Abstract
While singleshot detection of silicon spin qubits is now a laboratory routine, the need for quantum error correction in a largescale quantum computing device demands a quantum nondemolition (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 nondemolition manner by probing another electron spin in a neighboring dot Isingcoupled to the qubit spin. The high nondemolition 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 highfidelity (>99.6%) groundstate preparation. Our QNDbased 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.
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Introduction
Single electron spins confined in silicon quantum dots hold great promise as a quantum computing architecture with demonstrations of long coherence times^{1}, highfidelity quantum logic gates^{2,3,4}, basic quantum algorithms^{5}, and device scalability^{6}. However, the ability to measure a qubit in a singleshot QND manner has been lacking, despite its pivotal role in quantum error correction and quantum information processing, as well as its centrality to quantum science^{7,8,9}. An ideal singleshot QND readout process would, in addition to yielding an eigenvalue of the observable with projection probability for the input state (measurement), leave the system in the projected input state (nondemolition), meaning that the measurement is repeatable and that a posterior state can be predicted based on the eigenvalue obtained (preparation)^{7}. These features contrast with conventional readout schemes of a silicon spin qubit, which inherently demolish the spin state by mapping it to a more readily detectable, charge degree of freedom^{1,2,3,4,5,6,10}. Such spintocharge conversion techniques are employed to facilitate to measure the small magnetic moment of a single electron spin within its relaxation time, which, although exceptionally long for a solidstate quantum system, is limited to the millisecond timescale. A QND readout requires a mechanism to exquisitely expose the system to external circuitry for readout while maintaining the coherence and integrity of the qubit. Synthesizing an ancilla system which can be repeatedly initialized, controlled conditionally on the qubit state and separately measured, all on the microsecond timescale, constitutes a major challenge for the QND readout of a silicon electron spin qubit.
In this work we demonstrate repeatable measurements of a silicon electron spin qubit. We use a neighboring electron spin as an ancilla, with which we can perform a QND qubit readout at a 60 μs repetition cycle through a conditional rotation and spinselective tunneling. The highly QND nature is evidenced by the strong correlation between successive ancilla measurement outcomes. We take advantage of the repeatability and construct a QND qubit readout from n consecutive ancilla measurements to improve the overall performance. For complete characterization as a QND readout process, we identify and evaluate three key metrics^{7}: the nondemolition fidelity (F_{QND} = 99% for n = 1); the measurement fidelity (F_{M} = 95% for n = 20); the preparation fidelity (F_{P} = 92% for n = 20). (The numbers are the average of the spindown and up cases.) The nondemolition and preparation fidelities (F_{QND} and F_{P}) which are dissimilar to those in the destructive readout illustrate the distinct properties of the QND readout. We further show that the repetitive readout scheme allows us to preselect the cases where the qubit state is prepared with fidelities >99.6%.
Results
Isingcoupled qubitancilla system
Our qubit and ancilla are electron spins confined in a double Si/SiGe quantum dot (Fig. 1a) with natural isotopic abundance^{11}. Spin states can be discriminated and reinitialized within 30 μs relying on energyselective spintocharge conversion^{10,12} and the reflectometry response from a neighboring charge sensor (see Methods for details). An onchip micromagnet magnetized in an external magnetic field B_{ext} = 0.51 T separates the resonance frequencies of the qubit and ancilla spins by 640 MHz (centered around ~16.3 GHz). This enables frequencyselective electricdipolespin resonance rotations of individual spins at several MHz and ensures that the exchange interaction of ~MHz is well represented by the Ising type with minimal disturbance to the spin polarizations^{13,14}.
We correlate the ancilla and the qubit spins by a controlledrotation gate (Fig. 1b). During a square gatevoltage pulse for a duration t_{CZ} at a symmetric operation point, the ancilla spin acquires a qubitstatedependent phase due to enhanced exchange coupling^{3,15}. A Hahn echo sequence converts this phase to the ancilla spin polarization, in a robust manner against a slow drift of the ancilla precession frequency and the qubitstateindependent phase induced by the square gatevoltage pulse (~20π per μs) and the microwave bursts (~0.16π)^{16,17}. We extract the qubitdependent phase shift by changing the prepared qubit state by the microwave burst time t_{b} (Fig. 1c). The extracted phase grows linearly with t_{CZ} (Fig. 1d), consistent with an induced excess exchange coupling J of 0.94 MHz. Choosing t_{CZ} = 0.53 μs (=1/2J) and an appropriate projection phase θ, we can implement a conditional rotation which maps the qubit state to the ancilla spin, allowing for the ancillabased measurement of the qubit spin.
