GoKawiilNext we evaluate the controllability of the exchange interaction achieved through shuttling. To characterize the exchange interaction, we use a decoupled controlled-phase (DCPhase) sequence: after initialization, an R x (π/2) pulse is applied to Q2, followed by shuttling both qubits together to activate exchange coupling for a duration t/2 and shuttling the spins back to their starting position. R x (π) gates are then applied to both spins in their respective dots, followed by another exchange interaction period t/2. Finally, another R x (π/2) pulse is applied to Q2 before measuring both qubits using parity readout against their reference spins. Figure 2a presents the measured DCPhase oscillations versus conveyor cycles with a 65-mV pulsed offset applied to barrier gate B3. The oscillation frequency, which directly reflects the exchange amplitude J, increases smoothly as the potential minima are moved towards each other before plateauing and subsequently decreasing.
Fig. 2: Exchange and coherence of mobile spin qubits. The alternative text for this image may have been generated using AI. Full size image a, Two-dimensional map showing DCPhase oscillations as a function of wait time and conveyor cycles (left axis), with corresponding nominal potential displacement (right axis), with a 65-mV pulsed voltage offset on B3. See main text for further details. b, Exchange coupling J versus conveyor cycles (bottom axis), with corresponding nominal potential displacement (top axis), at different pulsed voltage offsets on V B3 and extracted from DCPhase oscillations as in panel a. The inset shows the peak exchange strength versus the voltage on B3 and an exponential fit. c, Simulated potential profiles showing the evolution of the two moving potential minima during shuttling, in which c represents the number of conveyor cycles applied. A cycle is defined by one full period of the primary frequency component f. d, Measured dephasing times \({T}_{2}^{\ast }\) for both qubits under different states of the other spin: Q2 (Q5 = |0⟩), Q2 (Q5 = |1⟩), Q5 (Q2 = |0⟩) and Q5 (Q2 = |1⟩), plotted against conveyor cycles (bottom axis), with corresponding nominal potential displacement (top axis). The right axis shows the exchange coupling J extracted from the measured difference in Ramsey frequency depending on the state of the other spin. A 9.5-mV pulsed offset is applied to B3, consistent with panel c. e, Simulated potential profiles at different conveyor cycles c for the elongated dot configuration. As the confinement potential evolves with increasing c, the elongated potential minima merge into a single even more elongated potential, supporting strong Coulomb interactions between the electrons, which favours the formation of a Wigner molecule. f, Exchange coupling and dephasing times \({T}_{2}^{\ast }\) for Q2 (Q5 = |0⟩), Q2 (Q5 = |1⟩), Q5 (Q2 = |0⟩) and Q5 (Q2 = |1⟩) in the merged configuration of panel e, showing exchange saturation at larger conveyor cycles. The exchange coupling is extracted from DCPhase oscillations in this condition.
Figure 2b shows the estimated exchange strength from DCPhase measurements performed at different B3 pulsed offsets (ranging from 65 to 115 mV). For each curve, J reaches a maximum and then slightly decreases. This is counterintuitive in the picture, in which two potential minima progressively move towards each other. It results from the fact that the central barrier gate B3 receives a lower conveyor amplitude (and a negative pulsed offset) compared with the other conveyor gates (Supplementary Information sections C and D). This keeps the minima from the two conveyor channels separated by a tunnel barrier while still allowing a controlled exchange coupling. Figure 2c illustrates this through simulated potential profiles at different conveyor cycles c. As seen in the inset to Fig. 2b, J increases exponentially with the pulsed offset on B3, consistent with the Fermi–Hubbard model describing tunnel-coupled quantum dots37. At the start of the conveyor (zero conveyor cycles), J is too small to measure. The exchange strength between the mobile qubits can be tuned up to 90 MHz through control of both the number of conveyor cycles c and the barrier voltage. Although higher exchange strengths are achievable, they were difficult to measure owing to their rapid decay under these conditions.
