High-fidelity sub-microsecond single-shot electron spin readout above 3.5 K

To realise these results, we manufacture a double multi-donor quantum dot device using scanning tunnelling microscopy hydrogen resist lithography²⁵ in a natural silicon substrate as shown in Fig. 1a. In contrast to previously reported devices^22,25, the quantum dots in this LSR device are precision engineered such that they are asymmetrically tunnel coupled to the SET. We achieve this by increasing the relative geometric distance between one of the dots and the SET reservoir as can be seen in Fig. 1b, where the top dot (t) is around 10 nm further away from the SET compared to the bottom dot (b), strongly suppressing electron tunnelling to the top dot. The device is mounted on a custom printed circuit board (PCB) with the source contact connected to a RC bias-tee and rf-tank circuit. The rf-tank circuit consists of a L₀ = 1.2 μH inductor and C_p = 0.4 pF parasitic capacitance (giving a resonance frequency of f_r ~ 225 MHz) chosen to impedance match the SET with the 50 Ω transmission line to ensure maximum signal contrast²³. This circuit probes the reflected input RF signal due to an admittance change of the rf-SET. The data from the device shown in this paper corresponds to a single rf-quadrature only, since the resonance frequency operating point is tuned such that the signal is maximised in one quadrature (Q) and minimised in the other (I). Figure 1c shows a 2D map of the reflected rf-signal as the left and right electrostatic gate voltages of the device are varied. The bright lines, running at 45^∘, correspond to where electrons tunnel through the SET causing a mostly dissipative response of the tank circuit (probing mostly rf-SET conductance)²³. The distinct breaks in the rf-signal (dashed white lines), correspond to charge transitions of electrons between the two quantum dots and the electron reservoir. The charge states (n_b, n_t) corresponding to the number of electrons on the bottom (n_b) and top (n_t) quantum dots in each region are also shown in the figure.

There are two types of breaks or discontinuities in the stability diagram in Fig. 1c. In addition to the distinct shifts in SET response due to a change in total electron number, there are also small shifts or discontinuities at the (1,1)  ↔ (2,0) region, highlighted by the green box, corresponding to the movement of an electron between the two quantum dots. These shifts are typically not observable in devices with symmetric dot-SET coupling²³. Since the total number of electrons is conserved across this transition, the capacitive shift of the reflected signal from the SET charge sensor due to this electron movement is small. Here the movement of electrons between the quantum dots shifts the conductance peak by less than a full peak width (from blue to red) as illustrated in Fig. 1d, resulting in a readout contrast (difference in SET signal) that is  <1. This is the operating regime for standard Pauli-spin blockade (PSB) readout²⁸. To maximise the readout contrast, we can leverage the strong-response SET regime by performing LSR in the (2,1) charge region instead, where the total electron number of the system changes by 1. The addition and removal of a whole electron results in a much larger capacitive shift of the conductance signal during readout by a full peak width, as illustrated in Fig. 1e (blue box). In this case, there is maximum readout contrast of 1.

The 3-level pulse sequence for LSR is shown in Fig. 2a. First, the system is randomly initialised into one of the four, two-electron spin states (\(\left\vert \uparrow \uparrow \right\rangle\), \(\left\vert \uparrow \downarrow \right\rangle\), \(\left\vert \downarrow \uparrow \right\rangle\), \(\left\vert \downarrow \downarrow \right\rangle\)) by pulsing from the (2,1) charge state to the (1,1) charge region at point A where the two electrons are weakly interacting. Next, the system is pulsed to point B in the (2,0) charge region where now due to the presence of the exchange interaction between the two donor dots, the two spin basis states form the singlet (S: \(\left\vert \uparrow \downarrow \right\rangle -\left\vert \downarrow \uparrow \right\rangle\)) and triplet (T₀: \(\left\vert \uparrow \downarrow \right\rangle+\left\vert \downarrow \uparrow \right\rangle\), T₊: \(\left\vert \uparrow \uparrow \right\rangle\), T₋: \(\left\vert \downarrow \downarrow \right\rangle\)) states. The pulse from A  → B passes through the S − T₋ anti-crossing, hence the finite rise time (~1 ns) of the arbitrary waveform generator used to generate the pulse results in a small adiabatic loss (~0.07%), as calculated by the Landau–Zener formula. The time spent at point B is short, such that the even parity states do not have enough time to relax to the (2,0) charge ground state, but long enough that the singlet-triplet (T₀) states mix together due to the strong hyperfine coupling present in donor devices²⁹. Hence at point B the even parity states (\(\left\vert \uparrow \uparrow \right\rangle\), \(\left\vert \downarrow \downarrow \right\rangle\)) remain in the (1,1) charge configuration due to Pauli spin blockade. Meanwhile the odd parity spin states form the singlet-triplet (T₀) states (\(\left\vert \uparrow \downarrow \pm \downarrow \uparrow \right\rangle\)) in the (2,0) region where both electrons are on dot b. Pauli spin blockade is lifted for the odd states, allowing these states to occupy the (2,0) charge configuration. Lastly, latched readout is performed by pulsing to point C in the (2,1) charge region and waiting for a settling time of 250 ns before integrating the SET signal. This settling time leads results in ~0.47% of the odd states relaxing to the even states in this period of time. Here, as shown by the orange section of the energy level diagram in Fig. 2b, an even parity state maps directly to the (2,1) charge configuration while an odd parity state is initially latched to the (2,0) charge state, due to the slow tunnel rate between the charge states (2,0)  ↔ (2,1) that has been engineered by moving the dot further away from the SET sensor, as illustrated in Fig. 1b. By biasing the readout position to the top of a conductance peak in the (2,1) region, we observe in Fig. 2c that the SET charge sensor trace produced by the even state (blue) remains high throughout the measurement period. While the odd state (red) starts with a low trace that becomes high as it relaxes to the ground state (dashed arrow in Fig. 2b).

