Real-time adaptive tracking of fluctuating relaxation rates in superconducting qubits

TL;DR

The paper presents a real-time, FPGA-implemented Bayesian estimator that adaptively tracks sub-millisecond relaxation-rate fluctuations in two fixed-frequency transmon qubits. By modeling $\Gamma_1$ with a gamma-distributed prior and updating via a fast on-device posterior approximation, the method achieves rapid $T_1$ estimation with adaptive waiting times $\tau_{i+1}=c\hat{T}_1$, uncovering telegraphic TLS-driven fluctuations with dwell times on tens of milliseconds. PSD and Allan deviation analyses reveal Lorentzian noise components and TLS switching rates up to $10\,\text{Hz}$, two orders of magnitude faster than previously observed in superconducting qubits. The approach reduces calibration timescales, enables rapid outlier identification for large qubit arrays, and offers a pathway toward real-time error mitigation and Hamiltonian learning in quantum processors. Overall, the work establishes a fast, robust framework for monitoring and exploiting fast relaxation dynamics to improve qubit performance and calibration cadence.

Abstract

The fidelity of operations on a solid-state quantum processor is fundamentally bounded by environmental decoherence. Characterizing environmental fluctuations is challenging because the acquisition time of nonadaptive experimental protocols limits temporal precision and can average out rapid features of the underlying dynamics. Here, we overcome this temporal-resolution limit by two orders of magnitude using a field-programmable gate-array (FPGA) powered classical controller that adaptively and continuously tracks the relaxation-time fluctuations of two fixed-frequency superconducting transmon qubits, which exhibit average relaxation times of approximately 0.17 ms and occasionally exceed 0.5 ms. We report events in which the relaxation time switches by nearly an order of magnitude over timescales of just tens of milliseconds, rather than minutes or hours as previously reported. Our real-time Bayesian estimation protocol estimates relaxation times within a few milliseconds, close to the decoherence timescale itself. Our statistical analysis further suggests that some of these fast fluctuations arise from two-level systems switching at rates up to 10 Hz, four orders of magnitude faster than earlier reports. These results redefine the timescales relevant for calibration in superconducting quantum processing units, establish a reference for rapid relaxation-rate characterization in device screening, and improve our understanding of fast relaxation dynamics.

Figures (4)

