Highly Tunable Two-Qubit Interactions in Si/SiGe Quantum Dots by Interchanging the Roles of Qubit-Defining Gates

Abstract

Silicon-quantum-dot spin qubits have become a promising platform for scalable quantum computing because of their small size and compatibility with industrial semiconductor manufacturing processes. Although Si/SiGe heterostructures are commonly used to host spin qubits due to their high mobility and low percolation density, the SiGe spacer creates a gap between the qubits and control electrodes, which limits the ability to tune the exchange coupling. As a result, residual coupling leads to unwanted single-qubit phase shifts, making multi-qubit control more difficult. In this work, we explore swapping the roles of overlapping nanogates to overcome this issue. By reconfiguring the gate voltages, we demonstrate in situ role switching while maintaining multi-qubit control. Additionally, this method improves the tunability of the exchange coupling by up to 3.6 times. This strategy reduces unintended single-qubit phase shifts and minimizes the complexity of multi-qubit control, supporting scalable growth with minimal experimental overhead.

Spin qubits based on semiconductor quantum dots have been extensively studied as a feasible technology for implementation in scalable quantum processors, largely motivated by their fabrication compatibility with industrial complementary metal oxide semiconductor technology. Several studies have demonstrated the operation of one- and two-dimensional spin-qubit arrays, − high-fidelity single- and two-qubit gates, − long-range coupling via a superconducting resonator, , and operation at elevated temperatures, which significantly boosts cooling power and offers advantages for quantum-classical chip integration. − Building on this progress, high-fidelity initialization, readout, and manipulation of spin qubits have now been demonstrated within a single device, as validated by observing the violation of Bell’s inequality. In parallel, the industrially mature fabrication technology has also motivated extensive research toward the high-throughput production of high-uniformity spin-qubit devices. −

The formation of quantum dots is a prerequisite for the realization of spin qubits. This involves confining electrons or holes to specific regions by leveraging the properties of the material in combination with the electric potential generated by applying voltage to nanofabricated metal gates. In particular, silicon-quantum-dot qubits have been extensively studied, due to the high natural abundance of zero-nuclear-spin isotopes. Furthermore, residual nuclear spins can be further removed via isotopic enrichment to provide a magnetically quiet environment for the spin qubits. With this platform, the overlapping metal gate architecture has been widely adopted to realize tight electron confinement. ,, In early device implementations, approaching the single-electron regime was a crucial objective, which prompted the fabrication of plunger gates with an appropriate width to ensure their energy controllability and sufficient coupling between quantum dots. In many studies, a tuning strategy was employed in which quantum dots are formed beneath the plunger gates located below the barrier gate layer, a configuration termed conventional tuning in this work. In this tuning scheme, gates with stronger capacitive coupling to the quantum dots are typically assigned to control the dot energy levels. Studies using this configuration have demonstrated that static tunnel coupling between quantum dots can be well-controlled. However, as research efforts have increasingly shifted toward high-fidelity multi-qubit control, which often requires dynamic control of the coupling via barrier pulses, the achievable on/off ratio of the coupling with barrier pulses exceeding 100 and stable operation under large barrier voltages have been reported to be limited in some studies. , These constraints have motivated alternative approaches, such as exploiting multiple barrier pulses or detuning. Meanwhile, spin shuttling, a potential pathway toward long-range coupling, has also been extensively studied in devices where plunger and barrier nanogates of nearly equal widths are used to form and move quantum dots continuously.

In line with these efforts, we fabricated a quantum-dot qubit device with equal-width nanogates and investigated the effect of changing the tuning strategy. This approach, termed interchanged tuning hereafter, assigns the nanogates closer to (farther from) the spin qubits as barrier (plunger) gates. Using this tuning method, the well-defined formation of quantum-dot single-spin qubits is demonstrated. Although previous studies on shuttling have shown that potential wells can be dynamically formed beneath each gate and moved continuously, shifting multiple wells and energy barriers with maintained multi-qubit controllability has not been demonstrated in a similar gate structure. Here, we show that both of these are feasible with the proposed tuning strategy. Furthermore, the tunability of the exchange coupling is enhanced by several orders of magnitude for all nearest-neighbor pairs with a maximum value of 16.6 dec/V, while the controllability of the spin qubits is simultaneously maintained. We further discuss how this enhancement may influence future spin-qubit experiments.