Demonstration of repetitive readout
We now demonstrate that the ancilla can be repeatedly entangled with the qubit and measured, using a sequence shown in Fig. 2a. After preparing the qubit state by microwave control, we repeat 30 cycles of a controlledrotation gate and the ancilla measurement and reinitialization, until we destructively read out and reinitialize the qubit. We use m_{i} and q to denote the outcomes of the ith ancilla measurement (with i = 1, 2, … 30) and the final qubit readout, respectively. Remarkably, all ancilla measurement outcomes show clear Rabi oscillations (Fig. 2b), indicating each functions as a singleshot QND readout of the qubit. Strong correlations between successive measurements, a hallmark of the QND readout, are verified from joint probabilities P(m_{1}m_{2}), see Fig. 2c.
The Rabi oscillation visibility of m_{i} is affected by both the probability distribution \(p_{i  1}^{ \downarrow \left( \uparrow \right)}\) of the prepared qubit spin state s_{i−1} and the ith QND measurement fidelity \(f_i^{ \downarrow \left( \uparrow \right)}\) given \(s_{i  1} = \downarrow ( \uparrow )\). We separate the error in the prepared qubit spin state (during the process of initialization, rotation, and preceding ancilla measurements) from the measurement infidelity^{18} by expressing the joint probability P(m_{i}m_{30}) as
Here \(g_i^{ \downarrow \left( \uparrow \right)}\) denotes the measurement fidelity of m_{30} for s_{i−1} prepared in ↓(↑), and \(\Theta _{s,m}\left( f \right)\) equals f when s = m and 1 − f when s ≠ m. We model \(p_{i  1}^{ \downarrow ( \uparrow )}\) by an exponentially decaying Rabi oscillation^{11} and obtain \(p_i^{ \downarrow \left( \uparrow \right)}\), \(f_i^{ \downarrow \left( \uparrow \right)}\), and \(g_i^{ \downarrow \left( \uparrow \right)}\) as a function of i (see Methods and Supplementary Fig. 2). We find that \(f_i^{ \downarrow ( \uparrow )}\) is essentially iindependent as expected, with the average 85% (75%) for i = 1–20.
Characterization of the QND readout
A distinct feature of the QND readout is that it is repeatable, meaning we can potentially gain more accurate information about the qubit state from consecutive measurements. In the following, we leverage this potential by constructing a cumulative QND readout from n outcomes, m_{n} = {m_{1}, m_{2},… m_{n}} which yields estimators σ for s_{0} (the input qubit state, projected to either spindown or up) and ς for s_{n} (the posterior qubit state), see Fig. 3a. We characterize its performance as a QND readout as a function of n, through three key fidelity figures of merit, F_{QND}, F_{M}, and F_{P}. These fidelities are, as depicted in Fig. 3a, defined by the correspondences between the estimators (σ and ς) and/or the qubit states before and after the process (s_{0} and s_{n}). Importantly, these together will enable us to test all key criteria that the QND readout should satisfy^{7}—i.e., nondemolition (F_{QND}), measurement (F_{M}), and preparation (F_{P}).
