Rapid single-shot parity spin readout in a silicon double quantum dot with fidelity exceeding 99 %

Silicon-based spin qubits offer a potential pathway toward realizing a scalable quantum computer owing to their compatibility with semiconductor manufacturing technologies. Recent experiments in this system have demonstrated crucial technologies, including high-fidelity quantum gates and multiqubit operation. However, the realization of a fault-tolerant quantum computer requires a high-fidelity spin measurement faster than decoherence. To address this challenge, we characterize and optimize the initialization and measurement procedures using the parity-mode Pauli spin blockade technique. Here, we demonstrate a rapid (with a duration of a few us) and accurate (with>99% fidelity) parity spin measurement in a silicon double quantum dot. These results represent a significant step forward toward implementing measurement-based quantum error correction in silicon.

The readout of a single-spin state typically relies on spin-to-charge conversion techniques, such as energy-selective tunneling [13] or Pauli spin blockade (PSB) [14,15].The energy-selective readout requires high bandwidth measurement with a long duration to detect stochastic tunneling events with various time scales.While this scheme has been employed for demonstrations of high-fidelity singlespin readout in the literature, the readout is typically orders of magnitude slower than phase coherence times [9,16].On the other hand, the PSB readout utilizes spin-selective tunneling between two quantum dots.In this case, the signal results from a stationary difference of charge states, and charge discrimination is much easier.Here, we demonstrate that it is possible to perform a single-spin readout with a fidelity well above 99 % with the standard PSB mechanism.The spin readout can be performed within 3 us and is considerably faster than the average spin echo coherence times of around 100 us.
We show coherent Rabi oscillations with visibility approaching 99.6 % (or a state preparation and measurement (SPAM) fidelity of 99.8 %), potentially compatible with measurement-based quantum error detection and correction protocols.
Figure 1a shows a scanning electron microscope image of our device.The overlapping aluminum gates [17] are fabricated on top of an isotopically enriched silicon/silicon-germanium wafer (heterostructure B in [18]).The sample is cooled down in a dilution refrigerator and an external magnetic field of 0.5 T is applied.In this work, we form two quantum dot spin qubits QL and QR under gates P3 and P4, respectively, while the left part of the device serves as an extended reservoir for QL.
Gate B3 is used to control the tunnel coupling between the left reservoir and QL, and gate B4 is used to control the inter-dot tunnel coupling  c .We monitor the occupancy of the double quantum dot by a nearby charge sensor quantum dot (indicated by the upper circle in Fig. 1a).The conductance of the charge sensor is measured by a radio-frequency reflectometry technique [19,20] with a bandwidth of approximately 10 MHz, which is limited by the quality factor of the tank circuit.In what follows, we use virtual gate voltages (vB3, vP3, vB4, vP4) to individually control quantum dot potential and tunnel coupling [21].The single-qubit manipulation is realized by an electric-dipole spin resonance (EDSR) technique using a micromagnet [22].The qubit manipulation is performed at the charge symmetry point, where the qubit energy is first-order insensitive to detuning charge noise [23,24].The residual exchange coupling is measured to be 21 kHz (see Supplementary Fig. 1).
We use the four-electron charge states (4,0) and (3,1) for the PSB readout, where ( L , R ) denotes the number of electrons in the left and right quantum dots.For the (4,0) configuration, the first excited state is energetically split by the orbital splitting rather than the valley splitting [11].This results in a larger readout window since the orbital excitation energy (400 eV for the four-electron state) is larger than the valley splitting (100-150 eV in this device).In Fig. 1b, we measure a charge stability diagram as a function of vP3 and vP4.We rapidly ramp vP4 from the (3,1) configuration towards the (4,0) configuration, causing the (3,1) triplet state to latch in the PSB regime.We choose an integration time of 10 s per point to ensure that relaxation of the triplet state is not significant.The PSB regime appears as an extended trapezoidal region showing (3,1) charge signal within the (4,0) configuration (see inside the white rectangle in Fig. 1b).
We begin with characterizing and optimizing the initialization protocol.Our initialization procedure starts by waiting near the (3,0)-(4,0) boundary where energy-selective tunneling to the (4,0) singlet state S(4,0) occurs.The waiting time is set to 5 s, and we pulse vB3 to increase the initialization rate.
The singlet-triplet excitation energy being much larger than the thermal broadening (~300 eV vs. ~4 eV) results in negligible population of higher excited states.We subsequently apply a voltage ramp to separate S(4,0) to |↑↓⟩ by a rapid adiabatic passage as illustrated by the dashed black arrow in Fig. 2a.For spin qubits with micromagnet, the primary concern in this procedure is the adiabatic transition through the S − T_ anti-crossing.The size of this anti-crossing is determined by the magnetic field gradient and  c , while only the latter is well controllable through gate voltages.To mitigate the adiabatic transition, we use a ramp procedure shown in Fig. 2b.Here the detuning is defined as vP4−vP3 using virtual gate voltages.We first pulse the system to the (3,1) regime close to the interdot boundary and increase  c by opening vB4.The concept is to make  c sufficiently small around the PSB regime and the S − T − anti-crossing, and large enough at the inter-dot transition, while ensuring a simple pulse shape for routine device calibration.The inter-dot coupling is then decreased during the ramp toward point O to adiabatically decrease the exchange coupling (see Supplementary Fig. 2).
