Machine Learning for Predictive Estimation of Qubit Dynamics Subject to Dephasing

Decoherence remains a major challenge in quantum computing hardware, and a variety of physical-layer controls provide opportunities to mitigate the impact of this phenomenon through feedback and feed-forward control. In this work, we compare a variety of machine-learning algorithms derived from diverse fields for the task of state estimation (retrodiction) and forward prediction of future qubit-state evolution for a single qubit subject to classical, non-Markovian dephasing. Our approaches involve the construction of a dynamical model capturing qubit dynamics via autoregressive or Fourier-type protocols using only a historical record of projective measurements. A detailed comparison of achievable prediction horizons, model robustness, and measurement-noise-filtering capabilities for Kalman filters (KFs) and Gaussian process regression (GPR) algorithms is provided. We demonstrate superior performance from the autoregressive KF relative to Fourier-based KF approaches and focus on the role of filter optimization in achieving suitable performance. Finally, we examine several realizations of GPR using different kernels and discover that these approaches are generally not suitable for forward prediction. We highlight the linkages between predictive performance and kernel structure, and we identify ways in which forward predictions are susceptible to numerical artifacts.

Figure 1

(a) A Ramsey experiment at $t=n\mathrm{\Delta }t$ with fixed wait time $\tau$ and time steps $n$, spaced $\mathrm{\Delta }t>\tau$ apart. A $\pi /2$ pulse rotates the qubit state to a superposition of $|d⟩$ states, $d\in \{0,1\}$; the qubit evolves via ${\widehat{\mathcal{H}}}_N(t)$, accumulating relative stochastic $f_n$ for nonzero environmental dephasing $\delta \omega (t)$. Jittering arrows depict the potential qubit-state vectors permitted for an (unknown) random $f_n$. The qubit state is measured as $d_n=d$ in the ${\widehat{\sigma }}_z$ basis after a second $\pi /2$ rotation. (b) Black dots depict $\{d_n\}$ against time steps $n$; data collection stops at $n=0$, separating past state estimation from future prediction (the blue region). The black solid line shows the true qubit-state likelihood $\propto h(f_n)$, and the red solid line shows the state estimate (prediction) for $n<0$ ($n>0$). A prediction horizon is for all $n<n^{*}\in [0,N_P]$, for which the dark-gray region between the red and black lines is minimized (the Bayes prediction risk) relative to predicting the mean of dephasing noise; algorithmic tuning occurs by minimizing the light-gray region (the Bayes state estimation risk). $\mathcal{Q}$ quantizes the black line into noisy qubit measurements, $d_n$, under the Gaussian uncertainty $v_n$. (c) Single-shot outcomes in (b) are preprocessed to yield noisy measurements $\{y_n\}$ (black dots); $y_n$ is linear in $f_n$, and $v_n$ represents additive white Gaussian measurement noise. Msmts. denotes measurements.

Figure 2

Comparison of the algorithmic structure between KF and GPR by superposing the lower panels of Fig. 1 with KF and GPR predictive frameworks. (a) KF: Purple distribution represents a prior, with mean $x_n$ and covariance $P_n$, propagated in time steps $n$, using Kalman dynamics ${\mathrm{\Phi }}_n$, and updated within each $n$ by the Kalman gain ${\gamma }_n$ to yield a posterior distribution (red) at $n$. The posterior at $n$ is the prior at $n+1$. The mean of a posterior distribution at each $n$ is used to derive predictions given by the red line using $h(x_n)$. In the blue region, the red posterior predictive distribution is propagated using ${\mathrm{\Phi }}_n$, but ${\gamma }_n\equiv 0$. Gaussian white Kalman “process” noise, $w_n$, is colored by ${\mathrm{\Phi }}_n$ to yield the dynamics for $x_n$. (b) Purple prior distribution defined over sequences $f$, with mean ${\mu }_f$, and variance ${\mathrm{\Sigma }}_f$ is constrained by the entire measurement record. The resulting posterior predictive distribution (red) is evaluated at test points in time, $n^{‡}\in [-N_T,N_P]$; state estimates (predictions) are for the mean, ${\mu }_{f^{‡}}$ at $n^{‡}<0$ ($n^{‡}>0$). A choice of kernel defines each element in ${\mathrm{\Sigma }}_f$, ${\mathrm{\Sigma }}_{f^{†}}$. In both (a) and (b), the purple shadow represents posterior state variance (the diagonal $P_n$ or ${\mathrm{\Sigma }}_{f^{‡}}$ elements) constrained by data and filtered measurement noise $v_n$.

