Efficient and versatile tomography of bosonic quantum states (Part I)*

The use of bosonic superconducting circuits for quantum information processing is promising due to their hardware efficiency and error-correction capabilities. In this setting, conventional methods for characterizing bosonic states, such as Wigner tomography, require many measurements and are not feasible for scaling up to multimode systems. Here, we present a Quantum Reservoir Processing (QRP) approach that uses ergodic dynamics on the bosonic systems with operations as simple as displacements. By identifying the action of the dynamics as a linear map, we perform state estimation from a few measured simple observables. We show that our approach allows for robust reconstruction of an arbitrary input state of dimension D with high fidelity from D²-1 measured photon number distribution, which is not only fewer than the measurements required for standard Wigner tomography but also simpler to execute. We provide systematic and thorough error analysis to show how coherent and decoherent errors affect the observables, and therefore, the state reconstruction. Our protocol offers an efficient, robust, and versatile tool for state tomography in bosonic superconducting circuits.