Coherent spin qubit transport in silicon

A fault-tolerant quantum processor may be configured using stationary qubits interacting only with their nearest neighbours, but at the cost of significant overheads in physical qubits per logical qubit. Such overheads could be reduced by coherently transporting qubits across the chip, allowing connectivity beyond immediate neighbours. Here we demonstrate high-fidelity coherent transport of an electron spin qubit between quantum dots in isotopically-enriched silicon. We observe qubit precession in the inter-site tunnelling regime and assess the impact of qubit transport using Ramsey interferometry and quantum state tomography techniques. We report a polarization transfer fidelity of 99.97% and an average coherent transfer fidelity of 99.4%. Our results provide key elements for high-fidelity, on-chip quantum information distribution, as long envisaged, reinforcing the scaling prospects of silicon-based spin qubits.


Supplementary Note 1: Transport verification
Here we provide experimental evidence that the qubit is indeed transported between the two sites every time the detuning is ramped across the zero detuning point in the fidelity characterization experiments. We note that while it is possible to simulate the expected diabatic tunneling probability within the model using the extracted Hamiltonian parameters and the detuning ramp rate, the existence of noise and excited levels may introduce a subtlety to the dynamics in the experiment, in which case the wavefunction (or position) of the electron may not adiabatically follow the ground state of the electric potential controlled via gate voltages (on gates A and B with typically swept between +/-10 mV). Furthermore, the actual gatevoltage pulse shapes at the device end may well deviate from the voltage pulses generated in the room temperature circuit, as they are sent through filtered coaxial cables inside the dilution refrigerator (designed for a bandwidth of 80 MHz in our setup). We find it impracticable to estimate the distorted pulse shape in the presence of the transmission non-idealities, such as the standing wave modes, non-linear phase responses and nanosecond-scale inter-gate skews, inside the cryostat. Therefore, we design the following control experiment to verify that the qubit alternates between sites A and B for the same number of times as the number of applied transfer ramps, especially when we use the same pulse shape as in the fidelity characterization experiments.
The central idea of our control experiment is that the spin phase acquired under detuning pulses is very sensitive to the time the qubit spends in each site, due to the site-dependence of the qubit frequency (given by the slightly different g-factors, see the main text). We can precisely determine the spin precession rate for a given charge configuration and detuning from the tunneling spectroscopy result (see Fig. 2c and Supplenentary Fig. 2c). When is swept between +/-10 mV, the difference in the phase precession rate is 32.4 MHz if the qubit switches its site following the (orbital) ground state as expected, with the dominant contribution coming from Δ AB . In the meantime, if the qubit somehow remains in the same site, the precession rate would change by at most 1.2 MHz, since the Stark shift and the tunneling hybridization effect for these values of are much smaller. This means that during a 56 ns interval between the transfer ramps, the amount of spin phase accumulation will be around 3.6π in the case of adiabatic transfer as opposed to 0.13π in the absence of electron transfer.
In the actual experiment, we employ pulses with different odd numbers of ramps, n = 1, 3 and 5 (see Supplenentary Fig. 1a) and compare the qubit phases after the pulses. We henceforth denote the difference in the post-transfer spin phase between the pulses with the odd numbers of ramps, n = i and i+2, by +2, . The phase difference +2, will be approximately 3.6π when the electron does return to site A and spends roughly 56 ns longer time there (or, equivalently, 56 ns shorter time in site B) after the i-th ramp is completed. Conversely, if the electron fails to change its site, +2, will be much smaller (~0.13π).
We measure the fringes after the detuning ramps and quantify +2, from the difference in the fringe phase as a function of 1 , where 1 is the specified value of at the ramp starting point (see Supplenentary Fig. 1a). By changing 1 in sufficiently small steps, this protocol allows us to evaluate +2, larger than 2π experimentally (with no ambiguity given +2, ≈ 0 for 1 = +10 mV). It is important that the microwave phases of the two ESR pulses in this Ramsey-type sequence (applied at different sites and thus with different tones) be defined consistently for different numbers of ramps. The fringes obtained for various values of 1 are exemplified in Supplenentary Fig. 1b-f. Figure S1g plots the extracted values of 3,1 and 5,3 as a function of 1 . The two traces agree well, suggesting the reliability of the phase measurement protocol and the consistency in the trajectories as n is incremented. When 1 is large (e.g. point f), the phase difference is constantly small as the electron is not shuttled between sites and stays in site B. At around 1 = 2 mV (point e), the spin phase starts to pick up the tunneling hybridization effect around the anticrossing. Around 1 = -10 mV (point b), +2, reaches 3.6π, which can only be accounted for by a decrease in time spent in site B by ~56 ns upon increasing n by 2, indicating that following the i-th ramp, the qubit moves from site B to A (the (i+1)-th ramp) and, after dwelling for ~56 ns in site A, goes back to site B (the (i+2)-th ramp). We therefore conclude that for 1 = -10 mV, the same value that was used for the transfer fidelity measurements, the electron consistently moves between the sites for n times.
The data is even more compelling when we compare them with the calculation based on the qubit spectrum. The traces are already well explained by assuming that the actual qubit detuning is identical to the specified detuning pulse and that the dynamics is completely adiabatic (see the grey solid curve in Supplenentary Fig. 1g). For illustration purposes, we plot the results for two other transfer functions between the specified detuning pulse and the actual qubit detuning, which yield a slightly better alignment with the data, compared to the unfiltered case: a Butterworth filter (6 th -order 40 MHz lowpass) and a Chebyshev filter (type 1, 1 st -order bandstop with a 25-50 MHz stopband and a 1dB ripple level). The trajectories of the qubit detuning and frequency for 1 = -10 mV are also shown for individual cases in Supplenentary  Fig. 1h-m. In all cases, the simulations verify that the qubit is transported between sites, further reinforcing our conclusion.