Figure 3

Approaches to construction of the KF dynamical model. Figure 2 is superimposed with Kalman dynamical models, $\mathrm{\Phi }\equiv {\mathrm{\Phi }}_n$, $\forall n$. (a) AKF or QKF. A set of autoregressive coefficients, $\{{\varphi }_{q^{\prime }\le q}\}$, define $\mathrm{\Phi }$ to yield $f_n$ as a weight sum of $q$ past measurements. (b) LKFFB. Red arrows with heights $\parallel x_n^j\parallel$ depict a set of basis oscillators for $j=1,\dots ,J^{(B)}$, probe the true purple spectrum of $f_n$, and yield time-domain dynamics of $f_n$ as a stacked system of resonators, ${\mathrm{\Theta }}_j$. Black $\mathsf{L}$-shaped arrows depict a single instance of $f_n$ at $n=0$ based on historical $\{f_{n-1},f_{n-2},\cdots \}$ values.

Figure 4

(a) Solid dots depict $y_n$ against time steps $n$, and data collection ceases at $n=0$. Optimized LSF, AKF, and LKFFB yield predictions $n>0$ in the blue region plotted as open, colored markers. A black solid line shows one realization of a true $f_n$, drawn from a flattop spectrum with $J$ true Fourier components spaced ${\omega }_0$ apart and uniformly randomized phases. Other parameters are ${\omega }_0/{\omega }_0^{(B)}\notin \mathcal{Z}$ (natural numbers); $J=45000$; ${\omega }_0/2\pi =\frac{8}{9}\times {10}^{-3}\mathrm{Hz}$, such that $>500$ true components fall between adjacent LKFFB oscillators; and $\mathrm{N}\mathrm{L}=10\%$. (b)–(d) The procedure in (a) is repeated for ensemble $M$ different realizations of $f$ and noisy data sets to compute ${\tilde{L}}_{\mathrm{BR}}$ for a LSF, an AKF, and a LKFFB. ${\tilde{L}}_{\mathrm{BR}}$ vs $n\in [0,N_P]$ is plotted; the dark-gray horizontal line marks ${\tilde{L}}_{\mathrm{BR}}\equiv 1$ for predicting the mean ${\mu }_f\equiv 0$ value. Vertical dashed lines mark the forward prediction horizon, $n^{*}$, where ${\tilde{L}}_{\mathrm{BR}}\lesssim 0.8<1$ for all prediction time steps $0<n\le n^{*}$ in outperforming the prediction of the noise mean. Marker color (dark indigo to pink) depicts the true $f$ cutoff, $J{\omega }_0$, varied relative to ${\omega }^{(B)}\equiv {\omega }_0^{(B)}{J}^{(B)}\approx r{\omega }_{(S)}$, with a fixed Nyquist $r\gg 2$, ${\omega }_0/2\pi =0.497\mathrm{Hz}$, $J=20$, 40, 60, 80, and 200; $\mathrm{N}\mathrm{L}=1\%$. In (a)–(d), a trained LKFFB is implemented with ${\omega }_0^{(B)}/2\pi =0.5\mathrm{Hz}$ and $J^{(B)}=100$ oscillators; trained AKF and LSF models are $q=100$, with $N_T=2000$, $N_P=50$ steps, $\mathrm{\Delta }t=0.001\mathrm{s}$, $M=50$ runs, and $K=75$ optimization trials.

Figure 5

Measurement noise filtering in AKF vs LSF. (a) Dashed lines with markers depict the ratio of ${\tilde{L}}_{\mathrm{BR}}$ for AKF to LSF against time steps $n>0$, for cases (i)–(iv) with $\mathrm{N}\mathrm{L}=0.1\%$, 1.0%, 10.0%, and 25.0%. The green trajectory shows that the LSF outperforms the AKF with a ratio $>1$ for $n\le n^{*}$; crimson trajectories show that the AKF outperforms the LSF with a ratio $<1$ for $n\le n^{*}$. (b) ${\tilde{L}}_{\mathrm{BR}}$ against $n$ is plotted for cases (i)–(iv), which confirms that a maximal forward prediction horizon marked by $n^{*}$ exists for all ratios in (a) for both the LSF and the AKF. In (a) and (b), the AKF and the LSF share identical $\{{\varphi }_q\}$. True $f$ is drawn from a flattop spectrum with ${\omega }_0/2\pi =\frac{8}{9}\times {10}^{-3}\mathrm{Hz}$, $J=45000$, $N_T=2000$, $N_P=100$ steps, $\mathrm{\Delta }t=0.001\mathrm{s}$, and $r=20$, such that Fig. 6 corresponds to case (ii). The AKF is optimized with $q=100$, $M=50$ runs, and $K=75$ trials.