The device was fabricated on an isotopically enriched ²⁸Si/SiGe heterostructure wafer. First, a 20-nm-thick Al₂O₃ layer was deposited by atomic layer deposition (ALD) to serve as insulation, followed by three layers of Ti/Pd metallic gates and oxide layers using e-beam lithography, e-beam evaporation, and ALD. The first of these layers consists of screening gates, which prevent the electric field from accessing the upper metal layers and locally deplete the two-dimensional electron gas (2DEG), thereby roughly defining a one-dimensional channel for quantum-dot formation. The second and third layers are used to control either the energy of the quantum dots or the tunnel coupling between them. A micromagnet was fabricated on top of the metal layers to introduce a magnetic field gradient to enable individual qubits to be addressed and to enhance the controllability of spin qubits via electric-dipole spin resonance. The micromagnet was magnetized by applying an external in-plane magnetic field of 0.44 T in the direction, as indicated in Figure a. The spin-qubit resonance frequency is typically in the range of 15–16 GHz. Single-qubit operation is achieved by applying microwave signals to the screening gate at the qubit frequency, which causes the spin qubit to experience an oscillating transverse magnetic field. High-bandwidth spin-qubit measurements were conducted by employing radio-frequency (RF) reflectometry to apply a carrier signal to the ohmic contacts, which are connected to LC tank circuits composed of a commercial inductor and parasitic capacitance. All gates were connected to direct-current voltage sources and to either an arbitrary waveform generator or a signal generator through on-board bias tees. The device was cooled in a dilution refrigerator, and the electron temperature was approximately 75 mK. Additional details of the experimental setup are provided in Supporting Information S1.

1.

²⁸Si/SiGe linear quantum-dot qubit device and schematic representation of tuning strategies. (a) Scanning electron microscopy image of the device. The squares with X symbols denote an ohmic contact through which a RF signal is injected via a tank circuit for RF reflectometry. The purple gate enclosed within the dashed line represents the screening gate on the first metal layer, to which microwaves are applied for single-qubit manipulation. The gates colored green (blue) correspond to the second (third) metal layer. The micromagnet fabricated on top of the gates is omitted for clarity. (b and c) Transmission electron microscopy image showing a cross section of a similar device fabricated in the same batch. The estimated electrostatic potential landscape and quantum-dot locations for both tuning strategies are shown. For each strategy, the plunger and barrier gates are colored green and blue, respectively.

Figure b shows the gate configuration for the conventional tuning method, along with a schematic of the corresponding potential landscape, where the green (blue) gates located closer to (farther from) the 2DEG act as plunger (barrier) gates. The combination provides a potential energy landscape for an experiment using five quantum dots (Q₁, Q₂, Q₃, Q₄, and Q₅) and two sensing dots for charge sensing, one on each side. Alternatively, the roles of the nanogates can be switched by assigning the role of barrier gates to the gates closer to the 2DEG. The corresponding schematic potential landscape is shown in Figure c. This approach allows the formation of four quantum dots (Q₁′, Q₂′, Q₃′, and Q₄′) with the given gate structure.

First, the interchanged tuning method is shown to enable the single-electron regime to be reached for all quantum dots suitable for multi-qubit operation. Figure a shows the charge stability diagrams for the few-electron regimes of Q₁′–Q₂′, Q₂′–Q₃′, and Q₃′–Q₄′ with physical gates. Although the lever arm of the newly mapped plungers is reduced, more than three-electron filling is achieved due to a high-quality 8-nm-thick intergate oxide layer, which prevents breakdown even under a few volts of potential difference between the gates. Parity readout was adopted using the Pauli spin blockade (PSB) , for spin-state readout, a recent and widely used approach due to its deterministic readout time, applicability to isolated quantum-dot qubits, and robust operation at elevated temperatures. ,, By setting the charge occupation to (3, 1, 1, 3), a wide PSB readout window was realized by exploiting orbital energy spacing rather than valley splitting. The expression (k, l, m, n) denotes the number of electrons in each quantum dot. Under this configuration, the newly designated barrier gates were observed to have the potential to vary the tunnel coupling between dots over a wide range, as qualitatively illustrated in the bottom insets to Figure a. Compared with the conventional tuning method, the outermost quantum dots are farther from the sensor dots. Therefore, we used an integration time of 2 μs, which is slightly longer than that of the conventional strategy, to maintain high-fidelity readout. Interchanged tuning produced a signal-to-noise ratio up to 10.6, which corresponds to a charge readout fidelity above 99.9%. (see Supporting Information S2)

2.

