Practical, Reliable Error Bars in Quantum Tomography
Abstract: Precise characterization of quantum devices is usually achieved with quantum tomography. However, most methods which are currently widely used in experiments lack a robust, well-justified error analysis, especially in the regime of finite data. For example, maximum likelihood estimation does not provide any estimation of the error of the tomography procedure, and is typically complemented by an ad hoc method such as resampling/bootstrapping. We propose a new method which provides well-justified error bars. The error bars are practical, in that the error bars are typically of the same order of magnitude as those obtained by a resampling analysis. The error bars are determined for a figure of merit (such as the fidelity to a target state) which can be chosen freely. Our method takes as input the measurement data from the experiment, and runs an analysis based on the concept of confidence regions. We then introduce a new representation of the output of the tomography procedure, the quantum error bars. This representation is (i) concise, being given in terms of few parameters, (ii) intuitive, providing a fair idea of the "spread" of the error, and (iii) useful, containing the necessary information for constructing confidence regions. We present an algorithm for computing this representation and provide ready-to-use software. Our procedure is applied to actual experimental data obtained from two superconducting qubits in an entangled state, demonstrating the applicability of our method.
Philippe Faist(1)(2) and Renato Renner(1)
(1) Institute for Theoretical Physics, ETH Zurich
(2) IQIM, Caltech
Our software can be downloaded at https://github.com/Tomographer/tomographer/