Supervised learning based on qubit-phase measurements

In the language of machine learning, we consider the qubit’s instantaneous phase which we would like to predict at a future discretized time, t k , as labels, φP(t k ), and an arbitrary number, n, of previous measurements, φ i M (indexed by i and obtained by any appropriate method), as their associated features. We then calculate a linear combination of the features with optimized weighting coefficients, w={w} i,k , as a prediction of the label, . Based on measured features, the entries of w are optimized for each time step, t k , reflecting the time-varying correlations in the dephasing process, captured through the power spectrum.

We demonstrate prediction of a qubit’s state subject to stochastic dephasing by performing experiments using the ground-state hyperfine states, |F=0, m F =0> and |F=1, m F =0>, in trapped 171Yb+ ions as a qubit with transition frequency near 12.6 GHz. A coherent superposition of the qubit states in the measurement basis induced by microwave control35 evolves freely under the influence of an engineered dephasing interaction larger than any intrinsic noise in our experimental system (Supplementary Methods). In general we work in a regime where the noise evolves slowly during a single measurement period T M , but we allow the rate at which measurements of qubit-phase evolution are taken—the sampling frequency ω s —to vary relative to the highest frequency in the noise power spectrum, ω c (c.f. Fig. 3f). The dephasing noise processes presented here are all derived from a flat-top frequency power spectrum with characteristic cut-off at ω c . More complex spectra are discussed in Supplementary Discussion and demonstrate similar performance.

Figure 3: Experimental comparison of long-term stabilization using traditional and predictive feedback. (a) Schematics showing the key aspects of our cyclic feedback implementation using overlapping measurements. (b–d) Demonstration of feedback accuracy for different sampling frequencies ω s quantified in units of ω c , presented through correlation plots (c.f. Fig. 1c) for traditional feedback (blue) and prediction (magenta). Data presented are derived from Fig. 1a. (e) Measured sample variance for various protocols as a function of the number of cycles. Data are normalized to the sample variance of the uncorrected (free-running) signal at 1,000 samples. Each line represents data taken for one particular noise realization and thick lines represent the ensemble average. The inset shows an example suppression of variance over measurement outcomes using predictive against traditional feedback (normalized to the noise amplitude). (f) Sample variance at N=1,000 as a function of sampling frequency ω s in units of ω c , normalized to the sample variance of the uncorrected signal. The measurement time is fixed and ω s varied through introduction of dead time between measurements. Dotted lines display simulations and markers the measurement results averaged over ten noise realizations. Error bars represent the s.d. of the mean and the shaded areas show the maximum spread of outcomes. For fixed noise parameters varying ω s serves as a proxy for changing the ratio of (Supplementary Discussion). Simulations and measurements in all panels use n=20. Full size image

An important aspect of our approach is that measurements providing data serving as features may be performed through any suitable protocol. For instance, performing a series of p projective measurements on a single qubit to obtain ensemble-averaged information simply sets the scale of the measurement period, , with the duration of a single experiment. Here, we employ a projective measurement that captures statistical information through a spatial ensemble. The impact of such differences is explicitly captured in the sampling frequency of the measurement process.

Forward prediction of stochastic qubit-phase evolution

We begin by accumulating a series of projective measurements of the qubit’s phase under engineered dephasing. These serve as training data for the algorithm to optimize the coefficients in w. We then perform another series of measurements (shown, Fig. 1a) under application of a different noise process possessing similar statistical characteristics as used in acquiring the training data. This approach ensures that our estimates of prediction accuracy are conservative and exhibit reasonable model robustness and generality. Performing the learning algorithm on a single data set can enhance performance of the prediction algorithm but introduces extreme sensitivity to the input model, ultimately reducing prediction efficacy in the presence of variations in the detailed form of the noise.

An example engineered noise trace in time with overlaid measurement outcomes, φM, is depicted in Fig. 1a, with 97% correlation between φM and the applied phase φA (Fig. 1b). Beyond time t 0 we predict future labels of qubit-phase evolution φP(t k ), up to step t 150 using a variable number, n, of past measurements and the trained coefficients in w. Calculated predictions approximate φA well, reproducing key features including inflection points, maxima and minima as a function of t k . Our knowledge of the noise is used exclusively for quantitative evaluation of prediction efficacy—it does not enter into the machine-learning algorithm in any form.