We first assess the nondemolition fidelity \(F_{{\mathrm{QND}}}^{ \downarrow \left( \uparrow \right)}\), which addresses the requirement that the measured observable (spindown or up) should not be disturbed. It represents the correlation between the projected input (s_{0}) and posterior (s_{n}) qubit states, and unlike the other two fidelities, it is expected to decrease as n is increased. \(F_{{\mathrm{QND}}}^{ \downarrow \left( \uparrow \right)}\) can be defined using the conditional probability of s_{n} given s_{0} as \(F_{{\mathrm{QND}}}^{ \downarrow ( \uparrow )} = P(s_n = s_0s_0 = \downarrow ( \uparrow ))\). It follows from this definition that \(p_n^ \downarrow = F_{{\mathrm{QND}}}^ \downarrow\; p_0^ \downarrow + ( {1  F_{{\mathrm{QND}}}^ \uparrow } )p_0^ \uparrow\). The results obtained from the fit to this equation is shown in Fig. 3b, where \(F_{{\mathrm{QND}}}^{ \downarrow ( \uparrow )}\) gradually decreases to 99% (61%) as n is increased up to 20. By modeling the n dependence of \(p_n^ \downarrow\) (see Methods), we estimate \(F_{{\mathrm{QND}}}^{ \downarrow ( \uparrow )}\) for n = 1 to be 99.92% (97.7%), corresponding to the longitudinal spin relaxation time \(T_1^{ \downarrow ( \uparrow )}\) of 78 ms (2.5 ms) given the 60 μs cycle time.
The second requirement for the QND readout is that the measurement result should be correlated with the input state following the Born rule. We test this through the measurement fidelity defined as \(F_{\mathrm{M}}^{ \downarrow \left( \uparrow \right)} = P(\sigma = s_0s_0 = \downarrow ( \uparrow ))\), where σ is the estimator for the input qubit state s_{0} based on measurement results m_{n}. When σ is the more likely value of s_{0}, \(P\left( {{\boldsymbol{m}}_ns_0 = \sigma } \right) > P\left( {{\boldsymbol{m}}_ns_0 = \bar \sigma } \right)\) with \(\bar \sigma\) denoting the spin opposite to σ. We calculate these likelihoods using a Bayes model that assumes spinflipping events (see Methods). σ shows larger Rabi oscillations as n is increased (Fig. 3c), demonstrating \(F_{\mathrm{M}}^{ \downarrow \left( \uparrow \right)}\) enhancement by repeating ancilla measurements in our protocol. We obtain \(F_{\mathrm{M}}^{ \downarrow \left( \uparrow \right)}\) (Fig. 3d) through \(P\left( {\sigma = \downarrow } \right) = F_{\mathrm{M}}^ \downarrow p_0^ \downarrow + ( {1  F_{\mathrm{M}}^ \uparrow } ) p_0^ \uparrow\). While \(F_{\mathrm{M}}^{ \downarrow \left( \uparrow \right)} = 88\hbox{\%}\) (73%) for n = 1, it reaches 95.6% (94.6%) for n = 20, well above the measurement fidelity threshold for the surface code^{8}.
The last feature of the QND readout to be evaluated is the capability as a state preparation device. In order to quantify how precisely our cumulative QND readout process prepares a definite qubit state, we define the preparation fidelity F_{P} as the conditional probability of s_{n} = ς given the estimator ς for the posterior qubit state s_{n}, i.e., \(F_{\mathrm{P}}^{ \downarrow \left( \uparrow \right)} = P\left( {s_n = \varsigma \varsigma = \downarrow \left( \uparrow \right)} \right)\). We emulate the most relevant situation of a completely unknown input^{7} by using data with 0.08 μs < t_{b} < 1.3 μs, for which \(p_0^ \downarrow = 0.500\). To optimally determine ς from m_{n}, we again apply the Bayes’ rule (Methods) and compare the likelihoods \(P\left( {{\boldsymbol{m}}_ns_n = \downarrow } \right)\) and \(P\left( {{\boldsymbol{m}}_ns_n = \uparrow } \right)\). We estimate s_{n} from another estimator σ′ and convert the conditional probability \(P\left( {\sigma^{ \prime} = \varsigma \varsigma = \downarrow \left( \uparrow \right)} \right)\) to \(F_{\mathrm{P}}^{ \downarrow \left( \uparrow \right)}\) using the measurement fidelity of σ′ for s_{n} (Methods). We obtain \(F_{\mathrm{P}}^{ \downarrow \left( \uparrow \right)} = 76\hbox{\%}\, \left( {83\hbox{\%} } \right)\) for n = 1, which increments to 95.9% (88.6%) for n = 20 (Fig. 3e).