After the qubit manipulation at O, we can perform a projective measurement by a reverse pulse sequence to M and charge discrimination.To optimize vB4 for the ramp pulse, we measure the triplet return probability as a function of vB4 at point R in Fig 2b .Here, too large vB4 results in adiabatic transition to T − , while too small vB4 results in diabatic inter-dot transition causing the (3,1) charge state latching signal.For the best parameter used in the experiment, we obtain roughly 0.2% triplet probability after pulsing to the (3,1) state (Fig. 2c). Figure 2d shows a comparison of the readout outcomes for the states right after S(4,0) initialization and after the ramp.We see that the S(4,0) initialization error is negligible, and the ramp to the (3,1) state causes an increased error counts.
In the PSB regime, there exist three possible spin-blocked states: T − , T + , and |↓↑⟩.Among these three states, |↓↑⟩ quickly mixes with the unblocked |↑↓⟩ state via the micro-magnet field gradient ( 1 ~100 ns).As a result, in practice, only the spin-polarized triplet states T − and T + are blocked, and the readout is the so-called parity readout.In Fig. 3a, b, we show measurement of the T1 relaxation times of triplet states T − and T + .These two triplet states are prepared by applying a  rotation to |↑↓⟩.We obtain  1 = 18.2 ± 0.2 (3.268 ± 0.001) ms for T − (T + ) by fitting the data to exponential decay.These long T1 values are achieved by reducing  c during the readout (see Supplementary Fig. 3).While the field-gradient components causing the spin relaxation are similar in magnitude for these two cases, there are several possible mechanisms that cause this difference.For example, the respective detuning parameters for the relevant spin-flipping anti-crossings (S(4,0) − T ± ) are different for these two states, and the associated energy differences being compensated by phonons, charge noise, and/or Johnson-Nyquist noise also differ [25,26].In addition, for the case of T + , sequential relaxation via the unpolarized spin states may occur.Pinpointing the exact mechanisms limiting these  1 values, including additional experiments on parametric dependence and related theory, is a subject for future study.Nonetheless, these are both long enough for high-fidelity parity spin measurement.
Next, we move on to characterizing the signal-to-noise ratio (SNR), which is another important metric for spin readout.Figure 3c shows the results of charge discrimination after preparing an equal superposition of |↑↓⟩ and T − .Two peaks in each histogram correspond to singlet and triplet readout outcomes.If the integration time  int is too short, the overlap of two Gaussian peaks causes a substantial infidelity (upper panel in Fig. 3c).However, if  int is too long, the relaxation of excited (3,1) state reduces the readout fidelity.The optimal  int to minimize readout infidelity is obtained by balancing these two factors.Figure 3d plots the readout infidelity taking into account both  1 and SNR.For both T − and T + , we obtain measurement infidelities below 0.1% with  int = 2 s.For experiments requiring measurement-based feedback, it might be important to reduce  int further to minimize the dephasing of idling qubits.For reduced  int = 0.5 s, the infidelities are higher but remain below 1 %.
Finally, we assess the SPAM fidelity by measuring coherent Rabi oscillations.Throughout these experiments, we use  int = 3 s.In Fig. 4a, we measure two resonance peaks for QL and QR, with a difference in resonance frequencies corresponding to a Zeeman energy difference of Δ z = 85.6 MHz.The Rabi frequency is chosen so that the amplitude of the off-resonant drive of the idle qubit is negligibly small.We measure Rabi oscillations by varying the microwave pulse time  p (Fig. 4b, c).
We find that the Rabi oscillation decay is negligible, and the oscillation visibilities are well above 99% for both qubits ( 99.6 ± 0.2 % for T-and 99.4 ± 0.2 % for T+).For the T − readout, the 0.2% initialization error due to the adiabatic S − T − transition limits the visibility.For the T + readout, the  1 ~3 ms accounts for ~ 0.1% of the additional visibility loss as compared with the case for T − .We speculate that the additional 0.1% of infidelity comes from a larger adiabatic transition probability through the S − T + anti-crossing where vB4 is larger than the S − T − anti-crossing, as seen in the pulse shape of Fig. 2b.A more complex pulse to reduce  c when crossing both degeneracy points could circumvent this problem.
For implementing measurement-based quantum protocols that require phase coherence, the total duration of readout is also crucial.More explicitly, measurement has to be faster than relevant  2 , not only  1 .Our spin qubits have  2 * ~ 8 s (see Supplementary Fig. 4), and these inhomogeneous phase coherence times can be extended to above 100 s using a single refocusing  pulse (Fig. 4b).On the other hand, the overall duration of our measurement procedure can be as short as 2.4 μs (0.13 μs ramp time + 0.25 μs reflectometry signal rise time + 2 μs integration time).When comparing this duration to the typical loss of coherence expected from the Hahn echo decay of QL and QR, we expect less than a 1 % loss of phase coherence during the measurement (0.1% (0.5%) for QL (QR)).While a practical way to combine the spin echo pulse and the spin readout scheme remains to be explored, our result suggests that, with an appropriate device and experimental configuration, it is possible to perform a single-spin qubit readout precisely and fast enough while maintaining phase information.
In conclusion, we have demonstrated high-fidelity parity PSB readout for electron spin qubits in a silicon double quantum dot.The readout fidelities compare favorably to the best values in other spin qubit platforms [16,27,28].The visibility of Rabi oscillation is mainly limited by spin state mapping, which could be improved by further optimization of pulse shape.The readout duration is much shorter than the spin echo dephasing time.We note that the implemented parity spin readout does not preserve the original eigenstates unlike the standard ZZ parity measurement, and we need a designated measurement ancilla quantum dot to readout a single-spin state.It may suggest that optimizing only the T − readout is sufficient for applications where we need single-spin projective measurement.
Overall, our results suggest that it is possible to perform a single-spin readout in a time scale where the phase coherence of idling qubits is well preserved, and it may open the door toward realizing measurement-based quantum error detection or correction protocols in silicon.

