Ising-coupled qubit-ancilla system

Our qubit and ancilla are electron spins confined in a double Si/SiGe quantum dot (Fig. 1a) with natural isotopic abundance11. Spin states can be discriminated and reinitialized within 30 μs relying on energy-selective spin-to-charge conversion10,12 and the reflectometry response from a neighboring charge sensor (see Methods for details). An on-chip 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 frequency-selective electric-dipole-spin 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 polarizations13,14.

Fig. 1: Qubit and ancilla system. a Schematic of a device. The qubit spin (blue) and the ancilla spin (red) are hosted in two singly-occupied dots in a silicon quantum well layer. A proximal single electron transistor serves as a charge sensor. Scale bar: 200 nm. b Control pulse. Two microwave tones (represented by different colors) are used to selectively rotate qubit and ancilla spins. A controlled-phase shift is induced by applying square pulses simultaneously to V T , V B as well as to V C . c Ancilla spin-up probability after an entangling gate pulse. Traces with and without a π pulse applied to the qubit spin are plotted with filled and open symbols, respectively. d Measured controlled-phase accumulation. The dotted line indicates t CZ = 0.53 μs used for a conditional rotation. Full size image

We correlate the ancilla and the qubit spins by a controlled-rotation gate (Fig. 1b). During a square gate-voltage pulse for a duration t CZ at a symmetric operation point, the ancilla spin acquires a qubit-state-dependent phase due to enhanced exchange coupling3,15. 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 qubit-state-independent phase induced by the square gate-voltage pulse (~20π per μs) and the microwave bursts (~0.16π)16,17. We extract the qubit-dependent 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 ancilla-based 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 controlled-rotation 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 i-th 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 single-shot 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.

Fig. 2: Repetitive readout. a Quantum circuit for repetitive measurements. b Spin-up probabilities of the i-th ancilla measurement (only i = 1, 5, 10, 20, and 30 are shown for brevity) and the final qubit readout (q) out of 1000 events. Note the oscillation visibility for q is influenced by the compromised sensor sensitivity. c Probabilities of the four joint outcomes for the first and second ancilla measurements. The triangle symbols represent the experimental data, and the solid lines are the fit results to the model which takes into account preparation and measurement imperfections. Full size image

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 i-th 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 infidelity18 by expressing the joint probability P(m i m 30 ) as

$$P\left( {m_im_{30}} \right) = \mathop {\sum }\limits_{s = \downarrow , \uparrow } p_{i - 1}^s{\mathrm{\Theta }}_{s,m_i}\left(\, {f_i^s} \right){\mathrm{\Theta }}_{s,m_{30}}\left( {g_i^s} \right).$$ (1)

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 oscillation11 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 i-independent 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 spin-down 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 satisfy7—i.e., non-demolition (F QND ), measurement (F M ), and preparation (F P ).

Fig. 3: Cumulative QND readout and fidelities. a Diagram for the cumulative readout protocol and fidelity definitions. We regard n consecutive ancilla measurements (with outcomes m 1 , m 2 … m n ) as a single QND readout (with estimators σ and ς). The projected input spin state s 0 (either ↓ or ↑) changes to the posterior state s n after the process. The ideal QND measurement would give s n, identical to s 0 (non-demolition); σ, identical to s 0 (measurement); and s n , identical to ς (preparation). F QND , F M , and F P quantify these properties. b \(F_{{\mathrm{QND}}}^ \downarrow\), \(F_{{\mathrm{QND}}}^ \uparrow\), and \(( {F_{{\mathrm{QND}}}^ \downarrow + F_{{\mathrm{QND}}}^ \uparrow } )/2\) after n repetitive measurements. The solid lines show the values expected from the extracted \(T_1^{ \downarrow ( \uparrow )}\). c Rabi oscillations of the qubit spin acquired from multiple ancilla measurements. Plotted with a dashed curve is the estimated true qubit spin-up probability, \(p_0^ \uparrow\), consistent with a Rabi oscillation at a 630 kHz frequency detuning. d \(F_{\mathrm{M}}^ \downarrow\), \(F_{\mathrm{M}}^ \uparrow\), and \(F_{\mathrm{M}}^{{\mathrm{avg}}} = ( {F_{\mathrm{M}}^ \downarrow + F_{\mathrm{M}}^ \uparrow } )/2\) as a function of n. State-dependent single-shot measurement fidelities \((f_i^ \downarrow > f_i^ \uparrow )\) produce even-odd effects of \(F_{\mathrm{M}}^{ \downarrow \left( \uparrow \right)}\), whereas the average \(F_{\mathrm{M}}^{{\mathrm{avg}}}\) increases monotonically. Note that \(F_{\mathrm{M}}^{{\mathrm{avg}}}\) can be related to the measurement visibility12 V through \(V = 2F_{\mathrm{M}}^{{\mathrm{avg}}} - 1\). e \(F_{\mathrm{P}}^ \downarrow\), \(F_{\mathrm{P}}^ \uparrow\), and \(( {F_{\mathrm{P}}^ \downarrow + F_{\mathrm{P}}^ \uparrow } )/2\) as a function of n. Full size image

We first assess the non-demolition fidelity \(F_{{\mathrm{QND}}}^{ \downarrow \left( \uparrow \right)}\), which addresses the requirement that the measured observable (spin-down 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 \(F_{{\mathrm{QND}}}^{ \downarrow ( \uparrow )} = P(s_n = s_0|s_0 = \downarrow ( \uparrow ))\). 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 \(F_{\mathrm{M}}^{ \downarrow \left( \uparrow \right)} = P(\sigma = s_0|s_0 = \downarrow ( \uparrow ))\), 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 , \(P\left( {{\boldsymbol{m}}_n|s_0 = \sigma } \right) > P\left( {{\boldsymbol{m}}_n|s_0 = \bar \sigma } \right)\) with \(\bar \sigma\) denoting the spin opposite to σ. We calculate these likelihoods using a Bayes model that assumes spin-flipping 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 code8.

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 input7 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 \(P\left( {{\boldsymbol{m}}_n|s_n = \downarrow } \right)\) and \(P\left( {{\boldsymbol{m}}_n|s_n = \uparrow } \right)\). 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 high-fidelity state preparation

It is worth noting that for n ≥ 2, these likelihoods \(P\left( {{\boldsymbol{m}}_n|s_n = \varsigma } \right)\) can signal events where we have higher confidence in the final spin state. To explore this potential of heralded high-fidelity 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 spin-down case, as expected from \(F_{{\mathrm{QND}}}^{ \downarrow \left( \uparrow \right)}\).