Heralded highfidelity state preparation
It is worth noting that for n ≥ 2, these likelihoods \(P\left( {{\boldsymbol{m}}_ns_n = \varsigma } \right)\) can signal events where we have higher confidence in the final spin state. To explore this potential of heralded highfidelity state preparation, we calculate the likelihood ratio \({\it{\Lambda}} ^\varsigma = P\left( {{\boldsymbol{m}}_{10}s_{10} = \varsigma } \right)/P\left( {{\boldsymbol{m}}_{10}s_{10} = \bar \varsigma } \right)\) (i.e., for n = 10) and select events with \({\it{\Lambda}} ^\varsigma\) above a certain threshold. The conditional probability \(P\left( {\sigma^{\prime} = \varsigma \varsigma = \downarrow \left( \uparrow \right)} \right)\) is then estimated following the procedure described above (but with more ancilla measurements, see Methods). Indeed, \(F_{\mathrm{P}}^ \downarrow\) increases from 94 to 99% at the median (for \({\it{\Lambda}} ^ \downarrow\) > 1), and \(F_{\mathrm{P}}^ \downarrow\) reaches 99.6% at the 76th percentile, see Fig. 4. The limiting value is higher for the spindown case, as expected from \(F_{{\mathrm{QND}}}^{ \downarrow \left( \uparrow \right)}\).
Discussion
In the present experiment, 30 ancilla measurements are feasible before we lose strong correlation between the input and the outcome (\(F_{{\mathrm{QND}}}^ \uparrow \;\lesssim\;\) 50%). This is limited by a relatively short electron spin lifetime, compared with single nuclear spins in silicon where 99.8% readout fidelity is achieved as a result of >99.98% nondemolition fidelity^{19,20}. We note that, while both F_{M} and F_{P} are successfully improved by the cumulative QND readout, the observed F_{QND} falls short of our earlier expectations^{18} and the overall QND performance is impacted by this. The ratio \(T_1^ \downarrow /T_1^ \uparrow = 31\) is deviated from the ideal thermal population ratio (=16) between the Zeeman sublevels at the electron temperature (~50 mK), and the measured \(T_1^ \uparrow\) is roughly 30 times shorter than nominal expectation for an idle spin away from the hotspot^{21}. Indeed, data imply that the qubit relaxation occurs predominantly during the ancilla readout process (see Supplementary Fig. 1). This effect is expected to be suppressed by further quenching the residual exchange coupling (~MHz), e.g., via an interdot gate electrode^{6} or by fast readout with an ancilla encoded in doubledot spin states^{22}. We anticipate that we will then improve F_{QND} and the QND readout in all aspects, as a higher F_{QND} should raise F_{M} and F_{P} that are achievable by repeating QND measurements.
F_{M} and F_{P} will also improve, particularly for small numbers of n, by decreasing singleshot QND measurement infidelities \(1  f_i^{ \downarrow ( \uparrow )}\), which are 15% (25%) on average for i = 1–20. We estimate the contribution of charge discrimination error to be 1% (7%) for the spindown (up) case (see Supplementary Fig. 3), which can be straightforwardly reduced by tuning the charge sensor sensitivity solely for the ancilla dot. The remaining singleshot infidelities (in converting the qubit spin state to an ancilla electron tunneling event) are more significant, 14% and 18% for the spindown and up cases, respectively. We believe that these arise from the qubitancilla conditional operation and the ancilla spintocharge conversion and initialization process, and can be addressed by optimizing the twoqubit gate operation and the spinselective tunneling process^{4,10}.
To conclude, we have demonstrated a QND readout of a single electron spin in silicon. The presented technique uses an electron spin in a neighboring dot as an ancilla, requiring no increased structural complexity to multipledot quantum information processing units. Central to the 99% nondemolition fidelity are a synthesized Ising type qubitancilla coupling and the rapid conditional ancilla rotation and measurement. The ancillabased QND readout is a crucial element in qubit error detection and correction protocols. More specifically, it should be naturally extensible to QND measurements of the parity of multiple qubits with proper choice of single and twoqubit gate operations, in contrast to the spin blockadebased singleshot readout of a single spin^{23}, which may also allow for repetitive readout^{24}. Combined with highfidelity single and twoqubit gates^{2,4}, the demonstrated results will pave the way toward faulttolerant quantuminformation processing in the silicon quantumdot platform.
