Tomography of parametrized quantum states

Characterizing quantum systems is a fundamental task that enables the development of quantum technologies. Various approaches, ranging from full tomography to instances of classical shadows, have been proposed to this end. However, quantum states that are being prepared in practice often involve families of quantum states characterized by continuous parameters, such as the time evolution of a quantum state. In this work, we extend the foundations of quantum state tomography to parametrized quantum states. We introduce a framework that unifies different notions of tomography and use it to establish a natural figure of merit for tomography of parametrized quantum states. Building on this, we provide an explicit algorithm that combines signal processing techniques with a tomography scheme to recover an approximation to the parametrized quantum state equipped with explicit guarantees. Our algorithm uses techniques from compressed sensing to exploit structure in the parameter dependence and operates with a plug and play nature, using the underlying tomography scheme as a black box. In an analogous fashion, we derive a figure of merit that applies to parametrized quantum channels. Substituting the state tomography scheme with a scheme for process tomography in our algorithm, we then obtain a protocol for tomography of parametrized quantum channels. We showcase our algorithm with two examples of shadow tomography of states time-evolved under an NMR Hamiltonian and a free fermionic Hamiltonian.