Experimental Single-Setting Quantum State Tomography

Abstract

Quantum computers solve ever more complex tasks using steadily growing system sizes. Characterizing these quantum systems is vital, yet becoming increasingly challenging. The gold-standard method for this task is quantum state tomography (QST), capable of fully reconstructing a quantum state without prior knowledge. The measurement and classical computing costs, however, increase exponentially with the number of constituents (e.g., qubits)—a daunting bottleneck given the scale of existing and near-term quantum devices. Here, we demonstrate a scalable and practical QST approach that only uses a single measurement setting, namely symmetric informationally complete (SIC) positive operator-valued measures (POVMs). We implement these nonorthogonal measurements on an ion trap quantum processor by utilizing additional energy levels within each ion—without requiring ancillary ions to assist in measurements. More precisely, we locally map the SIC POVM to orthogonal states embedded in a higher-dimensional system, which we read out using repeated in-sequence detections, thereby providing full tomographic information in every shot. Combining this SIC tomography with the recently developed randomized measurement toolbox (“classical shadows”) proves to be a powerful combination. SIC tomography alleviates the need for choosing measurement settings at random (“derandomization”), while classical shadows enable the estimation of arbitrary polynomial functions of the density matrix orders of magnitudes faster than standard methods. The latter enables in-depth entanglement characterization, which we experimentally showcase on a five-qubit absolutely maximally entangled state. Moreover, the fact that the full tomography information is available in every shot enables online QST in real time (i.e., while the experiment is running). We demonstrate this on an eight-qubit entangled state (which has 2^{8}⋅2^{8}−1=65535 degrees of freedom), as well as for fast state identification. All in all, these features single out SIC-based classical shadow estimation as a highly scalable and convenient tool for quantum state characterization.