Fast-feedback protocols for calibration and drift control in quantum computers

TL;DR

This work tackles the challenge of calibrating large-scale quantum processors under drift and SPAM by introducing two fast-feedback strategies: IOC, which updates control parameters shot-by-shot using indefinite-outcome circuits, and DOC, which updates after definite-outcome events such as quantum-error-correction syndromes. The IOC protocol achieves rapid, real-time drift compensation and extends to multi-parameter tuning with adaptive gain and circuit-depth scheduling; the DOC protocol enables in-situ calibration during error correction, leveraging syndrome data while addressing ambiguities in multi-parameter settings. Numerical simulations demonstrate rapid convergence and robustness to decoherence and SPAM for 1- and 2-qubit gates, and show that DOC can stabilize logical qubits in the 5-qubit code under drift. The paper also outlines practical considerations for hardware latency, parallelization, and extensions to spectator qubits and reinforcement-learning-based strategies, highlighting a path toward low-downtime, scalable quantum calibration in real devices.

Abstract

We introduce two classes of lightweight, adaptive calibration protocols for quantum computers that leverage fast feedback. The first enables shot-by-shot updates to device parameters using measurement outcomes from simple, indefinite-outcome quantum circuits. This low-latency approach supports rapid tuning of one or more parameters in real time to mitigate drift. The second protocol updates parameters after collecting measurements from definite-outcome circuits (e.g.~syndrome extraction circuits for quantum error correction), balancing efficiency with classical control overheads. We use numerical simulations to demonstrate that both methods can calibrate 1- and 2-qubit gates rapidly and accurately even in the presence of decoherence, state preparation and measurement (SPAM) errors, and parameter drift. We propose and demonstrate effective adaptive strategies for tuning the hyperparameters of both protocols. Finally, we demonstrate the feasibility of real-time in-situ calibration of qubits performing quantum error correction, using only syndrome data, via numerical simulations of syndrome extraction in the [[5,1,3]] code.

Figures (16)

• Figure 1: IOC and DOC calibration protocols compared to Rabi calibration. Many calibration protocols run batches of circuits, take many measurements, and require significant data processing to determine parameter updates. Such approaches are insensitive to short-time drift, and relying on them can reduce the amount of time available to run useful circuits on the quantum computer. Our proposed IOC and DOC protocols are diagrammed in panels (a) and (b), respectively, in the context of a simple example calibration of a $\mathsf{G}_x$ gate. As shown in panel (c), these protocols can significantly outperform a batched Rabi curve-fitting calibration protocol in terms of tuned gate infidelity and experiment uptime. Further details of this simulation comparison can be found in App. \ref{['App:TraditionalCal']}.
• Figure 2: IOC gain effects on convergence behavior. Numerical simulations analyzing convergence behavior of the mean and variance of the miscalibration, $\Delta\eta_t$, as a function of the value of the IOC gain parameter, $g$, for calibrating the rotation angle of a $\mathsf{G}_x$ gate as discussed in Sec. \ref{['sec:IOCsimpleexample']}. Results are computed over 10,000 realizations of the IOC protocol with $r=1$ and no drift. Different colors correspond to different values of $g$. Panel (a) presents the convergence of the mean, $\mu_t$, of the parameter deviation, $\Delta\eta_t$, as a function of the step, $t$, of the IOC protocol, when the initial offset is set to $\Delta\eta_0 = 0.3$ for all trajectories. We see that the rate of convergence increases with $g$ and all curves converge to $\mu_t\rightarrow 0$ for large $t$, as predicted by Eq. (\ref{['eq:mean']}). Panel (b) shows the convergence of the variance, $\sigma^2_t$ of the parameter deviation as a function of $t$ for initial offset sampled uniformly randomly from $\Delta\eta_0\in [-0.3, 0.3]$. Here, we observe that while the speed of convergence increases with $g$, the stationary value of $\sigma_t^2$ decreases with $g$. This tradeoff between rapid convergence and low stationary variance motivates the heuristics in Sec. \ref{['Sec:IOCAdaptiveStrategies']} for dynamically scheduling $g$ to be large at the outset, promoting rapid transient convergence, and then reduce over time to promote lower variance.
• Figure 3: Single-parameter IOC calibration for $\mathsf{G}_x$ gate. Results comparing the performance of the baseline IOC calibration protocol (blue) against an uncalibrated baseline (red). Here, the nominal value of the control parameter, $\eta_{{{\rm opt},}}$, drifts according to a discrete random walk, per Eq. (\ref{['Eq:RandomWalkUpdate']}), with $\ell=0.001$. The IOC circuit depth is set to $r=13$ to coherently amplify miscalibration errors for faster convergence, and the gain is set as $g=\ell s$. This simulation also incorporates per-gate depolarization with $p=0.001$ and depolarizing SPAM with $p_{SPAM}=0.01$. Panel (a) plots the magnitude of the miscalibration, $\Delta\eta_t$, as a function of shot, $t$, with $\Delta\eta_0=0.2$. The solid curve shows the mean and the shading shows associated standard deviation, computed over 50 trajectories. Panel (b), meanwhile, shows the miscalibration dynamics of a single trajectory. We observe that IOC calibration leads to rapid reduction and stabilization of the miscalibration error relative to the uncalibrated, drifting baseline.
• Figure 4: Two-parameter IOC calibration for $\mathsf{G}_x$ and $\mathsf{G}_y$ gates. Performance of the multiparameter IOC protocol applied to the two-parameter example described in Sec. \ref{['Sec:multiparameter']}. The magnitude of the parameter miscalibrations (a) $\Delta\eta_{\theta,t}$ and (b) $\Delta\eta_{\phi,t}$ are plotted as a function of shot, $t$ for simulations with IOC calibration (blue) and without any calibration (red). In these simulations, the nominal values, $\eta_{\theta,\text{opt}},\eta_{\phi,\text{opt}}$ of our tunable parameters each experience random walk drift with $\ell = 0.001$. These simulations also incorporate SPAM error with $p_{SPAM}=0.01$ and per-gate depolarization with $p=0.001$. The solid curves show the mean behavior over a set of 50 trajectories, and the shading shows the associated standard deviation. Both circuits are repeated $r=5$ times per calibration step to coherently amplify error rates with $g=0.001$. We observe that the multiparameter IOC protocol is successful in jointly tuning $\eta_\theta,\eta_\phi$ to reduce initial miscalibration error and maintain stable calibration in the presence of drift.
• Figure 5: Two-qubit, three-parameter IOC calibration for $\mathsf{CZ}\xspace$ gate. Performance of the multiparameter IOC protocol, used here to stabilize three drifting phases of a two-qubit $\mathsf{CZ}$ gate. Solid curves show mean $\mathsf{CZ}$ gate infidelity as a function of shot, $t$, and shading shows associated standard deviation, taken over 50 realizations of the protocol. These simulations include drift in all three parameters, modeled as independent random walks that advance every shot with a step size of $\ell = 0.001$. Parameter offsets are initialized by setting $\eta_{\theta_{{ZI}},opt,0}= \eta_{\theta_{{IZ}},opt,0}= \eta_{\theta_{{ZZ}},opt,0}=-0.25$. The gain used for the IOC protocol is $g = 2.5\times10^{-4}$ and $r=1$ repetition is used for each circuit. We see that IOC calibration is effective at jointly tuning the three control parameters to reduce and stabilize the $\mathsf{CZ}\xspace$ gate infidelity.