Figures:
Figure 1 (a) False-colored scanning electron microscope image of the device.The green, purple, and brown gates correspond to plunger/accumulation, barrier, and screening gates, respectively.The lower channel is used for defining the right and left quantum dots under the plunger gates P3 and P4 (lower panel).A micromagnet is fabricated on top of the fine gate stack.The right quantum dot in the upper channel is used as a charge sensor quantum dot.The right bath for the charge sensor is connected to a radio frequency tank circuit.The white circles indicate where quantum dots are formed.(b) Charge stability diagram around the (4,0) -(3,1) charge degeneracy point. rf is the reflectometry signal from the charge sensor.The vP4 voltage sweep is performed from the right to left so that the triplet (3,1) state is latched in the spin blockade regime.

Figure 2 .
Figure 2. Initialization procedure.(a) Energy diagram of the two-electron states around the inter-dot charge transition.The zero detuning point is defined as the S(4,0)-S(3,1) degeneracy point.The dashed line represents the ideal rapid adiabatic passage for initializing |↑↓⟩ .We use parameters Δ ST = 400 μeV, average Zeeman energy of 18 GHz, Zeeman energy difference of 100 MHz, and  c = 2 GHz for the illustration.(b) Schematic of initialization pulse trajectory in the vB4-detuning plane.The durations are 15 ns (ramp time from M to R), 15 ns (waiting time at R), and 100 ns (ramp time from R to O).The position of S − T ± crossings is illustrated for  c changing exponentially as a function of vB4 (solid gray lines).(c) Measurement of triplet return probability as a function of the virtual barrier gate voltage vB4 at R. (d) Histograms of the charge sensor signal for 10,000 shots of measurement before and after the ramp to the (3,1) charge state.The solid lines show a fitting to a Gaussian function.The (3,1) counts for the configuration right after initialization is too small to fit.

Figure 3
Figure 3 Relaxation, SNR, and readout fidelity.(a) Relaxation times.The blue (red) trace shows the measurement result for the initial state T-(T+).The data is obtained by normalizing the measured  rf so that  rf ( w = 0) = 1 and  rf ( w = ∞) =  rf (S(4,0)) = 0 .The S(4,0) signal is obtained by measuring  rf at M directly after the initialization.The solid black lines are the exponential fit curves  rf ( w ) =  0 exp(− w / 1 ), where  0 ~1 is the fitting parameter to account for the signal contrast.(b) Zoom up of the energy diagram plotted with relevant relaxation channels (left panel: S to T − , right panel: S to T + ).(c) Histograms for two different measurement integration times  int = 0.5 s and 2 s.The blue points show histogram counts for measured data.The solid black lines show fitting to a double Gaussian function.(d) Infidelities as a function of  int .The SNR-inferred infidelity is calculated from the overlap of two Gaussian peaks, and the  1 -inferred infidelity is defined as 1 − exp(− int / 1 ).

Figure 4 .
Figure 4. Spin qubit control and SPAM fidelity.(a) EDSR spectra of two spin qubits.The blue (red) trace shows the measured data of QL (QR).The resonance frequencies are 19.1506GHz for QL and 19.2362 GHz for QR.(b) Spin echo measurements.The evolution time refers to the total duration of waiting time between microwave pulses.The blue (red) points show the measured data of QL (QR).The black lines show the fit with exponential decay ( evol ) = exp(−( evol / 2 echo )  ) from which we extract  2 echo = 105 ± 1 (138 ± 8) μs ,  = 1.89 ± 0.07(1.3± 0.1) , and  = 0.972 ± 0.002 (0.969 ± 0.002) for QL (QR).The data points and fit curve for QR are offset by −0.25 for clarity.The non-ideal visibility is due to the imperfect calibration of microwave pulses.The dashed black line represents  evol = 2.4 s.(c) Rabi oscillation of QL.The blue points are measured data, and the black line is a sinusoidal fit.(d) Zoom up of a peak and dip in c. (e,f) measurements similar to c and d, but for QR.