Methods
Measurement setup
The device is a dualgated accumulationmode Si/SiGe quantum dot reported in ref. ^{11} and is measured in a dry dilution refrigerator (Oxford Instruments Triton 200). A Tektronix AWG5014 arbitrary waveform generator is used to generate threechannel gate pulses (applied to V_{C}, V_{T}, and V_{B}). To ensure the adiabaticity of the pulses, they are filtered through Bessel analog filters with a 3 dB cutoff frequency at 39 MHz. The AWG5014 triggers a Tabor WX2184 waveform generator which produces I/Q modulation waveforms for two Keysight 8267D microwave sources. We use singlesideband modulation at 20 MHz to suppress the effects from leakage and spurious modes. In order to maintain the device in the symmetric condition throughout a controlledrotation operation, the pulse heights for V_{C}, V_{T}, and V_{B} are chosen to be +30.0 mV, −23.1 and −21.0 mV, respectively^{15}. Each spin is read out using the energyselective tunneling to the adjacent reservoir. (The ancilla and qubit electrons tunnel in and out from different reservoirs.) The reflectometry signal (at 205 MHz) is demodulated to baseband, sampled by an AlazarTech digitizer ATS9440 at 10 MSPS, filtered at 1 MHz using a second order Butterworth digital filter and decimated at 2 MSPS for post processing. The lengths of individual traces are 45 μs for experiments in Fig. 1c and 30 μs for those in Fig. 2. Peaktopeak values (the difference between the maximum and the minimum readings in individual traces) are used to detect the tunneling events.
Bayesian models
We construct a cumulative QND readout from n consecutive outcomes (m_{n}) of ancilla measurements (Fig. 3) based on the performance of singleshot QND measurements (m_{i}) characterized using Eq. (1) as described in the main text. In order to analyze all joint probabilities in a consistent manner, the fitting is performed in the following steps. First, the joint probability P(m_{i}m_{30}) for each value of i is fit to Eq. (1) with \(p_{i  1}^ \downarrow \left( { = 1  p_{i  1}^ \uparrow } \right)\), \(f_i^{ \downarrow \left( \uparrow \right)}\), and \(g_i^{ \downarrow \left( \uparrow \right)}\) as fitting parameters, assuming that \(p_{i  1}^ \downarrow\) is an exponentially decaying Rabi oscillation^{11}, similarly to ref. ^{18}. We then model the idependence of \(p_i^ \downarrow\) as \(p_{i\, +\, 1}^ \downarrow = \rho ^ \downarrow p_i^ \downarrow + \left( {1  \rho ^ \uparrow } \right)p_i^ \uparrow\), where \(\rho ^{ \downarrow \left( \uparrow \right)} = {\mathrm{exp}}\left( {  60\,\upmu {\mathrm{s}}/T_1^{ \downarrow \left( \uparrow \right)}} \right)\) is the spin conservation probability and \(p_0^ \downarrow\) is parametrized using an exponentially decaying Rabi oscillation again. Finally, we fix \(p_{i  1}^ \downarrow\) to the values calculated from \(p_0^ \downarrow\) and \(\rho ^{ \downarrow ( \uparrow )}\) in the model above and fit P(m_{i}m_{30}) for each value of i with only \(f_i^{ \downarrow \left( \uparrow \right)}\) and \(g_i^{ \downarrow \left( \uparrow \right)}\) as fitting parameters. Values extracted in these initial and later analysis steps are compared in Supplementary Fig. 2.
