Adaptive Bayesian phase estimation for quantum error correcting codes

Abstract

Realisation of experiments even on small and medium-scale quantum computers requires an optimisation of several parameters to achieve high-fidelity operations. As the size of the quantum register increases, the characterisation of quantum states becomes more difficult since the number of parameters to be measured grows as well and finding efficient observables in order to estimate the parameters of the model becomes a crucial task. Here we propose a method relying on application of Bayesian inference that can be used to determine systematic, unknown phase shifts of multi-qubit states. This method offers important advantages as compared to Ramsey-type protocols. First, application of Bayesian inference allows the selection of an adaptive basis for the measurements which yields the optimal amount of information about the phase shifts of the state. Secondly, this method can process the outcomes of different observables at the same time. This leads to a substantial decrease in the resources needed for the estimation of phases, speeding up the state characterisation and optimisation in experimental implementations. The proposed Bayesian inference method can be applied in various physical platforms that are currently used as quantum processors.

Experimental adaptive Bayesian estimation of multiple phases with limited data

“…In that paper, given an arbitrary state, prior knowledge and number of repetitions of the experiment, explicit recipes for the optimal measurements are provided in the case where the estimators commute. Further perspectives include the study of different multiparameter scenarios, as well as practical applications to quantum sensing of delicate samples 81 and quantum error correcting algorithms [82][83][84] .…”

Experimental Bayesian calibration of trapped ion entangling operations

“…We describe the selection process of the measurement settings to improve convergence of the Bayesian parameter estimation protocol and investigate the effect on the experimental run time of the algorithm. The choice of measurement setting determines the amount of information that will be gained, and can have significant effects on the number of measurements required [27,45,[64][65][66]. We finally experimentally verify the performance of the algorithm by checking the consistency of the final gate parameters returned by the algorithm.…”

“…At first sight, the problem of calibrating multiple control parameters would not appear difficult if their action on the quantum system could be independently measured and the parameter corrected accordingly. For example, Ramsey spectroscopy in both frequentist [24][25][26] and Bayesian [24,[26][27][28][29] form can be used to determine the mismatch between a qubit transition frequency and the driving field. Indeed, combined with a Rabi frequency measurement [30] to determine the applied field strength, single-qubit operations can be efficiently calibrated and traced in time using the minimum number of experimental measurements to correct for drifts [31,32].…”

Bayesian parameter estimation using Gaussian states and measurements

“…Historically, phase estimation has been closely associated with interferometry [61], but nowadays, phase estimation is usually considered in a broader context. In particular, Bayesian phase estimation has been studied for a variety of applications, see, e.g., [62][63][64]. We therefore want to focus on a special case of Bayesian phase estimation, where there is no prior information on the phase and local estimation hence cannot be employed in a meaningful way.…”