Experimental Single-Setting Quantum State Tomography

Abstract

Quantum computers solve ever more complex tasks using steadily growing system\nsizes. Characterizing these quantum systems is vital, yet becoming increasingly\nchallenging. The gold-standard is quantum state tomography (QST), capable of\nfully reconstructing a quantum state without prior knowledge. Measurement and\nclassical computing costs, however, increase exponentially in the system size -\na bottleneck given the scale of existing and near-term quantum devices. Here,\nwe demonstrate a scalable and practical QST approach that uses a single\nmeasurement setting, namely symmetric informationally complete (SIC) positive\noperator-valued measures (POVM). We implement these nonorthogonal measurements\non an ion trap device by utilizing more energy levels in each ion - without\nancilla qubits. More precisely, we locally map the SIC POVM to orthogonal\nstates embedded in a higher-dimensional system, which we read out using\nrepeated in-sequence detections, providing full tomographic information in\nevery shot. Combining this SIC tomography with the recently developed\nrandomized measurement toolbox ("classical shadows") proves to be a powerful\ncombination. SIC tomography alleviates the need for choosing measurement\nsettings at random ("derandomization"), while classical shadows enable the\nestimation of arbitrary polynomial functions of the density matrix orders of\nmagnitudes faster than standard methods. The latter enables in-depth\nentanglement studies, which we experimentally showcase on a 5-qubit absolutely\nmaximally entangled (AME) state. Moreover, the fact that the full tomography\ninformation is available in every shot enables online QST in real time. We\ndemonstrate this on an 8-qubit entangled state, as well as for fast state\nidentification. All in all, these features single out SIC-based classical\nshadow estimation as a highly scalable and convenient tool for quantum state\ncharacterization.\n