Practical and Reliable Error Bars in Quantum Tomography

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.

Figure 1

Setup of quantum tomography. Measurements are taken on $n$ copies of a quantum system. The outcomes allow us to infer what the state of the quantum system is. In this example a qubit is measured using Pauli operators. Here, the experimental data are most consistent with the state being $|\uparrow ⟩$, located at the top of the Bloch sphere (in green). However, because only finite data are collected, there is an uncertainty associated with this statement. In the method of Ref. [28], a distribution ${\mu }_{B^n}(\rho )$ (the red gradient) is determined from the data, from which confidence regions can be constructed (delimited by the dotted line). These are regions in state space in which the state lies with high probability.

Figure 2

Construction of confidence regions from the distribution $\mu (f)$ on the figure of merit. High weight intervals with respect to $\mu (f)$ (left plot) correspond to high weight regions in state space with respect to ${\mu }_{B^n}(\rho )$ (right diagram) which are (essentially) confidence regions, according to Ref. [28].

Figure 3

Analysis of measurement data from two superconducting qubits prepared in a Bell state. We determined effective measurement operators which model the noise in the measurement process. The histogram of the fidelity to the target state $|\psi ⟩$ (the blue data points), produced using our procedure, fits well to our theoretical model in Table 1. The quantum error bars are a concise, intuitive, and precise characterization of the fit model, which is interpreted as a skewed Gaussian function. The parameter $f_0$ is the peak maximum, $\mathrm{\Delta }$ is the half width of the original Gaussian, and $\gamma$ characterizes the skewing in terms of the displacement of the sides of the peak from the exact Gaussian at the relative height $1/e$. This example involving experimental data demonstrates a good level of practical applicability of our method.