We present and test a protocol to learn the matrix-product operator (MPO) representation of an experimentally prepared quantum state. The protocol takes as an input classical shadows corresponding to local randomized measurements, and outputs the tensors of a MPO which maximizes a suitably-defined fidelity with the experimental state. The tensor optimization is carried out sequentially, similarly to the well-known density matrix renormalization group algorithm. Our approach is provably efficient under certain technical conditions which are expected to be met in short-range correlated states and in typical noisy experimental settings. Under the same conditions, we also provide an efficient scheme to estimate fidelities between the learned and the experimental states. We experimentally demonstrate our protocol by learning entangled quantum states of up to qubits in a superconducting quantum processor. Our method upgrades classical shadows to large-scale quantum computation and simulation experiments.

The simulation of quantum field theories, both classical and quantum, requires regularization of infinitely many degrees of freedom. However, in the context of field digitization (FD) -- a truncation of the local fields to discrete values -- a comprehensive framework to obtain continuum results is currently missing. Here, we propose to analyze FD by interpreting the parameter as a coupling in the renormalization group (RG) sense. As a first example, we investigate the two-dimensional classical -state clock model as a Z FD of the -symmetric -model. Using effective field theory, we employ the RG to derive generalized scaling hypotheses involving the FD parameter , which allows us to relate data obtained for different -regularized models in a procedure that we term (FDS). Using numerical tensor-network calculations at finite bond dimension , we further uncover an unconventional universal crossover around a low-temperature phase transition induced by finite , demonstrating that FDS can be extended to describe the interplay of and . Finally, we analytically prove that our calculations for the 2D classical-statistical Z clock model are directly related to the quantum physics in the ground state of a (2+1)D Z lattice gauge theory which serves as a FD of compact quantum electrodynamics. Our study thus paves the way for applications of FDS to quantum simulations of more complex models in higher spatial dimensions, where it could serve as a tool to analyze the continuum limit of digitized quantum field theories.