Prediction accuracy increases with n, as the algorithm learns more about the temporal correlations in φA. For values of k≳n, corresponding to prediction times exceeding the range over which the algorithm possesses knowledge about the noise features, the prediction quality diminishes. In addition, over very large values of t k the prediction tends towards the mean of the noise. Comparing predictive estimation to a ‘traditional feedback’ model, in which future estimates are based simply on the last measured value φM(t 0 ), the algorithm shows a distinct advantage as it allows for temporal evolution of the noise in the future.

The quantitative benefits of predictive estimation relative to traditional feedback, and the large t k behaviour of the predictive algorithm are succinctly captured in the root-mean-square (r.m.s.) prediction error averaged over the entire data set, , and calculated as a function of t k and n (Fig. 1c). This demonstrates that even over a large ensemble of predictions the algorithm’s advantages remain robust. We now move on to provide examples of real-time qubit stabilization in which the incorporation of future state prediction shows significant advantages over existing techniques.

Time-division multiplexed decoherence suppression

As described above, a reliance on feedback involving frequent projective measurements renders a qubit effectively useless for quantum information or other applications, but omission of stabilization techniques in the presence of dephasing noise may lead to phase errors and eventually to total decoherence. To mitigate the effect of dephasing, we tailor an approach in which we temporally multiplex the necessary measurement and actuation operations in distinct probe and stabilization periods respectively (Fig. 2a,b). During the probe period, a fixed number of measurements are taken and processed in real time. From these measurement outcomes the algorithm produces a prediction of the future time-dependent evolution of the noise during the subsequent stabilization period up to some t k ; the qubit is dedicated exclusively to measurement of the dephasing process in the probe period. During the stabilization period, corrections are applied during each discrete time step to compensate the predicted stochastic phase evolution, but no measurements are conducted; this permits periods of unsupervised evolution during which the qubit is useful and stabilised against dephasing.

As an example we set the objective of maintaining zero net qubit-phase accumulation (in the rotating frame) during each time step of the stabilization period such that arbitrary high-fidelity operations may be conducted on the qubit; here we apply only the identity. Diagnostic measurements are performed after a variable number of corrections to demonstrate the efficacy of this approach but would not ordinarily be required. Two representative S probe/stabilization cycles are displayed in Fig. 2b showing a reduction in integrated phase error of about 70% after a stabilization delay of t 50 during the first cycle and a reduction of about 85% during the second. These improvements are partially limited by measurement fidelity, as illustrated in the ensemble-averaged data (Fig. 2c). Predictive compensation in all tested regimes is superior to corrections based only on traditional feedback down to measurement fidelity limits. Compared against numerical simulations we see that for small t k the algorithm can provide large relative gains.

Predictive estimation inside a periodic feedback loop

In a second application we employ real-time predictive control in a metrological context. Qubits realised in atoms are frequently used as stable references against which local oscillators (LOs) may be disciplined36. However, stochastic evolution of the LO frequency between interrogations leads to imperfect corrections in the feedback loop. This scenario is commonly encountered when classical processing, actuation and system reinitialisation introduce dead time, producing an effective lag in the feedback loop which degrades the long-term stability of the locked oscillator37. The impact of rapid fluctuations in the LO frequency relative to dead time is generally referred to as the Dick effect38, and represents a significant limiting phenomenon in passive frequency standards using atomic references. The correspondence between LO-induced instabilities in frequency references and dephasing in qubits39 thus invites the application of predictive control in a setting where periodic interrogation and projective measurement are native to the feedback loops used in precision frequency metrology.

The usefulness of predictive estimation in improving correction accuracy inside a feedback loop is demonstrated in Fig. 3b–d, where we plot the predicted phase φP(t k ) (based on two different techniques) against the applied phase error φA(t k ). A prediction with unity correlation to the applied noise would form a diagonal line along φP=φA (similar to Fig. 1b), while imperfect predictions—hence imperfect corrections—result in a dispersion of points around this line in an ellipse.