Figure 6

Comparison of KF performance under various imperfect learning scenarios. (a)–(d) True noise properties are varied to introduce pathological learning with respect to a fixed algorithmic configuration: ${\omega }_0/2\pi =0.5$, 0.499, $\frac{8}{9}\times {10}^{-3}$, $\frac{8}{9}\times {10}^{-3}\mathrm{Hz}$, and $J=80$, 80, 45 000, and 80 000 respectively. The relationship between the LKFFB basis and the true noise spectrum is shown schematically above the columns. (a) Perfect learning. (b) Imperfect projection on the LKFFB basis. (c) Finite computational Fourier resolution. (d) Relaxed basis bandwidth assumption. (a)–(d) ${\tilde{L}}_{\mathrm{BR}}$ against time steps $n>0$ is shown for the LKFFB, the AKF, and the LSF. (e)–(l) Optimization results for (top row) the LKFFB and (bottom row) the AKF in each of the four regimes in (a)–(d). The gray dots depict $K$ random (${\sigma }^2$, $R$) pairs, where $M$ realizations of $f$, $\mathcal{D}$ are used to calculate ${\tilde{L}}_{\mathrm{BR}}$ for each pair. Purple (crimson) circles represent low-loss regions where the risk value in Eq. (45) for (${\sigma }^2$, $R$) is $<10\%$ of the median risk value during state estimation (prediction) for $-n_L<n<0$ ($n_L>n>0$), with $n_L=50$. The black star, (${\sigma }^{*}$, $R^{*}$), minimizes risk values over the purple circles during state estimation. A KF filter is “tuned” if an optimal (${\sigma }^{*}$, $R^{*}$) value lies in the overlap of low-loss regions for state estimation (purple) and prediction (crimson); disjoint regions in (h) show LKFFB tuning failure. KF algorithms set up with $q=100$ for the AKF; $J^{(B)}=100$ and ${\omega }_0^{(B)}/2\pi =0.5\mathrm{Hz}$ for the LKFFB, with $N_T=2000$, $N_P=100$ steps, $\mathrm{\Delta }t=0.001\mathrm{s}$, and $r=20$; and $\mathrm{N}\mathrm{L}=1\%$.

Figure 7

(a)–(d) Blue (red) open markers plot the LKFFB (AKF) spectrum estimates, and the true spectrum (flattop) of $f$ is plotted as a black solid line. The dashed black vertical line marks the true noise cutoff, $J{\omega }_0$, and this cutoff is varied relative to a measurement sampling rate, ${\omega }_{(S)}$, and ${\omega }^{(B)}\equiv {\omega }_0^{(B)}{J}^{(B)}\approx {\omega }_{(S)}/r$ in the LKFFB, such that ${\omega }_0/2\pi =0.497\mathrm{Hz}$, $J=20$, 40, 80, and 200. For the LKFFB, the blue open markers are $\propto \parallel {\widehat{x}}_n^j{\parallel }^2$ in a single run with ${\omega }_0^{(B)}/2\pi =0.5\mathrm{Hz}$ for $j\in J^{(B)}=100$ oscillators; the dashed blue vertical line marks the edge of the LKFFB basis. For the AKF, red markers are $\widehat{S}(\omega )$ computed using the learned $\{{\varphi }_{q^{\prime }\le q}\}$ and optimized ${\sigma }^{*}$ values, with order $q=100$. In all plots, the zeroth Fourier component is omitted on the log scale. $N_T=2000$, $N_P=50$ steps, $\mathrm{\Delta }t=0.001\mathrm{s}$, and $r=20$, with $M=50$ runs and $K=75$ trials. $\mathrm{N}\mathrm{L}=1\%$.

