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.
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.
Physical models of quantum gates can directly incorporate experimental degrees of freedom, such as laser power or microwave phase, but the forward simulation required to use these models can result in impractical overhead in computational time.In this poster, we develop a method for efficient, accurate interpolation of quantum process matrices over physical parameter space, and we apply our method to study the sensitivity of quantum applications at both the gate level and circuit level.We then implement an extension of gate-set tomography (GST) that provides a comprehensive view into the errors suffered by a quantum information processor.In its standard form, GST fits a general Markovian model to experimental data in terms of process matrices that are often difficult to interpret.Our extension to this method uses process matrices interpolated over physical parameters in the optimization loop of GST to directly estimate physical model parameters, providing a clear description of the errors in a quantum processor, while reducing the required number of GST sequences by orders of magnitude compared to its standard form.
Quantum error correction uses the measurement of syndromes and classical decoding algorithms to estimate the location and type of errors while protecting the encoded quantum bits. Here we consider how prior information and Bayesian updates can play a critical role in improving the performance of QEC in the scenario of a particularly noisy qubit. This allows for leveraging even distance codes, which typically are less valued in QEC, to handle the noisy qubit, changing the power-law scaling of the logical error rate with the baseline physical error rate. A crucial component of this is updating the prior by real time feeding of decoder outputs into a approximate Kalman filter. Thus our approach provides a bootstrap to the actual error rates. We show this via simulation of the full closed-loop system: starting from uniform priors, the update procedure gradually learns site-specific error rates, enabling the decoder to outperform a fixed-prior baseline. In turn, we show that this enables in situ calibration of unitary operations via injection of gate set tomography operations with only moderate overhead in the more typical scenario of low noise qubits.
We present a novel quantum tomographic reconstruction method based on Bayesian inference via the Kalman filter update equations. The method not only yields the maximum likelihood/optimal Bayesian reconstruction but also a covariance matrix expressing the measurement uncertainties in a complete way. From this covariance matrix the error bars on any derived quantity can be easily calculated. This is a first step towards the broader goal of devising an omnibus reconstruction method that could be adapted to any tomographic setup with little effort and that treats measurement uncertainties in a statistically well-founded way. In this first part, we restrict ourselves to the important subclass of tomography based on measurements with discrete outcomes (as opposed to continuous ones), and we also ignore any measurement imperfections (dark counts, less than unit detector efficiency, etc), which will be treated in a follow-up paper. We illustrate our general theory on real tomography experiments of quantum optical information processing elements.
The calibration of high-quality two-qubit entangling gates is an essential component in engineering large-scale, fault-tolerant quantum computers. However, many standard calibration techniques are based on randomized circuits that are only quadratically sensitive to calibration errors. As a result, these approaches are inefficient, requiring many experimental shots to achieve acceptable performance. In this work, we demonstrate that robust phase estimation can enable high-precision, Heisenberg-limited estimates of coherent errors in multi-qubit gates. Equipped with an efficient estimator, the calibration problem may be reduced to a simple optimization loop that minimizes the estimated coherent error. We experimentally demonstrate our calibration protocols by improving the operation of a two-qubit controlled-Z gate on a superconducting processor, and we validate the improved performance with gate set tomography. Our methods are applicable to gates in other quantum hardware platforms such as ion traps and neutral atoms, and on other multi-qubit gates, such as CNOT or iSWAP.
An important step in building a quantum computer is calibrating experimentally implemented quantum gates to produce operations that are close to ideal unitaries. The calibration step involves estimating the error in gates and then using controls to correct the implementation. Quantum process tomography is a standard technique for estimating these errors, but is both time consuming, (when one only wants to learn a few key parameters), and requires resources, like perfect state preparation and measurement, that might not be available. With the goal of efficiently estimating specific errors using minimal resources, we develop a parameter estimation technique, which can gauge two key parameters (amplitude and off-resonance errors) in a single-qubit gate with provable robustness and efficiency. In particular, our estimates achieve the optimal efficiency, Heisenberg scaling. Our main theorem making this possible is a robust version of the phase estimation procedure of Higgins et al. [B. L. Higgins, New J. Phys. 11, 073023 (2009)].
Abstract In quantum physics, a stochastic master equation (SME) estimates the state (density operator) of a quantum system in the Schrödinger picture based on a record of measurements made on the system. In the Heisenberg picture, the SME is a quantum filter. For a linear quantum system subject to linear measurements and Gaussian noise, the dynamics may be described by quantum stochastic differential equations (QSDEs), also known as quantum Langevin equations, and the quantum filter reduces to a so-called quantum Kalman filter. In this article, we introduce a quantum extended Kalman filter (quantum EKF), which applies a commutative approximation and a time-varying linearization to systems of nonlinear QSDEs. We will show that there are conditions under which a filter similar to a classical EKF can be implemented for quantum systems. The boundedness of estimation errors and the filtering problem with ‘state-dependent’ covariances for process and measurement noises are also discussed. We demonstrate the effectiveness of the quantum EKF by applying it to systems that involve multiple modes, nonlinear Hamiltonians, and simultaneous jump-diffusive measurements.