qubits Q1 and Q6. The drive amplitudes were adjusted in order to obtain uniform Rabi frequencies of 5 MHz. b, Qubit frequency for each of the six qubits. c, Rabi frequency of each qubit as a function of the applied microwave power. d, Randomized benchmarking results for each qubit, using a 5 MHz Rabi frequency. The reported fidelity is the average single-qubit gate fidelity, and the uncertainties (2σ) are calculated using the covariance matrix of the fit. e, Table showing the dephasing time T₂*, Hahn echo decay time (T₂^h) and visibilities (vis.) for each qubit.

We manipulate the qubits by EDSR³⁹. A micromagnet located above the gate stack is designed to provide both qubit addressability and a driving field gradient (Fig. 1a and Supplementary Information). We can address each qubit individually and drive coherent Rabi oscillations as depicted in Fig. 2. We observe no visible damping in the first five periods. The data in Fig. 2b shows that the qubit frequencies are not spaced linearly, deviating from our prediction based on numerical simulations of the magnetic field gradients (Supplementary Fig. 1). However, the smallest qubit frequency separation of approximately 20 MHz is sufficient for selective qubit addressing with our operating speeds varying between 2 and 5 MHz. The Rabi frequency is linear in the driving amplitude over the typical range of microwave power used in the experiment (Fig. 2c). We operate single-qubit gates sequentially, to ensure we stay in this linear regime and to keep the calibration simple. Simultaneous rotations would involve additional characterization and compensation of crosstalk effects (see also Extended Data Fig. 5). We characterize the single-qubit properties of each qubit separately. Figure 2d shows the results of randomized benchmarking experiments. All average single-qubit gate fidelities are between 99.77 ± 0.04% and 99.96 ± 0.01%, which demonstrates that, even within this extended qubit array, we retain high-fidelity single-qubit control. The coherence times of each qubit are tabulated in Fig. 2e. We expect spin coherence to be limited by charge noise coupled in by the micromagnet⁴⁰.

Two-qubit gates are implemented by pulsing the (virtual) barrier gate between adjacent dots while staying at the symmetry point. Pulsing the barrier gate leads to a ZZ interaction (throughout, X, Y and Z stand for the Pauli operators, I for the identity and ZZ is shorthand for the tensor product of two Pauli Z operators, and so on), given that the effect of the flip-flop terms of the spin exchange interaction is suppressed because of the differences in the qubit splittings⁴¹. The quantum circuit in Fig. 3a measures the time evolution under the ZZ component of the Hamiltonian only, as the single-qubit π pulses in between the two exchange pulses decouple any IZ/ZI terms⁴². The measured signal oscillates at a frequency J/2 (Fig. 3b–f) as a function of the barrier gate pulse duration, corresponding to controlled phase (CPhase) evolution. When pulsing only the barrier gate between the target qubit pair, the desired on/off ratio of J_ij (>100) could not be achieved. We solve this, without sacrificing operation at the symmetry point, by using a linear combination of the virtual barrier gates (vB1–vB6). Specifically, the barrier gates around the targeted quantum dot pair are pulsed negatively to push the corresponding electrons closer together and thereby enhance the exchange interaction (Extended Data Table 1). The exponential dependence of J_ij on the virtual barrier gates is seen in Fig. 3h. In Fig. 3g we investigate the residual exchange of idle qubit pairs, while one qubit pair is pulsed to its maximal exchange value within the operating range. The results show minimal residual exchange amplitudes in the off state between the other pairs.

Fig. 3: Two-qubit gate characterization.

a, Quantum circuit used to measure CPhase oscillations between a pair of qubits. b–f, Measured spin probabilities as a function of the total evolution time 2t for neighbouring qubit pairs Q1–Q2 (b), Q2–Q3 (c), Q3–Q4 (d), Q4–Q5 (e) and Q5–Q6 (f) for different virtual barrier gate voltages (with 0 and 1 corresponding to the exchange switched off and at its maximum value). g, Maximum exchange coupling measured for each qubit pair, and the corresponding residual exchange coupling for the other pairs, achievable within the AWG pulsing range without retuning of the static gate voltages. Bottom row: J_ij with all exchange couplings switched off (see Supplementary Information for error bars). h, Exchange coupling versus virtual barrier gate voltage for all qubit pairs. i, Schematic showing the energy levels in the absence (left) and presence (middle, right) of the effective Ising ZZ interaction under exchange (see text). Owing to the ZZ coupling term, the antiparallel spin states are lowered in energy, and pick up an additional phase as a function of time, resulting in a CPhase evolution. The shifted energy levels also enable conditional microwave-driven rotations (CROT), which we use during initialization and readout. j, Pulse shape of the exchange amplitude throughout a gate voltage pulse used for the CZ gate, and the corresponding pulse shape converted to gate voltage.