• Figure 1: Device and Bayesian adaptive decay-rate estimation. (a) Optical micrograph of transmon qubits ($\text{Q}_1$ and $\text{Q}_2$) nominally identical to the ones used in this work. Each transmon ($\text{Q}_j$) is individually controlled by microwave pulses (XY$_j$) and read out through independent resonators (RO$_j$). (b) Experimental scheme for adaptively estimating the qubit decay rate $\mathit{\Gamma}_1$ (purple box) on the FPGA in real time over $N$ probe cycles. In each cycle, labeled $i$, the controller initializes $\text{Q}_j$ to the excited state (init), waits a time $\tau_i$ adaptively chosen based on $\hat{T}_1 \equiv 1/ \langle \mathit{\Gamma}_1\rangle$ from the previous Bayesian distribution ${\cal P}_{i-1}(\mathit{\Gamma}_1)$, then updates the probability distribution ${\cal P}_i(\mathit{\Gamma}_1)$ based on the measurement outcome $m_i$. (c) Example of a nonadaptive estimation of the relaxation time. From the normalized fraction $\Tilde{p}(\ket{1})$ of excited states as a function of linearly stepped probing waiting times $\tau_{\text{lin}}$, $\mathit{\Gamma}_1$ is estimated by an exponential fit. The total elapsed time is $\approx 1s$. (d) Example of an adaptive estimation of $\hat{T}_1$ by Bayesian statistics implemented on the controller. Each circle is a single-shot measurement outcome $\ket{0}$ (red) or $\ket{1}$ (blue) which updates the current estimate of $\hat{T}_1$ and the subsequent adaptive waiting time $\tau_i$. The total elapsed time for the entire estimation is $\approx 11ms$. (e) Evolution of the probability distribution ${\cal P}$ during each probe cycle $i$ of the estimation algorithm. The current estimate is $\hat{\mathit{\Gamma}}_1$ (gray dashed line). On each probing cycle, the estimate of $\hat{\mathit{\Gamma}}_1$ is updated according to two possible likelihood functions (dot–dashed lines), multiplied by the prior distribution (dashed). This yields the posterior distribution (solid), whose estimate is shifted left or right depending on the measurement outcome, while the uncertainty is reduced on average. (f) Convergence of ${\cal P}$ as a function of the $i^{\text{th}}$ probe cycle. The line shows the resulting estimate $\hat{T}_1$ and the shaded area marks the 90% credible interval. As in panel (d), each circle is a single-shot outcome $m_i$ with corresponding waiting time $\tau_{i}$.
• Figure 2: Protocol for tracking and validation of the decay-rate fluctuations by adaptive estimation on the controller. (a) Experimental results for the adaptive tracking protocol with $N= 100$ to estimate $\hat{T}_1$ (purple dots, downsampled by $D=30$) and its 68% credible interval (shaded area). Each purple point in this plot required an average estimation time of $\approx20ms$. The dashed yellow line indicates the mean value of all estimates $\overline{T}_1\approx 350µs$. The black line is a moving mean over 100 samples and the black shaded area is its 68% confidence interval (see main text). (b) Estimated $\hat{T}_1$ at $\approx 1876s$ of panel (a) (see purple dashed lines), where $\hat{T}_1$ shows telegraphic switching with timescales on the order of tens or hundreds of milliseconds. (c) The interleaved estimation sequence for $\mathit{\Gamma}_1$ used to validate the adaptive protocol. Each of the $N$ probe cycles, labeled $i$, consists of parts contributing to the adaptive (purple) and nonadaptive (gray) estimates. Each adaptive probe cycle is followed by the nonadaptive part of the cycle, where the qubit is again initialized in the excited state, the wait time is fixed to $\tau_{\text{lin},i}=i\tau_0$, and the measurement outcome is stored for offline post-processing. After the $N$ probe cycles, the final adaptively obtained distribution ${\cal P}_N(\mathit{\Gamma}_1)$ is saved. (d) Experimental results for the adaptive tracking protocol, interleaved with nonadaptive measurements. Main panel: The estimate $\hat{T}_1$ (purple dots) and 68% credible interval (shaded area) corresponding to the final probability distribution $\mathcal{P}_{50}(\mathit{\Gamma}_1)$ of the 2,000 adaptive estimates performed during the $\approx 60s$ of the experiment. The black line shows a moving mean over 5 samples and the black shaded area is its 68% confidence interval. The purple arrow indicates the mean of all the adaptive estimates $\hat{T}_1$. The dashed line is the value extracted from the fit shown in the inset. Inset: Experimental results for the nonadaptive estimate using linearly sampled waiting time $\tau_{\text{lin}, i}$. Error bars represent the standard error.
• Figure 3: Frequency and time domain analysis of $\hat{T}_1$ fluctuations on a 72-hour timescale (a) Estimated $\hat{T}_1$ (purple dots, downsampled by $D=30,000$) as a function of laboratory time by real-time adaptive tracking with $N= 49$ and sampling speed of $\approx7ms$ over 72 hours. The dashed yellow line shows the mean value of all estimates $\overline{T}_1\approx 168µs$. The black line is a moving mean with a window of size 20,000 over the original estimates. Lower panel: Histogram of the moving mean. (b-c) Frequency and time domain analysis of the $\hat{T}_1$ fluctuations shown in panel (a), obtained from 2.8-hour running windows with 80% overlap (black dashed lines): (b) Power spectral density. Lower panel: PSD of the full time trace. (c) Allan deviation on a logarithmic scale; arrows denote the zoom-in regions of Fig. \ref{['fig:fig5']}. Lower panel: Allan deviation of the full time trace. (d) Amplitudes of the Lorentzian ($A_{\text{L}}$), 1/$f$ ($A_{1/f}$), and white noise ($A_\text{w}$) contributions, extracted from the full time trace, by a simultaneous PSD and Allan deviation fit to the analytical formulas from Table \ref{['tab:noise']}. The purple dashed line is the standard deviation $\delta \hat{T}_1$ (see main text) of the Bayesian posterior distribution. Lower panel: Cumulative histogram of the fitted $A_i$ and their medians (dashed lines).
• Figure 4: Time domain analysis of $\hat{T}_1$ fluctuations on sub-minute observation times. Zoom-ins of the regions indicated by the arrows in Fig. \ref{['fig:fig4']}(c). (a) Estimated $\hat{T}_1$ (purple dots, downsampled by $D=3,000$) as a function of laboratory time. The dashed yellow line represents the mean value of all estimates $\overline{T}_1$ of Fig. \ref{['fig:fig4']}(a). The black line is a moving mean with a window size of 3,000 over the original estimates. (b) Allan deviation in logarithmic scale of the $\hat{T}_1$ time trace with a running window of about 7 minutes and 80% overlap. (c) Amplitudes of the Lorentzian ($A_\text{L}$), $1/f$ ($A_{1/f}$), and white ($A_\text{w}$) noise contributions in each interval extracted from panel (a) by a simultaneous fit to the Allan deviation and PSD using analytical models from Table \ref{['tab:noise']}. (d) Fitted switching rate $\gamma\approx 10Hz$ over more than one hour of the Lorentzian component. (e-h) Same as (a-d) with a running window of 17 minutes and 80% overlap. The switching rate fit in panel (h) reveals a stable Lorentzian process with $\gamma\approx 100mHz$ for more than 3 hours.