Figure 8

Normalized risk against $n>0$ plotted for a QKF in open markers; the dark-gray line at ${\mu }_f\equiv 0$ depicts performance underpredicting the noise mean. QKF outperforms predicting the mean if open markers lie in the green regions. Marker color (dark indigo to pink) depicts true noise cutoff varied by $J{\omega }_0/{\omega }_{(B)}=0.2$, 0.4, 0.6, and 0.8 for $f$ defined identically to Fig. 7 with ${\omega }_0/2\pi =0.497\mathrm{Hz}$, $J=20$, 40, 60, and 80; $\mathrm{N}\mathrm{L}=1\%$. (a) We obtain $\{{\varphi }_{q^{\prime }\le q}\}$, $q=100$ coefficients from AKF or LSF acting on a linear measurement record generated from true $f$. A new truth, $f^{\prime }$, is generated from an $\mathrm{AR}(q)$ process using $\{{\varphi }_{q^{\prime }\le q}\}$, $q=100$ as true coefficients and by defining a known, true $\sigma$. Quantized measurements from $f^{\prime }$ are obtained; data are corrupted by measurement noise of a true, known strength $R$. (b) We use $\{{\varphi }_{q^{\prime }\le q}\}$, $q=100$ coefficients from (a), but we generate quantized measurements from the original, true $f$. QKF noise design parameters are optimized for (${\sigma }_{\mathrm{AKF}}^{*}\le {\sigma }_{\mathrm{QKF}}$, $R_{\mathrm{AKF}}^{*}\le R_{\mathrm{QKF}}$) with $M=50$ runs and $K=75$ trials. For (a) and (b), $N_T=2000$, $N_P=50$ steps, and $\mathrm{\Delta }t=0.001\mathrm{s}$, $r\gg 2$.

Figure 9

(a) ${\tilde{L}}_{\mathrm{BR}}$ vs $n^{‡}$ (in units of number of time steps) are plotted for GPR with a periodic kernel. The dark-gray horizontal line at unity for ${\mu }_f\equiv 0$ marks ${\tilde{L}}_{\mathrm{BR}}$ underpredicting the mean; GPR outperforms predicting the mean if data fall below this line. The gray-black markers correspond to optimization within physical bounds for $\kappa \le 0$ (kernel resolution at or above Fourier resolution); crimson markers and lines depict optimization within unphysical regimes, $\kappa >0$, with solid lines in the regime with high values of $\kappa \gg 0$. The remaining $\{R,\sigma ,l\}$ values are optimized for non-negative values. (Inset) ${\tilde{L}}_{\mathrm{BR}}$ vs $n^{‡}$ of a periodic kernel (PER) with $\kappa \approx {10}^3$ is plotted against results from naively trained Gaussian kernels (RBF, RQ); a Matern kernel (MAT32) and a quasiperiodic kernel (QPER). (b)–(d) True state $f_n$ vs $n$ (the black solid line) and GPR predictions ${\widehat{\mu }}_{f^{‡}}$ vs $n^{‡}$ (the open markers) plotted for a periodic kernel for tracking a sinusoid with frequency ${\omega }_0$; the noisy data record (not shown) ceases at $n=0$. We fix $\kappa =0$ and 70; triangles plot predictions for manually tuned $\{R,\sigma ,l\}$ values; circles plot predictions for the optimized $\{R,\sigma ,l\}$ values. Vertical dashed lines mark $n=\kappa$, where we overlay true $f$ at the beginning of the data record as a red dashed line. (b) Perfection projection is possible: ${\omega }_0/{\omega }_0^{(B)}\in \mathcal{Z}$ (natural numbers) and ${\omega }_0/2\pi =3\mathrm{Hz}$. (c) Imperfect projection, with ${\omega }_0/{\omega }_0^{(B)}\notin \mathcal{Z}$, ${\omega }_0/2\pi =3\frac{1}{3}\mathrm{Hz}$, and $\kappa =0$. (d) We moderately raise $\kappa >0$, such that ${\omega }_0/{\omega }_0^{(B)}\gg 0\notin \mathcal{Z}$ for the original ${\omega }_0/2\pi =3\mathrm{Hz}$. (e) We test (c) and (d) for $\kappa >0$, ${\omega }_0/{\omega }_0^{(B)}\notin \mathcal{Z}$, and ${\omega }_0/2\pi =3\frac{1}{3}\mathrm{Hz}$. For (b)–(e), $N_T=2000$, $N_P=150$ steps, and $\mathrm{\Delta }t=0.001\mathrm{s}$; $\mathrm{N}\mathrm{L}=1\%$.