Streaming quantum gate set tomography using the extended Kalman filter

TL;DR

This work addresses real-time calibration of quantum gate sets by applying an extended Kalman filter (EKF) to gate set tomography (GST), treating the gate-error parameters as a static vector $x$ and updating estimates $\hat{x}_k$ with uncertainty $P_k$ using the Jacobian $H_k$ of the observation function $h_k(x)$. By modeling the GST outcomes with a Gaussian likelihood via the central limit theorem and employing gauge-invariant (FOGI) representations, the authors demonstrate that streaming EKF GST can achieve accuracy comparable to batched maximum-likelihood estimation while dramatically reducing computational load. The method yields online uncertainty quantification and processes circuit outcomes on standard hardware at practical rates, enabling closed-loop calibration possibilities. Extensions to non-Markovian noise, singular covariances, and memory-efficient variants (e.g., sigma-point or square-root Kalman filters) are discussed, highlighting EKF GST as a viable building block for real-time control of quantum processors.

Abstract

Closed-loop control algorithms for real-time calibration of quantum processors require efficient filters that can estimate physical error parameters based on streams of measured quantum circuit outcomes. Development of such filters is complicated by the highly nonlinear relationship relationship between observed circuit outcomes and the magnitudes of elementary errors. In this work, we apply the extended Kalman filter to data from quantum gate set tomography to provide a streaming estimator of the both the system error model and its uncertainties. Our numerical examples indicate extended Kalman filtering can achieve similar performance to maximum likelihood estimation, but with dramatically lower computational cost. With our method, a standard laptop can process one- and two-qubit circuit outcomes and update gate set error model at rates comparable with current experimental execution.

Figures (4)

• Figure 1: Closed-loop control techniques benefit from online filters that can update model parameters in real time as data is collected. This work adapts the extended Kalman filter for the purpose of streaming estimation of error rates (and their uncertainties) in gate-model quantum processors. This approach offers an alternative to the batched maximum likelihood estimation utilized in, eg. gate set tomography.
• Figure 2: Kalman iteration
• Figure 3: Influence of the prior distribution on Bayesian updates. The top two plots show the prior and likelihood for a short and a long circuit for a simplified 1-parameter model, and the bottom two plots show the resulting posterior when calculated via Bayes rule. In the case of the short circuit, the wide circuit prior moves closer to the true value than the narrow circuit prior. In the case of the long circuit, the wide prior produces a multi-modal distribution when multiplied with the likelihood, which violates the assumptions of the Kalman filter, and the narrow prior results in a unimodal distribution that can be well approximated as a Gaussian. This example illustrates that, in order to assume Gaussian priors and Gaussian likelihoods needed for Kalman filtering, the length of the circuit should be selected such that the likelihood is unimodal on the principle support of the prior.
• Figure 4: Numerical performance of streaming GST. Plots (a)-(d) compare the convergence rates of the Kalman filter's point estimate with batched MLE point estimates under a metric of mean square error (MSE) and mean absolute error (MAE) between the point estimate and the parameters of the data generating model. The $x$-coordinates of the gray lines correspond to batches of germs of fixed power, and the batched MLE point estimates are calculated based on the observations from all data up-to and including the current batch. These plots also compare the evolution in the filters' MSE and MAE with their expected evolution given by $\text{Tr}(P_k)$ and $\text{Tr}(\sqrt{P_k})$ respectively. MSE is the natural metric for a Kalman filter since Kalman filters minimize the square of the expected error in the estimate, but MAE is a stronger performance metric that is more sensitive to small differences in parameters. Plots (e) and (f) display error in the estimate of particular Hamiltonian parameters that correspond to the types of errors we expect could be reduced with improved calibration. The dotted lines denote the "true" parameters that were used to generate the data.