We vary the sampling frequencies ω s as a proxy for introducing a variable dead time in the feedback loop (Supplementary Discussion). In a regime where the LO-induced dephasing process evolves slowly, quantified as ω s ≫ω c , both φM(t 0 ) and the predicted phase φP(t k ) show positive correlation to φA(t k ) (Fig. 3b). As we decrease ω s , noise evolution during the dead time leads to diminishing correlation between the prediction and actual noise, causing the ellipses to rotate and broaden—a manifestation of the Dick effect.

Predictive estimates are compared with the traditional feedback model described above. For ω s approaching the Nyquist limit we observe that the traditional prediction can become anticorrelated with the rapidly evolving applied noise (blue ellipse, Fig. 3d), which in real-world applications would lead to an unstable system under feedback. By contrast, using optimized predictions, the decrease in correlation is much slower and the machine-learning algorithm prevents the prediction from ever becoming anticorrelated with the applied dephasing noise. In circumstances tested we always find the optimal prediction correlation r P >r T for traditional feedback. Corrections used to discipline the qubit or LO based on predictive estimation can therefore possess enhanced average accuracy relative to traditional feedback.

We now implement real-time evaluation of φP(t k ) inside a feedback loop, demonstrating the ability to improve the individual corrections and ultimately achieve improved long-term stability of the locked qubit. In our experiment we set n=20, calculate φP(t k ) on the fly, and cyclically correct based on these predictions (Fig. 3a), again comparing against traditional feedback. The long-term stability achieved under both methods is calculated via the sample variance40 over a variable number of feedback cycles (Fig. 3e).

Over the range of dead times explored experimentally, the use of optimized predictive feedback, in which future estimates are updated as new measurements are acquired in real time, yields net enhancements over the free-running LO (Fig. 3e,f). This includes regimes near the Nyquist limit where rapid evolution of the noise can result in feedback-induced instability as in Fig. 3d. Over most of this range and for the noise parameters we have employed, performance gains over traditional feedback are ∼2 × using optimized predictive feedback—a metrologically significant improvement using only enhanced software in the stabilization. Similar performance enhancements have been observed for a wide range of noise spectra and parameters (Supplementary Discussion).

Predictive estimation applied to intrinsic system noise

Finally, with quantitative evaluation of these techniques in hand using engineered noise, we move on to a study of the intrinsic dephasing noise in our system, which arises due to a combination of LO phase noise and magnetic field fluctuations. We perform thousands of sequential projective measurements on the free-running qubit–LO system and process predictions offline. The spectrum of measured fluctuations combines a 1/f2 type low-frequency tail with an approximately white plateau, resulting in significant spectral weight near the measurement cycle time. We perform an analysis similar to that presented in Fig. 1, with prediction accuracy quantified using the r.m.s. error between predictions and the future measurement outcomes as a function of t k (Fig. 4a).

Figure 4: Application of predictive qubit state estimation to intrinsic system noise. (a) r.m.s. errors between predictions, φP and actual values φ(A) for various numbers of past measurements and discrete steps forward in time, averaged over the whole set of validation data. The r.m.s. values are normalized to the r.m.s.d. of the uncorrected data from zero. The bottom row (1*) corresponds to traditional feedback. (b) Sample variance of the corrected measurements averaged over 5,000 cycles, as a function of past measurements used for prediction, normalized to the sample variance of the uncorrected system. The expected sample variance obtained by performing traditional feedback is added for comparison. Data are split into two subsets, where the first 70% serve for training purposes and the remaining 30% are used for validation. (Inset) Power spectrum of a series of projective measurements on the free-running qubit–LO system. The data is overlaid with a smoothed version to visualize the general trend. The maximum frequency in the spectrum corresponds to our sampling frequency and is about 1.7 Hz. Full size image

Our machine-learning algorithm enhances the prediction of future qubit evolution by ∼30% relative to the r.m.s. error of the uncorrected measurements. We achieve similar performance gains relative to both traditional feedback and the free-running system in calculated sample variance over thousands of correction cycles based on predicted qubit phase, Fig. 4b. In this case the rapid evolution of the noise causes traditional feedback to produce a larger sample variance than free evolution—a situation similar to that experienced in Fig. 3d. The calculated performance enhancements of our method on the intrinsic system noise are significant and show that our algorithm possesses the capability to improve the stability against the noise background in our system.