When we regard a cumulative QND readout process as a measurement device, it should estimate from m_{n} the input spin state s_{0} to be either ↓ or ↑. Our goal is to precisely determine the estimator σ for s_{0} such that s_{0} is more likely σ given m_{n}, i.e., \(P\left( {s_0 = \sigma {\boldsymbol{m}}_n} \right) > P\left( {s_0 = \bar \sigma {\boldsymbol{m}}_n} \right)\), assuming no prior knowledge about the probability distribution of P(s_{0}), i.e., \(P\left( {s_0 = \downarrow } \right) = P\left( {s_0 = \uparrow } \right) = 1/2\). From the Bayes theorem, we see that for the more likely value of σ
meaning that the input state estimation comes down to comparing likelihoods \(P\left( {{\boldsymbol{m}}_ns_0 = \downarrow } \right)\) and \(P\left( {{\boldsymbol{m}}_ns_0 = \uparrow } \right)\). For optimal performance, we should consider all (2^{n−1}) possible spin trajectories \(\left\{ {s_1,s_2, \ldots s_{n  1}} \right\}\) following s_{0}, with the realization probabilities taken into account. Using the spin transition probability to calculate the realization probabilities, the likelihood P(m_{n}s_{0}) can be computed as
for n = 1 and for n > 1
When we instead view a cumulative QND readout process as a state preparation device, it prepares the posterior spin state s_{n} in a specific state, either ↓ or ↑, as determined from m_{n}. Our task is then to construct the estimator ς for s_{n} such that s_{n} is more likely ς given m_{n}, i.e., \(P\left( {s_n = \varsigma {\boldsymbol{m}}_n} \right) > P\left( {s_n = \bar \varsigma {\boldsymbol{m}}_n} \right)\), assuming no prior knowledge about the probability distribution of P(s_{n}), i.e., \(P\left( {s_n = \downarrow } \right) = P\left( {s_n = \uparrow } \right) = 1/2\). From the Bayes theorem, it follows for the more likely value of ς
This means that we should now compare the likelihoods P(m_{n}s_{n} = ↓) and P(m_{n}s_{n} = ↑). Again, for optimal performance, we need to sum up probabilities of all (2^{n}) possible spin trajectories \(\left\{ {s_0,s_1, \ldots s_{n  1}} \right\}\) ending with s_{n} = ς as
In the main text, these Bayesian models are used to determine σ and ς.
Conversion between conditional probabilities
As explained in the main text, \(F_{\mathrm{P}}^{ \downarrow \left( \uparrow \right)}\) is defined by the conditional probability \(P\left( {s_n = \varsigma \varsigma = \downarrow \left( \uparrow \right)} \right)\). Experimentally, s_{n} can only be estimated with a finite fidelity from the ancilla measurement results \({\boldsymbol{m}}_{j,k}^\prime = \left\{ {m_j,m_{j\, +\, 1}, \cdots ,m_{j\, +\, k  1}} \right\}\) with j ≥ n + 1. We denote such an estimator for s_{n} as σ′ and its measurement fidelity as \(F_{\mathrm{M}}^{\prime s_n} \equiv P(\sigma^{\prime} = s_ns_n).\) We can then convert \(P(\sigma^{\prime} = \varsigma \varsigma )\) and \(F_{\mathrm{M}}^{\prime s_n}\) to \(F_{\mathrm{P}}^\varsigma\), by noting that for ς = ↓ and ↑,
Larger j − n and k would make the result more robust against correlated measurement errors and statistical shot noise, but j + k cannot exceed 30 (in the current experiment). We employ (j, k) = (n + 4,5) for Fig. 3e so that we can measure up to n = 20, and (n + 4,15) for Fig. 4 in order to obtain statistically more reliable results for high percentiles.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank Microwave Research Group at Caltech for technical assistance. Part of this work was financially supported by CREST, JST (JPMJCR15N2, JPMJCR1675), the ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan), MEXT Quantum Leap Flagship Program (MEXT QLEAP) Grant Number JPMXS0118069228, JSPS KAKENHI Grants Nos. 26220710, 17K14078, 18H01819, and 19K14640, RIKEN Incentive Research Projects and The Murata Science Foundation.
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J.Y. acquired and analyzed the data and wrote the paper. K.T. and A.N. set up the measurement hardware with assistance from J.Y., T.N., and S.L. T.N. contributed to the data analysis. K.T. fabricated the device with help from J.K. and T.K. S.T. supervised the project.
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Yoneda, J., Takeda, K., Noiri, A. et al. Quantum nondemolition readout of an electron spin in silicon. Nat Commun 11, 1144 (2020). https://doi.org/10.1038/s41467020148188
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DOI: https://doi.org/10.1038/s41467020148188
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