Streaming Quantum Gate Set Tomography Using the Extended Kalman Filter for Real-Time Characterization and Calibration of Quantum Processors

Core Concepts

The extended Kalman filter can be used to efficiently perform streaming quantum gate set tomography, providing real-time estimates of error rates and their uncertainties for closed-loop calibration and control of quantum processors.

Abstract

This work presents a method for applying the extended Kalman filter to the problem of quantum gate set tomography (GST). GST is a technique for precisely estimating the error rates in the elementary quantum operations (gates) of a quantum processor.
The key insights and highlights are:

Kalman filtering provides a streaming, online estimation approach that can update the gate set error model and its uncertainties with each new circuit outcome, in contrast to the standard batch-based maximum likelihood estimation.

The authors make several approximations to embed the nonlinear GST model within the Kalman filtering framework, including linearizing the observation model and assuming Gaussian noise in the circuit outcomes.

Numerical simulations demonstrate that the extended Kalman filter can achieve estimation accuracy comparable to maximum likelihood estimation, but with dramatically lower computational cost, enabling real-time processing of circuit data.

The method provides not only point estimates of the error parameters, but also uncertainty estimates, which are crucial for closed-loop calibration and control of quantum processors.

The authors discuss several extensions and alternative approaches, such as dealing with singular covariance matrices, incorporating non-Markovian noise, and optimizing the computational and memory efficiency of the algorithm for deployment on embedded hardware.

Overall, this work presents a promising approach for integrating real-time characterization and calibration into the closed-loop control of quantum computers, a key requirement for scaling up these systems.

Stats

The number of qubits in the gate set is n.
The number of distinct quantum gates in the gate set is NG.
The number of possible measurement outcomes is NE = 2^n.

Quotes

"Efficient, closed-loop stabilization protocols that utilize active experimental feedback will be necessary for future quantum processors to enable rapid calibration and to maintain error rates below the threshold for fault tolerance."
"Recursive filters, such as the Kalman filter, offer a compelling alternative and have a long history of performing the streaming parameter estimation that underlies many industrial control techniques."

Key Insights Distilled From

Streaming quantum gate set tomography using the extended Kalman filter

by J. P. Marcea... at arxiv.org 03-29-2024

https://arxiv.org/pdf/2306.15116.pdf

Deeper Inquiries

How could the method be extended to handle non-Markovian noise and time-varying error parameters in the quantum processor?

To extend the method to handle non-Markovian noise and time-varying error parameters in the quantum processor, the assumption of static dynamics in the Kalman filter model would need to be relaxed. This would involve incorporating a dynamic state transition model that allows for changes in the error parameters over time. The Q covariance matrix, which models stochastic drift in the state, could be modified to capture the time evolution of the error parameters. By introducing non-zero elements in the Q matrix, the filter would be able to account for drift and changes in the error rates over time. This adaptation would enable the filter to track and estimate the evolving error parameters in a non-Markovian quantum system.

What are the potential limitations or drawbacks of relying on the Gaussian approximation to the multinomial circuit outcome distributions?

While the Gaussian approximation to the multinomial circuit outcome distributions simplifies the estimation process and allows for the application of the Kalman filter, there are potential limitations and drawbacks to consider:

Validity of the Approximation: The Gaussian approximation may not hold true for all scenarios, especially when the number of samples is limited or when the underlying distribution deviates significantly from a Gaussian distribution. In such cases, the approximation may introduce errors in the estimation process.
Loss of Information: Gaussian approximations can smooth out the true distribution, leading to a loss of information about the actual variability and shape of the data. This loss of detail could impact the accuracy of the parameter estimates.
Singularity of Covariance: The Dirichlet covariance matrix used in the Gaussian approximation can be singular, posing challenges in matrix inversion and computation. Dealing with the singularity of the covariance matrix may require additional computational resources or alternative approaches.
Assumption of Independence: The Gaussian approximation assumes independence between the observations, which may not always hold in practice. Correlations between observations could affect the accuracy of the estimates derived from the Gaussian model.

Could the insights from this work on streaming quantum tomography be applied to other quantum characterization and control tasks beyond gate set tomography?

The insights from streaming quantum tomography using the extended Kalman filter can be applied to various other quantum characterization and control tasks beyond gate set tomography. Some potential applications include:

Quantum State Estimation: The methodology can be adapted for real-time estimation of quantum states, enabling continuous monitoring and feedback for quantum state preparation and manipulation.
Quantum Error Correction: The approach can be utilized for online error correction and fault tolerance schemes in quantum error correction protocols, enhancing the stability and reliability of quantum computations.
Quantum Sensing and Metrology: The techniques developed for streaming quantum tomography can be employed in quantum sensing applications, such as precision measurements and quantum metrology, to improve the estimation of physical parameters.
Quantum Control Systems: The real-time estimation capabilities can be integrated into quantum control systems for adaptive feedback control, optimizing the performance of quantum devices and processes.

By leveraging the principles and methodologies of streaming quantum tomography, researchers can enhance a wide range of quantum characterization and control tasks, leading to advancements in quantum technology and applications.