Quantum tomography is a widely applicable method for reconstructing unknown quantum states and processes. However, its applications in quantum technologies usually also require estimating the difference between prepared and target quantum states with reliable confidence intervals. In this work we suggest a computationally efficient and reliable scheme for determining well-justified error bars for quantum tomography. We approximate the probability distribution of the Hilbert-Schmidt distance between the target state and the estimation, which is given by the linear inversion, by calculating its two moments. We also present a generalization of this approach for quantum process tomography and deriving confidence intervals for affine functions. We benchmark our approach for a number of quantum tomography protocols using both simulation and demonstration with the use of a cloud-accessible quantum processor. The obtained results pave the way for the use of the suggested scheme for the complete characterization of quantum systems of various natures.
Quantum tomography is a widely applicable method for reconstructing unknown quantum states and processes. However, its applications in quantum technologies usually also require estimating the difference between prepared and target quantum states with reliable confidence intervals. In this work we suggest a computationally efficient and reliable scheme for determining well-justified error bars for quantum tomography. We approximate the probability distribution of the Hilbert-Schmidt distance between the target state and the estimation, which is given by the linear inversion, by calculating its two moments. We also present a generalization of this approach for quantum process tomography and deriving confidence intervals for affine functions. We benchmark our approach for a number of quantum tomography protocols using both simulation and demonstration with the use of a cloud-accessible quantum processor. The obtained results pave the way for the use of the suggested scheme for the complete characterization of quantum systems of various natures.
Precise characterization of quantum devices is usually achieved with quantum tomography. However, most methods which are currently widely used in experiments, such as maximum likelihood estimation, lack a well-justified error analysis. Promising recent methods based on confidence regions are difficult to apply in practice or yield error bars which are unnecessarily large. Here, we propose a practical yet robust method for obtaining error bars. We do so by introducing a novel 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. The statements resulting from our method are formulated in terms of a figure of merit, such as the fidelity to a reference state. 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.
Quantum state tomography is the task of inferring the state of a quantum system from measurement data. A reliable tomography scheme should not only report an estimate for that state, but also well-justified error bars. These may be specified in terms of confidence regions, i.e., subsets of the state space which contain the system's state with high probability. Here, building upon a quantum generalization of Clopper-Pearson confidence intervals-a notion known from classical statistics-we present a simple and reliable scheme for generating confidence regions. These have the shape of a polytope and can be computed efficiently. We provide several examples to demonstrate the practical usability of the scheme in experiments.
In quantum tomography, a quantum state or process is estimated from the results of measurements on many identically prepared systems. Tomography can never identify the state or process exactly. Any point estimate is necessarily "wrong" -- at best, it will be close to the true state. Making rigorous, reliable statements about the system requires region estimates. In this article, I present a procedure for assigning likelihood ratio (LR) confidence regions, an elegant and powerful generalization of error bars. In particular, LR regions are almost optimally powerful -- i.e., they are as small as possible.
Abstract In the present work, we propose a generalization of the confidence polytopes approach for quantum state tomography (QST) to the case of quantum process tomography (QPT). Our approach allows obtaining a confidence region in the polytope form for a Choi matrix of an unknown quantum channel based on the measurement results of the corresponding QPT experiment. The method uses the improved version of the expression for confidence levels for the case of several positive operator-valued measures. We then demonstrate how confidence polytopes can be employed for calculating confidence intervals for affine functions of quantum states (Choi matrices), such as fidelities and observables mean values, which are used both in QST and QPT settings. As we propose, this problem can be efficiently solved using linear programming tools. We also study the performance and scalability of the developed approach on the basis of simulation and experimental data collected using IBM cloud quantum processor.
Current techniques in quantum process tomography typically return a single point estimate of an unknown process based on a finite albeit large amount of measurement data. Due to statistical fluctuations, however, other processes close to the point estimate can also produce the observed data with near certainty. Unless appropriate error bars can be constructed, the point estimate does not carry any sound operational interpretation. Here, we provide a solution to this problem by constructing a confidence region estimator for quantum processes. Our method enables reliable estimation of essentially any figure of merit for quantum processes on few qubits, including the diamond distance to a specific noise model, the entanglement fidelity, and the worst-case entanglement fidelity, by identifying error regions which contain the true state with high probability. We also provide a software package, QPtomographer, implementing our estimator for the diamond norm and the worst-case entanglement fidelity. We illustrate its usage and performance with several simulated examples. Our tools can be used to reliably certify the performance of, e.g., error correction codes, implementations of unitary gates, or more generally any noise process affecting a quantum system.
Quantum state tomography is currently the standard tool for verifying that a state prepared in the lab is close to an ideal target state, but up to now there have been no rigorous methods for evaluating the precision of the state preparation in tomographic experiments. We propose a new estimator for quantum state tomography, and prove that the (always physical) estimates will be close to the true prepared state with a high probability. We derive an explicit formula for evaluating how high the probability is for an arbitrary finite-dimensional system and explicitly give the one- and two-qubit cases as examples. This formula applies for any informationally complete sets of measurements, arbitrary finite number of data sets, and general loss functions including the infidelity, the Hilbert-Schmidt, and the trace distances. Using the formula, we can evaluate not only the difference between the estimated and prepared states, but also the difference between the prepared and target states. This is the first result directly applicable to the problem of evaluating the precision of estimation and preparation in quantum tomographic experiments.