Tensor network states (TNS), such as the matrix product states (MPS) and projected entangled pair states (PEPS), are paradigmatic states to describe the ground states of gapped Hamiltonians, and they include important states such as the cluster, W, GHZ, AKLT states. There is thus increasing interest in finding ways of preparing them in quantum computers or quantum simulators, either for quantum information applications like computing, metrology, communication, and networking, or as variational states for the study of many-body quantum systems. It is well-known that the MPS of the system size N can be prepared with circuit depth T=O(N). However, two pressing questions have not been fully resolved: Which subclass of high-dimensional PEPS can be efficiently prepared, and what is the optimal scaling for preparing different kinds of TNS?

Our works [1,2,3] is a significant step toward the efficient preparation of TNS. In [1,2], we develop schemes to efficiently prepare high-dimensional PEPS. In [1], we introduce Plaquette-PEPS, which is a subclass of PEPS that encodes important states such as the graph states and topologically ordered states. We provide an efficient scheme to prepare them with the circuit depth that scales with the linear dimension of the system, which is optimal for preparing states with topological order. In [2], we discover an adiabatic path that allows preparing short-range correlated TNS with evolution time T~ polylog N, which serves as the first deterministic method to create the high-dimensional AKLT states. In [3], we prove that one needs at least a logarithmically growing circuit depth to prepare short-range correlated MPS faithfully and develop an algorithm to prepare these MPS with such optimal scaling. By further utilizing measurements, we can prepare almost arbitrary MPS with circuit depth T=O(log log N), which is exponentially faster than known methods to prepare them.

The Kondo model and the Hubbard model are two cornerstones in understanding strongly correlated electron systems. However, it is largely unexplored what will happen when a Kondo impurity is coupled to a Hubbard bath. Such a setup is expected to possess rich (non)-equilibrium competitions of charge, magnetic, and superconducting orders, and capture the central behavior of a magnetic impurity coupled to a superconductor. 

In [4], we employ the non-Gaussian variational formalism to study the ground-state properties and out-of-equilibrium dynamics when a Kondo impurity couples with 1D and 2D attractive Hubbard bath. We find the ground state exhibits first-order quantum phase transition, and observe a dissipative spin-wave emission during the relaxation dynamics of the impurity. We further explore the finite-bias charge transport between two 1D leads connected by the impurity, where we find a multitude of phenomena, including the Josephson effect within the weak-coupling regime, the occurrence of ballistic charge transport in the strong-coupling regime, and intricate dynamical competition behaviors. The latter includes a dynamical transition from superconducting ordering to charge-density-wave ordering and the dynamical breaking of the Kondo singlet under a large bias voltage. Overall, our study discovers many theoretically intriguing phenomena arising from the interplay between two paradigmatic interactions and carries practical implications for the quantum simulation of superconductor-quantum dot devices using ultracold atoms.