The parity-based readout can also be used for readout of an individual spin qubit (top quantum dot) via coupling to the ancilla (bottom quantum dot loaded deterministically with a down electron). The high signal contrast observed arises due to the addition of a whole electron with associated large capacitive shift of the charge sensor, allowing for high-fidelity readout¹². While we focus on characterising the fidelity to discriminate between parity states, in principle, we could extend the analysis to include other aspects required for the fidelity of the individual qubit readout relevant for quantum error correction. However, this would involve determining additional experimental parameters such as electron spin resonance efficiency and initialisation fidelity for each electron spin which we could not measure in this device without the use of isotopically pure Si-28 and the addition of an on-chip microwave antenna for magnetic drive.

Latched spin readout offers two main advantages over energy-selective measurements (ESM)^30,31: high-temperature operation and non-stochastic tunnel events. Energy-selective readout relies on the spin qubit energy being much larger than the temperature of the system which typically limits the operating temperature to be ≪1 K³² for qubit energies of ~20–40 GHz^6,33. The lower thermal constraints on LSR also mean more power can be applied to the readout sensor, further increasing the readout signal. Additionally, ESM inherently involves stochastic tunnelling processes such that the sensor signal cannot be filtered using standard integration filters³², thus ultimately lowering the maximum achievable readout fidelity. On the other hand, for LSR, the SET signal begins in the desired measurable state, either even or odd (see Fig. 2c), hence the measurement window can be applied straight away, reducing the integration time needed. Finally, since LSR directly measures the two-spin state we can use it to characterise the tunnel coupling, t_c between quantum dots via a spin-funnel measurement by plotting the detuning position of the S–T minus anti-crossing³⁴. For this device we find t_c = 3 GHz at 0.02 K with an estimated gradient magnetic field of ~50 MHz shown in Supplementary I²⁵.

Ultimately, the fidelity of LSR strongly depends on the tunnelling transport characteristics of the SET and how it varies with temperature. In Fig. 3a we show the SET signal as a function of gate voltage while the bath temperature of the dilution refrigerator was varied from 0.1 K to 3.7 K, while keeping the rf signal constant. The SET signal peak height does not vary considerably even up to 3.7 K. Here, the tank circuit is being driven at high rf-power, already effectively power-broadening the SET response. This power-broadening only weakly affects the LSR fidelity (due to the strong capacitive coupling of the quantum dot to the SET which shifts the signal by a full peak width) and hence we can operate in the power-broadened regime for all temperatures²⁶.

For readout in the strong response regime²³, we are most interested in the change in peak height since we operate at the maximum signal point and not the maximum sensitivity point. To further illustrate this we plot the measured SET signal, \({V}_{contrast}=\max ({V}_{Q})-\min ({V}_{Q})\) in Fig. 3b. The maximum SET signal of 0.7 mV is maintained up to~1 K, and then drops to only half of its maximum value (0.35 mV) as the temperature is increased to 3.7 K. As the readout speed is critical for large-scale quantum processors, we also performed temporal measurements of the rf-SET tank circuit transient response at 0.02 and 3.7 K by applying 500 ns square pulses to one of the electrostatic gates between lifted and full Coulomb blockade as shown in Fig. 3c, d. By fitting exponential transients of the average rf-SET response, we find the characteristic (1/e) rise–fall constants to be τ_0.02K = 48–56 ns and τ_3.7K = 35–39 ns corresponding to quality factors (Q = 2πf_rτ) of Q = 34–39 at 0.02 K and Q = 25–27 at 3.7 K. These Q-factors are comparable to previously reported values in phosphorus-doped silicon²³ allowing for high measurement bandwidths. The signal contrast and speed can be improved with further improvements in the readout circuit setup. The noise in the system is currently limited by the cold preamplifier²³, such that the signal contrast can be further increased by using lower noise parametric cold amplifier^35,36. The speed can be increased by optimising the resonant circuit to have a higher resonance frequency (\({f}_{{{{\rm{res}}}}}\)) while maintaining the same quality factor (Q) as the bandwidth is \({f}_{{{{\rm{res}}}}}\)/Q.