Through suitable timing, we use the CPhase evolution to implement a controlled-Z (CZ) gate. Figure 3j shows the pulse shape that is used to ensure a high degree of adiabaticity throughout the CZ gate⁶. We use a Tukey window as waveform, with a ramp time of ${\tau }_{{\rm{ramp}}}=\frac{3}{\sqrt{\delta {B}^{2}+{J}_{\max }^{2}}}$(ref. ⁴³). This pulse shape is defined in units of energy and we convert it into barrier voltages using the measured voltage to exchange the energy relation⁶.

One of the challenges when operating larger quantum processors is to track and compensate for any dynamical changes in qubit parameters to ensure high-fidelity operation, initialization and readout. Another challenge is to keep track of and compensate for crosstalk effects imparted by both single- and two-qubit gates on the phase evolution of each qubit. We perform automated calibrations, as shown in Fig. 1f, and correct 108 parameters in total. The detailed description of each calibration routine is included in the Methods and Extended Data Fig. 4. Twice a week, we run the full calibration scheme, which takes about one hour. Every morning, we run the calibration scheme leaving out the phase corrections for single-qubit operations and the dependence of J_ij on the virtual barrier gates vB_ij. Sometimes, specific calibrations, especially qubit frequencies and readout coordinates, are re-run throughout the day, as needed. Supplementary Fig. 3 plots the evolution of the calibrated values for a number of qubit parameters over the course of one month.

With single- and two-qubit control established across the six-qubit array, we proceed to create and quantify pairwise entanglement across the quantum dot array as a measure of the quality of the qubit control (Fig. 4a–e). These experiments benefited from a high level of abstraction in the measurement software, allowing us to flexibly program a variety of quantum circuits acting on any of the qubits, drawing on the table of 108 calibration parameters that is kept updated in the background and on the detailed waveforms to achieve high-fidelity gates. The parity readout of the outer qubits yields a native ZZ measurement operator. We measure single-qubit expectation values by mapping the ZZ operator to a ZI/IZ operator, as shown in Fig. 4g. This allows full reconstruction of the density matrix. The state fidelity is calculated using $F=\left\langle \psi \right|\rho \left|\psi \right\rangle$, where ψ is the target state and ρ is the measured density matrix. The target states are maximally entangled Bell states. The obtained density matrices measured across the six-dot array have state fidelity ranging from 88% to 96%, which is considerably higher than the Bell state fidelities of 78% to 89% (all state preparation and measurement (SPAM) corrected, see Methods) reported on two-qubit quantum dot devices just a few years ago^42,44,45.

Fig. 4: Bell state tomography.

a–e, Measured two-qubit density matrices for qubits Q1–Q2 (a), Q2–Q3 (b), Q3–Q4 (c), Q4–Q5 (d) and Q5–Q6 (e), after removal of SPAM errors (see Supplementary Information for the uncorrected density matrices). The target Bell states are indicated and outlined with the wireframes. f, Colour wheel with phase information for the density matrices presented in a–e. g, Quantum circuits used for converting parity readout (ZZ) into effective single-qubit readout (IZ and ZI). h, State fidelities of the measured density matrices with respect to the target Bell states and the concurrences for the measured density matrices. Error bars (2σ) are derived from Monte Carlo bootstrap resampling^9,44,59. State fidelities without readout error removal: qubits Q1–Q2, 88.2%; Q2–Q3, 83.8%; Q3–Q4, 78.0%; Q4–Q5, 91.3%; Q5–Q6, 91.3%.

As a final characterization of the qubit control across the array, we prepare Greenberger–Horne–Zeilinger (GHZ) states, which are the most delicate entangled states of three qubits^46,47. Figure 5a shows the quantum circuit we used to prepare the GHZ states. The full circuit, including initialization and measurement, contains up to 14 CROT operations, 2 CZ operations, 42 parity measurements, 16 single-qubit rotations conditional on real-time feedback and 5 single-qubit X₉₀ rotations (Extended Data Fig. 2). The measurement operators for quantum state tomography are generated in a similar manner as for the Bell states. In order to reconstruct three-qubit density matrices, we perform measurements in 26 (for qubits Q2–Q3–Q4 and Q3–Q4–Q5) or 44 (for qubits Q1–Q2–Q3 and Q4–Q5–Q6) different basis and repeat each set 2,000 times to collect statistics. A full dataset consisting of 52,000 (88,000) single-shot repetitions takes about 5 min to acquire, thanks to the efficient uploading of waveforms to the waveform generator (Methods) and the short single-shot cycle times. Figure 5b–e shows the measured density matrices for qubits Q1–Q2–Q3, Q2–Q3–Q4, Q3–Q4–Q5 and Q4–Q5–Q6. The obtained state fidelities range from 71% to 84% (see Methods for a brief discussion of dephasing effects from heating). For comparison, the record GHZ state fidelity reported recently for a triple quantum dot spin qubit system is 88% (ref. ⁹). The same dataset from ref. ⁹ analysed without readout correction yields 45.8% fidelity, whereas our results with no readout error removal range from 52.8% to 67.2% (Supplementary Information). The reduction in state fidelities compared to the two-qubit case (especially when involving qubits Q3 and Q4) is mainly due to increased SPAM errors. From the same data sets, we calculate entanglement witnesses, which clearly demonstrate three-qubit entanglement (Supplementary Information).