Charge stability diagram of double-quantum-dot pairs and coupling. (a) Charge stability diagram of pairs of double quantum dots. P _i ′ denotes the voltage on the plunger gates in the interchanged configuration. Insets: Variation of the charge stability diagram near interdot tunneling as the voltage applied to the barrier gates becomes increasingly positive, with the text in each inset indicating the voltage increments. The label ΔB _i ′ denotes the magnitude of the voltage increase on the barrier gates. For Q₂′–Q₃′, the tunneling rate to the reservoir is slow compared with the time scale of voltage rastering, so that the transition lines appear latched. (b) Simulated electric potential for the conventional and interchanged gate configurations. Left: Device cross section for the two tuning strategies. Right: Corresponding simulated potential profile at the qubit location as the additional voltage on B₃ (B₃′) is varied from 0 to 0.3 V. The data for the conventional tuning strategy are vertically offset for clarity of comparison. (c) Tunnel coupling calculated from the simulated data for both tuning strategies.

The potential profile on 2DEG for the tuning strategies was simulated using COMSOL MULTIPHYSICS (Figure b). For both tuning methods, we calculated the potential profiles by varying the voltage applied to one of the additional barrier gates to a maximum of 0.3 V, considering the typical voltage range used in the experiment. As the barrier gate voltage of B₃ (B₃′) increases, the potential barrier between Q₂ (Q₂′) and Q₃ (Q₃′) gradually decreases, which leads to stronger tunnel coupling. The change is more pronounced in the interchanged gate configuration than in the conventional one. Based on the simulation result, the tunnel coupling was calculated as a function of the barrier gate voltage by solving the single-particle Hamiltonian between the quantum dots. As is evident from Figure c, the change in tunnel coupling exhibits a steeper gradient in the case of the interchanged tuning strategy, suggesting that a wider dynamic range of the coupling strengths is accessible.

Subsequently, we attempted to experimentally determine how the enhanced tunability of tunnel coupling, as revealed by simulation, translates into the tunability of exchange coupling. The spin qubits were manipulated at the symmetric point in the charge stability diagram, where the first derivative of the exchange coupling (J) with respect to detuning (Δ) vanishes (i.e., ∂J/∂Δ = 0). In this and all subsequent experiments, we employed virtual gates to achieve independent control of each quantum dot’s energy, allowing stable operation at the symmetric point. Figure a shows that high-quality single-qubit control and addressing can be demonstrated under the interchanged tuning strategy. During the experiments, two pairs of qubits were initialized to odd-parity by adiabatically changing the detuning to move the single electron from the doubly occupied singlet state to the other quantum dot (see Supporting Information S2). Variation of the burst duration (t _MW) of the microwave signal applied to the screening gate enabled the high-quality oscillations in the blockade probability to be observed. Although the Rabi oscillation quality of Q₄′ was limited, this was attributed to strong coupling with a nearby reservoir.

3.

Single-qubit controllability and exchange coupling spectroscopy. (a) Rabi oscillations of four qubits and pulse sequence in the interchanged tuning method as a function of the microwave duration t _MW. (b) Pulse sequence for exchange spectroscopy. (c and d) Exchange spectroscopy with conventional and interchanged tuning. vB _i (vB _i ′) denotes the ith virtual barrier gate voltage, while f _MW corresponds to the microwave frequencies applied to the target qubit. X (X ²) denotes a π/2 (π) rotation about the x axis. Q_C (Q_C′) and Q_T (Q_T′) denote the control and target qubits, respectively, for each experiment.

Exchange spectroscopy was used to quantify the achievable exchange coupling strength. The pulse sequence is illustrated in Figure b. First, the control qubit was rotated by π/2 about the x axis. Then, a virtual barrier pulse vB _i (vB _i ′) was applied to the gates between the qubits to increase their exchange coupling, and the microwave frequency (f _MW) was swept to determine the dependence of the resonance frequency on the exchange coupling. Parts c and d of Figure show the trend in exchange coupling, defined as the frequency difference between the two branches in a given exchange spectroscopy map, for the conventional and interchanged gate configurations, respectively. The global slope of the branches with respect to the barrier pulse amplitude arises from the change in location of the quantum dot under the spatial gradient of B _z generated by the micromagnet. For conventional tuning, the exchange coupling increased to only about 10 MHz with a 0.4 V barrier pulse. In contrast, the interchanged strategy could significantly enhance the coupling; for example, J ₁₂′ reached about 125 MHz at a 0.28 V barrier pulse for Q₁′–Q₂′. Even at the maximum achievable exchange coupling of 125 MHz, we did not observe unintended transition lines indicative of excessive crosstalk. Although the degree of tunability varied among different qubit pairs, the tunability was clearly enhanced overall.