The two-qubit gate fidelity that can be achieved depends on the balance between J and \({T}_{2}^{\ast }\). Figure 2d presents the \({T}_{2}^{\ast }\) dephasing times of Q2 and Q5, along with the corresponding exchange strength J, as a function of the number of conveyor cycles. As c increases, leading to a stronger exchange interaction and faster gate operation, we observe a decrease in \({T}_{2}^{\ast }\) for both qubits. The difference in \({T}_{2}^{\ast }\) depending on the state (|0⟩ or |1⟩) of the other qubit can be attributed to the spatial magnetic field gradient in the device15. We note that \({T}_{2}^{\ast }\) during shuttling in the device typically exceeds the static \({T}_{2}^{\ast }\) measured at fixed positions along the channel9. This is because of motional averaging, as the qubit samples different local environments at a rate faster than the correlation time of both the nuclear field fluctuations and charge noise8,38. On the basis of this characterization, we identify an operating regime that balances a strong enough exchange coupling for fast gates while maintaining sufficient coherence times. This condition, corresponding to 0.9 conveyor cycles and an exchange coupling of 33 MHz, enables the high-fidelity two-qubit operations demonstrated in Fig. 3.
Fig. 3: Fidelity benchmarking of shuttling-based CZ gate. The alternative text for this image may have been generated using AI. Full size image a, Schematic illustration of the gate electrodes and simulated potential profiles with static dots (dark line) and the conveyor during the interaction phase (faint line). b, Time evolution of the exchange coupling strength J (left axis) and the nominal potential displacement expressed in conveyor cycles (right axis) throughout the CZ gate operation. The dotted lines indicate the loading and unloading phases between the initial positions of the conveyor’s potential minima and the static dots. The pulse shape is designed to achieve adiabatic control of the exchange interaction. c, Calibration measurements of the CZ gate showing parallel spin probability oscillations for Q2 (left) and Q5 (right) as a function of an applied virtual phase shift θ. The measurements are performed with the other qubit initialized in either the |0⟩ or the |1⟩ state, as indicated. d, Results of interleaved randomized benchmarking, showing the return probability versus the number of Clifford operations for both reference (red) and interleaved CZ gate (blue) sequences. The schematic illustrates the interleaved randomized benchmarking protocol, in which shuttling-based CZ gates (‘Shuttle CZ’) are interleaved between random Clifford operations (C 1 to C N ) for the interleaved measurements, whereas reference measurements are performed with only the red Clifford operations.
Before proceeding to quantifying the gate fidelity, we investigate the exchange strength and spin coherence in a configuration in which two elongated travelling potential minima are created by applying a single sinusoidal wave having a spatial period corresponding to eight gate electrodes (once again with phase offsets that increase symmetrically from the outer gates towards the centre; here B3 receives a pulse offset of −10 mV). When the two elongated potentials overlap substantially at the centre, the system transitions away from a double-dot regime in which the Fermi–Hubbard description is applicable. Numerical simulations (Fig. 2e) reveal how the potential profile transitions from a barrier-controlled double-well configuration into a single highly elongated potential.
Figure 2f shows that, in this regime, J initially exhibits the standard exponential increase expected for a double quantum dot (up to approximately 0.86 conveyor cycles), in which the potential barrier dictates the coupling. Beyond this point, as the two elongated potentials merge, J no longer increases exponentially but instead saturates. This saturation may originate from strong electron–electron interactions within the merged elongated dot32,33,34. Notably, once in this merged regime, J becomes relatively insensitive to barrier voltage variations, enabling an enhancement of \({T}_{2}^{\ast }\) while still maintaining substantial exchange coupling. Unfortunately, before the conveyor has reached this favourable condition, it passes through a region in which \({T}_{2}^{\ast }\) is very short, resulting in overall low two-qubit gate fidelities. In fact, when measuring Q2’s \({T}_{2}^{\ast }\) with Q5 initialized in |0⟩, or Q5’s \({T}_{2}^{\ast }\) with Q2 in |1⟩, we cannot even accurately estimate \({T}_{2}^{\ast }\) because coherence is lost rapidly before J reaches the saturation regime. In the region with very short \({T}_{2}^{\ast }\), the growing electric dipole increases sensitivity to charge noise and, also, the frequency shifts because both the magnetic field gradient and the exchange interaction add constructively, as voltage fluctuations or charge noise simultaneously affect both the Zeeman energy splitting and the exchange interaction in a correlated way15. Notably, in this configuration of interacting mobile spins, tuning the exchange strength does not require independent dynamic control of the central barrier (B3) potential. Instead, the barrier gate is part of the travelling-wave potential. Although a static barrier voltage can be used in practice to adjust the exchange coupling, comparable exchange values can, in principle, be obtained only by adjusting the shuttling distance.