Fig. 5: Three-qubit GHZ state tomography.

a, Circuit diagram used to prepare the GHZ states. The U_map operation is the unitary that is executed in case we measure the IZ or ZI projections on qubits Q1–Q2 and Q5–Q6, similar to the Bell state experiments. b–e, Density matrices of the prepared GHZ states using qubits Q1–Q2–Q3 (b), Q2–Q3–Q4 (c), Q3–Q4–Q5 (d) and Q4–Q5–Q6 (e), obtained using quantum state tomography, after removal of SPAM errors (see Supplementary Information for the uncorrected density matrices). The black wireframes correspond to the ideal GHZ state. f, Colour wheel with phase information for the density matrices presented in d and e. g, Table showing the state fidelities and entanglement witness values for the different qubit sets. We choose ϕ in $\left|{\psi }_{{\rm{GHZ}}}\right\rangle =(\left|000\right\rangle +{{\rm{e}}}^{{\rm{i}}\varphi }\left|111\right\rangle )\sqrt{2}$, with respect to the highest state fidelity. State fidelities without SPAM removal: qubits Q1–Q2–Q3, 64.3%; Q2–Q3–Q4, 52.8%; Q3–Q4–Q5, 52.7%; Q4–Q5–Q6, 67.2%.

The demonstration of universal control of six qubits in a ²⁸Si/SiGe quantum dot array advances the field in multiple ways. While scaling to a record number of qubits for a quantum dot system, we achieve Rabi oscillations for each qubit with visibilities of 93.5–98.0%, implying high readout and initialization fidelities. The initialization uses a new scheme relying on qubit measurement and real-time feedback. Readout relies on PSB and QND measurements. This combination of initialization and readout allows the device to be operated while retaining the six electrons in the linear quantum dot array, alleviating the need for access to electron reservoirs. All single-qubit gate fidelities are around 99.9% and the high quality of the two-qubit gates can be inferred from the 89–95% fidelity Bell states prepared across the array. The development of a modular software stack, efficient calibration routines and reliable device fabrication have been essential for this experiment. Future work must focus on understanding and mitigating heating effects leading to frequency shifts and reduced dephasing times, as we find this to be the limiting factor in executing complicated quantum circuits on many qubits. The use of simultaneous single-qubit rotations and simultaneous two-qubit CZ gates will keep pulse sequences more compact, at the expense of additional calibrations. This will require accounting for crosstalk effects, which we anticipate will be easiest for the two-qubit gates. We estimate that the concepts used here for control, initialization and readout can be used without substantial modification in arrays that are twice as long, as well as in small two-dimensional arrays (Supplementary Information). Scaling further will require additional elements such as cross-bar addressing to control dense two-dimensional arrays^48,49 and on-chip quantum links to connect local quantum registers together^3,50,51,52.

Methods

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 ($\left|\uparrow \downarrow \right\rangle$) or a triplet ($\left|\downarrow \downarrow \right\rangle$, 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

$${P}_{s}(t)=\frac{{\varOmega }^{2}}{{\varOmega }^{2}+{\varDelta }^{2}}{\sin }^{2}\left(\frac{\sqrt{{\varOmega }^{2}+{\varDelta }^{2}}}{2}t\right)$$

(1)

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

$${P}_{s}(t)=\frac{A}{2}\sin (\omega t-{\varphi }_{0}){{\rm{e}}}^{-\frac{t}{\tau }}+B$$

(2)

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,

$${P}_{s}(x)=\alpha {{\rm{e}}}^{-\frac{{(x-\mu )}^{2}}{2{\sigma }^{2}}}\ .$$

(3)

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

$${P}_{s}(\varphi )=-\frac{A}{2}\cos (\varphi -{\varphi }_{0})+B$$

(4)

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 $\left|\downarrow \downarrow \right\rangle$ is located below the ground valley state with $\left|\uparrow \downarrow \right\rangle$. We assume in the diagram that the (2,0) singlet state ($\left|{\rm{S,\; 0}}\right\rangle$) 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 $\left|\uparrow \downarrow \right\rangle$ and $\left|\downarrow \uparrow \right\rangle$ 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.