The tunability of the exchange coupling strength was quantified more precisely by measuring the decoupled controlled-Z (dCZ) oscillation. Evaluation of the tunability makes it necessary to examine a coupling strength smaller than the resonance linewidth. Considering that the initial condition without the barrier pulse may already have a relatively large J, the time-domain dCZ oscillations were used to verify the coupling behavior. In the pulse sequence in Figure a (left), intervening π pulses (X ²) applied to both qubits cancel out the unwanted single-qubit phase accumulation, thus allowing us to observe the phase oscillations by J. The phase oscillation data shown in Figure a as a function of t _evol and ΔvB ₃ ′ can be fitted to a Gaussian decaying sinusoidal function to extract the frequency, which corresponds to J/2. The decay time of the evolution (T ₂ ^dCZ) and the corresponding quality factor can also be extracted. The quality factor, defined as $J{T_2}^{\mathrm{dCZ}}$ , was evaluated for each qubit pair at various barrier pulse amplitudes. The corresponding values are provided in Table . Although the decay time decreases with increasing barrier pulse amplitude, the accompanying increase in the coupling strength results in an improved quality factor. A comparison with the conventional tuning strategy over the same range of J shows that the interchanged tuning strategy yields comparable quality factors and even higher values at some coupling strengths (see Supporting Information S5). This indicates that the interchanged tuning strategy does not introduce additional charge noise that would degrade the quality factor.

4.

Enhanced tunability of exchange coupling. (a) dCZ oscillations of the qubits by applying a varying pulse amplitude to the barrier gates. X or X ² gates are performed on the Q_C′ (control qubit) or Q_T′ (target qubit) in the interchanged tuning strategy. A pulse with amplitude ΔvB _i ′ is applied to turn on the exchange coupling. For the echo sequence, the pulse is split into two equal halves of duration t _evol/2. (b) Exchange coupling as a function of the barrier pulse amplitude. The blue (orange) trace shows data from an interchanged (conventional) gate configuration. The extracted tunabilities are 16.6, 11.4, and 15.4 dec/V (7.25, 3.87, and 4.32 dec/V) for the interchanged (conventional) gate configuration, obtained by fitting to J = Ae ^Bv . The error bars represent the 1σ uncertainties obtained from the fit.

1. Quality Factors under the Interchanged Tuning Strategy as a Function of the Barrier Pulse Amplitude.

ΔvB i ′ (V)	Q1′–Q2′	Q2′–Q3′	Q3′–Q4′
0.08	8.69	3.01	4.27
0.12	16.1	6.38	9.04
0.16	22.3	9.47	12.6

Figure b compares the exchange coupling tunability between the two tuning methods on the basis of the measured results. With the interchanged tuning strategy, the dependence of the coupling strength on the barrier pulse amplitude is noticeably stronger. The data were fit to an exponential form by focusing on the few megahertz range typically used in experiments. The fitted tunability values are provided in Table . Using the interchanged tuning method, the tunability increases by several orders of magnitude for all-nearest-neighbor pairs, significantly exceeding that in the conventional case. Interestingly, the trend of exchange coupling does not follow a single exponential form over the entire data range. A previous study showed that tunnel coupling asymptotically approaches a constant as the two quantum dots merge into a single larger quantum dot because the potential barrier is lowered by the barrier pulse. This result is itself a consequence of the enhanced tunability of coupling. This high tunability allows us to explore the entire coupling regime and, henceforth, is expected to help to accurately calibrate the actual profile of exchange coupling as a function of the barrier-pulse amplitude. This tuning strategy still allows us to construct virtual gates and suppress crosstalk, thereby enabling reliable multi-qubit control (see Supporting Information S4 and S5). These results indicate that the enhanced tunability can be maintained without introducing crosstalk that is sufficiently large to hinder qubit operation.

2. Fitted Tunabilities (dec/V) for the Qubit Pairs under Both Tuning Strategies.

conventional		interchanged
Q1–Q2	7.25	Q1′–Q2′	16.6
Q2–Q3	3.87	Q2′–Q3′	11.4
Q3–Q4	4.32	Q3′–Q4′	15.4

The most plausible explanation for the increased tunability is a difference in the lever arm. Indeed, when the lever arm ratios from the charge stability diagrams (see Supporting Information S2) for each tuning strategy are compared, the ratios are 1.75, 3.18, and 2.71, whereas the corresponding ratios for coupling controllability are 1.71, 2.72, and 2.10, respectively, for the case in which J ranges from 1 to 10 MHz. These results show that tunability is strongly affected by the lever arm and are consistent with the simulation results presented in Supporting Information S5. The simulation further shows that the lever arm is significantly affected by the bottom width of the gates, which can differ from the intended value because of fabrication-induced modifications (see Supporting Information S5).