Now that we have established the performance of the SET as a function of temperature, we demonstrate LSR at the mixing chamber base temperature of 0.02 K and at 3.7 K to investigate the low-temperature and high-temperature single-shot spin readout fidelity. In Fig. 4a we show the measured parity readout error (1 − fidelity) as a function of integration time (using a boxcar filter) of the SET signal at 0.02 and 3.7 K by fitting to the equations described in Barthel et al.³⁷, see Supplementary II. The model takes into account the unlatching of the (2,0) odd state to the (2,1) charge state due to an electron tunnelling from the reservoir to the top dot (t) during the integration period. This can be seen as the relaxation of the odd state with a T₁ time of 53 ± 1 μs at 0.02 K and 53 ± 2 μs at 3.7 K, extracted from the average odd parity RF signal shown in Supplementary III. Since it is a tunnelling process, it is not dominated by the temperature of the system in the experiments and therefore remains consistent as the temperature is increased. For short integration times, the fidelity is limited by the signal-to-noise ratio (SNR) of the SET in which the cold (HEMT) preamplifier noise dominates the signal²³. For longer integration times the tunnelling induced relaxation rate of the (2,0) odd parity states into the (2,1) charge state begins to dominate the fidelity and leads to an exponential increase in readout error seen in Fig. 4a. The optimal integration time is, therefore, a trade-off between these two effects: SNR and relaxation. The fidelity is significantly better at 0.02 K than at 3.7 K for short integration times due to the lower SET signal at higher temperatures. For integration times longer than 1 μs, the fidelities at both temperatures begin to approach each other. This is due to the fact that in this regime, relaxation errors dominate, and the measured relaxation time of the singlet at 3.7 K (~53 μs) is the same as the relaxation time at 0.02 K. Therefore, the main difference in the readout fidelities at the two temperatures is simply due to the SNR of the SET.

The SNR of the SET is plotted as a function of integration time at 0.02 and 3.7 K in Fig. 4b. We can see that the SNR values of the SET is significantly larger at 0.02 K compared to 3.7 K for any given integration time as expected from the decrease in the size of the SET signal peak measured in Fig. 3b. From extrapolating the SNR values vs integration time for both temperatures, we can estimate the measurement time τ_m of the sensor, defined as the integration time required to achieve a SNR of 2³⁸. Separate to the integration time, the measurement time is a metric that accounts for the noise and bandwidth of the charge sensor, giving an indication of the quality, and should be as short as possible for high-fidelity readout. Here, we measure τ_m = 8 ± 1.5 ns (0.02 K) and τ_m = 120 ± 12 ns (3.7 K). At elevated temperatures of 3.7 K, the phosphorus-doped silicon SET still acts as a high contrast, fast charge sensor.

The maximum fidelity at 0.02 K was calculated to be 99.44  ± 0.05% with an integration time of 175 ns with the signal histogram (dark blue) shown in Fig. 4c. As we wait 250 ns settling time, the total measurement time is 425 ns and we have accounted for spin relaxation in the fidelity over this total period. We load 6000 random states (at point A in Fig. 2a) with single-shot readout traces taken, resulting in two equal-sized probability peaks corresponding to both the odd and even states. We can deterministically load an odd state by waiting longer at point B in Fig. 2a where an even state will relax to the lower energy odd state. This results in a single peak, as illustrated by the light blue signal histogram overlaid. The latched parity readout error at 0.02 K (0.5%) is half the 1% fault-tolerant threshold and 3 orders of magnitude faster than the longest reported dephasing time (270 μs⁶) for an electron spin qubit using phosphorus-doped silicon. These readout metrics are comparable to the fastest recorded superconducting qubit readout times^39,40 and faster than the highest reported times for spin qubits (99% at 1.6 μs⁴¹) while demonstrating one of the highest fidelities to date. In Fig. 4d we show the fidelity as a function of the signal threshold which is defined as the level at which any signal above is classified as an even state and any signal below is an odd state. High-fidelity readout was maintained while decreasing the integration times further, with 99.15% achieved at the minimum measured integration time of 50 ns.

At 3.7 K, the maximum readout fidelity obtained was 97.87 ± 0.05% in 1.5 μs. Figure 4e shows the optimal measured signal histogram for latched readout (red) where we can see the asymmetry in the peaks due to the relaxation of the singlet state and the histogram for deterministically loaded odd states (orange) where only one peak is visible, as expected. In Fig. 4f, we plot the individual state fidelities as a function of the threshold voltage used to distinguish the two histograms. Compared to 0.02 K we find a reduction of only less than 2% fidelity when operating the device at 3.7 K. This fidelity is currently limited by the SNR of the charge sensor which can be increased further by better amplification of the SET signal and additional optimisation of the SET tunnel junctions to increase conductance, which if implemented in the future, can help push high temperature latched readout fidelity above the fault tolerant threshold of 99%. The current readout fidelity obtained at 3.7 K also implies that temperatures above 1 K (where the use of a dilution unit is not required) should be readily achievable for qubit readout above 99% allowing for high-temperature qubit operation.