Quantum non-demolition readout of an electron spin in silicon

While single-shot detection of silicon spin qubits is now a laboratory routine, the need for quantum error correction in a large-scale quantum computing device demands a quantum non-demolition (QND) implementation. Unlike conventional counterparts, the QND spin readout imposes minimal disturbance to the probed spin polarization and can therefore be repeated to extinguish measurement errors. Here, we show that an electron spin qubit in silicon can be measured in a highly non-demolition manner by probing another electron spin in a neighboring dot Ising-coupled to the qubit spin. The high non-demolition fidelity (99% on average) enables over 20 readout repetitions of a single spin state, yielding an overall average measurement fidelity of up to 95% within 1.2 ms. We further demonstrate that our repetitive QND readout protocol can realize heralded high-fidelity (>99.6%) ground-state preparation. Our QND-based measurement and preparation, mediated by a second qubit of the same kind, will allow for a wide class of quantum information protocols with electron spins in silicon without compromising the architectural homogeneity.


Supplementary Note 1: Qubit Relaxation
To investigate the mechanism of the observed qubit state relaxation, we perform several control experiments ( Supplementary Fig. 1a). In these experiments, we first control the qubit spin by a 10 s-long microwave frequency-chirped by 10 MHz. After applying different numbers of ancilla measurement pulses, we read out the qubit spin and record the maximum (on-resonance) and minimum (off-resonance) spin-up probabilities, ↑ max and ↑ min . Each ancilla measurement is 45 s long and repeated every 75 s (= cycle ). Their difference = ↑ max − ↑ min is expected to follow = ( ↓ + ↑ − 1) 0 , where 0 is a constant that explains the state preparation and measurement errors. From the fit, we obtain the total spin relaxation time 1 tot defined through exp(− cycle / 1 tot ) = ↓ + ↑ − 1.
In Supplementary Fig. 1b, we plot along with the fitting results in three different conditions. In all cases, the exchange pulses for a controlled rotation are deactivated. In cases ii and iii, we turn off the reflectometry carrier signals during the ancilla readout process. In case iii, we offset the gate-voltage pulse level at the ancilla measurement stage from the reservoir Fermi level by 5 mV (corresponding to ~ 0.1 meV). 1 tot shows a noticeable improvement from 1 tot = 2.4 ms in the QND readout condition (calculated from 1 ↓(↑) ) only in case iii, which indicates enhanced qubit relaxation during the ancilla readout process, presumably due to cotunneling. Further studies may be necessary to reveal the microscopic mechanisms of the phenomenon.

Supplementary Note 2: Joint Probability Analysis with the Sensor Signal Distribution
The qubit-state dependence of our QND readout performance ( M and P ) is partly explained by qubit-state dependent ( ↓ = 85% and ↑ = 75%), see Supplementary Fig. 2b.
A possible cause of this is ancilla-state dependent fidelity of the ancilla spin readout. The ancilla readout is a two-stage process, as is usually the case with single-shot destructive readout of single electron spins 10 . The first stage correlates the spin state to the presence and absence of tunneling events (spin-to-charge conversion). In the second stage, we detect a blip in the sensor signal as a result of such tunneling events, typically by setting a threshold (charge discrimination). Both stages are prone to errors which, in principle, may be modelled through device and setup parameters. While such detailed characterization is out of the scope of this study, one can anticipate that the charge discrimination infidelity overwhelms the relaxation and thermal broadening effects in our case, given the short measurement time (compared to the relaxation time) and the large Zeeman splitting (with respect to the thermal energy).
To evaluate the effect of charge discrimination errors, we calculate the joint probability P(mim30) using the histograms of sensor signals, u and d , for traces with and without blips, respectively (see Supplementary Fig. 3a and Supplementary Note 3). extracted in this manner, ̃ (rather than using thresholded values as in the main text) would be free from the charge discrimination errors and describe the errors in converting the qubit spin state to an ancilla electron tunnelling event. We obtain these qubit-to-ancilla-blip conversion fidelities as ̃↓ = 86% and ̃↑ = 82% (Supplementary Fig. 3c). Improvements from are consistent with the charge discrimination fidelities ( th d = 98.2% and th u = 90.0% at the threshold of 20.6 mV, see Supplementary Fig. 3b).
Note that these can be used (in place of ↓, ( ↓ ) and ↑, ( ↑ )) to calculate the likelihoods of initial as well as posterior spin states, however, the fidelity enhancement we observe is only marginal. It is illustrative to see how well approximated the ratio ↓ / ↑ is by the stepfunction as used in the main text (see Supplementary Fig. 3d), because of large qubit-to-ancillablip conversion infidelities 1 −̃ compared to charge discrimination errors 1 − th d (u) . We note that the threshold of 20.6 mV is in fact chosen such that ↓ = ↑ at the threshold.

Supplementary Note 3: Sensor Signal Distribution Estimation
This section describes our procedure to estimate the distribution functions u and d of the peak-to-peak charge sensor signal with and without detectable blips due to tunneling events, respectively. We denote the probability of finding below the threshold th by th d , and introduce the fidelity th u ( th d ) to describe the probability that is above ( respectively. c, ↓(↑) as a function of i. Crosses and dots are from the initial and last steps, respectively. Unlike ↓(↑) and ↓ , ↑ shows a monotonic increase as a function of i, consistent with the qubit state relaxation from ↑ to ↓ between the i-th and 30th ancilla measurements.