Residual exchange coupling induces unwanted ZZ operation, which complicates the resulting Hamiltonian. When a microwave signal is used to manipulate one of the qubits in a two-qubit system with residual coupling, the resulting time-evolution operator includes not only the single-qubit operation term but also contains an additional term proportional to $\frac{J}{\mathrm{\Omega }}ZZ$ . Here, Ω denotes the Rabi frequency. This residual coupling limits the achievable single-qubit fidelity. Mitigation of this effect requires the coupling to be much smaller than the Rabi frequency (J ≪ Ω) as a fundamental requirement to achieve high single-qubit gate fidelity. However, to realize both single- and two-qubit operations with high fidelity, the coupling must be dynamically controlled on a short time scale, which is typically challenging to achieve with insufficient tunability. Even though a calibration routine could compensate for the undesired phase accumulation, this becomes increasingly challenging as the scale of the qubit system grows. Therefore, achieving exchange interaction such that it has sufficient tunability is crucial and requires not only careful design of the gate geometry but also an optimal gate configuration for the experiment. From this perspective, the suggested tuning strategy provides a practical and efficient means to enhance the tunability of the exchange coupling simply by changing the voltage configuration within the same qubit device, without requiring any modification to the experimental setup.

In conclusion, we investigated the effectiveness of an interchanged quantum-dot tuning method in which the roles of the nanogates defining the quantum dots are interchanged. Using the proposed strategy, we demonstrated the formation of quantum dots and a single-qubit operation. Notably, the tunability of exchange coupling is enhanced by several orders of magnitude across all nearest-neighbor qubit pairs. The result is significant because it demonstrates the realization of tunability of approximately 16 dec/V in Si/SiGe devices, which has hitherto only been achieved in Si-MOS devices with a larger lever arm. Conventionally, the spin qubits in Si/SiGe devices are located far from the electrically noisy semiconductor–dielectric interface; however, the inherent distance between the qubits and metal electrodes limits the tunability of the coupling. By contrast, the proposed interchanged tuning strategy allows us to effectively exploit the advantages of the Si/SiGe platform, particularly for devices with limited interaction tunability without degrading the quality of exchange coupling. The enhanced tunability originates mainly from the increased lever arm, consistent with our simulation results. Beyond this relationship, we find that the lever arm strongly depends on the effective bottom width of the gates and that fabrication-induced changes to the gate geometry can significantly alter the relative influence of the two gate layers on the quantum dots from that intended in the original design. This suggests that not only the gate layout in the top view but also factors such as the intervening oxide thickness and gate pitch, which modify the gate geometry, should be carefully considered.

In addition, these results show that quantum-dot formation is not restricted to a single tuning strategy within a given device architecture, allowing the tuning method to be selected according to the experimental requirements. This flexibility allows the device properties to be studied more intensively by varying the location of the quantum dot used as a probe, for example, the magnetic gradient field to optimize micromagnet design, valley splitting as a function of position, and the spatial characteristics of noise originating from impurities and sensor dots. The prior works , have demonstrated parameter probing using conveyor-mode shuttling. In contrast, our strategy offers an alternative based on a static quantum-dot configuration, compatible with long-duration protocols such as Carr–Purcell–Meiboom–Gill (CPMG) sequences. Although our strategy employs a multilayer gate structure in this work, its underlying principle arises from differences in the capacitive coupling between the gates and quantum dots. Such differences can also be present in advanced single-layer architectures, , suggesting that the strategy demonstrated here is not limited to multilayer gate structures. In addition, Si-MOS-based qubit devices realized by using advanced fabrication techniques often employ overlapping gate structures. This implies that our study is broadly applicable to quantum-dot spin qubits across different material platforms because key quantities related to gate controllability, such as the lever arm and exchange coupling tunability, are ultimately governed by the engineering of capacitive coupling between gates and quantum dots regardless of the host material.

Further studies of fabrication strategies, including thinner gate oxides and advanced gate geometries, would contribute to bolstering the advantages of this strategy. This approach may provide a way to achieve sufficient coupling tunability with modest experimental effort, thereby facilitating convenient multi-qubit control, for example, by optimizing the PSB readout, increasing the efficiency of the resonant SWAP gate operation, and accurately engineering the Hamiltonian for Si/SiGe quantum-dot spin qubit processors defined by the gate overlay.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.nanolett.6c00044.

• Experimental setup, qubit initialization and measurement methods, simulation of the electric potential and tunnel coupling, virtual gate configuration, comparison of the exchange coupling quality, and simulation of the metal gate geometry on the lever arm and exchange coupling tunability (PDF)

The authors declare no competing financial interest.