Tensor Network Simulation Of Quantum Many-Body Problems:From Closed Systems To Open Systems

The quantum many-body problem is one of the core problems in the field of condensed matter physics and quantum information.The numerical simulations have always played a very important role in exploring the quantum many-body problem.Since the Hilbert space of the quantum many-body wave function grows exponentially with the size of the system,currently we can only strictly represent and simulate the quantum many-body state of the smaller system.On the other hand,the quantum Monte Carlo method can solve some quantum many-body problems efficiently,but this method suffers from sign problems for frustrated systems and fermionic systems.In general,for most quantum many-body problems,we can generally only use variational methods to solve them.With the in-depth study of the entanglement structure of quantum many-body states,it has been found that the entanglement entropy of quantum many-body states of interest satisfies the area law rather than the volume law.We can use this prior knowledge to propose a new type of variational wave function,namely the tensor network state.The Density Matrix Renormalization Group(DMRG)algorithm based on one-dimensional tensor network representation has become the workhorse for solving one-dimensional quantum many-body problems.In recent years,many works have extended the tensor network algorithms to two-dimensional quantum many-body systems,allowing us to simulate quantum many-body systems with frustration.Based on the method of vectorizing the density matrix,we can also directly apply many tensor network algorithms that simulate closed systems to open quantum many-body systems.But unlike many closed systems we know that the entanglement entropy of their quantum states have area law,the exploration of the entanglement structure of open quantum many-body systems is still very rare.In this thesis,we use the tensor network algorithm to study closed and open quantum many-body systems in two dimensions.The innovative results we obtained are as follows:.1.We found a Z2 vortex phase in a quantum frustration model with anisotropic exchange on a triangular lattice.The ground state of a two-dimensional frustrated system can have many exotic quantum phases.Recently,anisotropic exchange model on triangular lattices has been used to describe many materials,but whether there are multiQ phase in the quantum phase diagram of this model is controversial.In order to solve this problem,we utilize variational methods based on entangled pair projected states to simulate the quantum frustration systems with anisotropic exchange on triangular lattices.We gived the quantum phase diagram of this model and found that in addition to stripe phases and a 120° phase,there is also a multi-Q phase that DMRG did not find.We identify this multi-Q phase as the Z2 vortex phase.Our work provides theoretical support for studying materials relevant to this model,and it also shows that the twodimensional tensor network method based on entangled pair projected states is very suitable for studying incommensurate quantum phases.2.Theoretically reveal the characteristics of the evolution of operator entanglement entropy of noisy random quantum circuits on two dimensions.The evolution of random quantum circuits is a computational task that is achievable by noisy quantum computing devices in the near future and has the potential to achieve quantum advantage.It can be viewed as quantum chaotic dynamics in an open environment.To explore its entanglement structure and classical simulability,we use a tensor network representation based on matrix product operators to study the evolution of twodimensional noisy random quantum circuits.We explore the entanglement structure of the evolution process by calculating the operator entanglement entropy.With comprehensive numerical simulation and theoretical analysis,we find that when the noise rate does not vary with system size,the maximum achievable operator entanglement entropy of the system always obeys the area law.The maximum achievable operator entanglement entropy of the system satisfies the volume law only if the noise rate decreases with the size of the system.For a system of fixed size,the maximum achievable operator entanglement entropy also increases as the noise decreases.Our study suggests that the primary goal of improving the computational power of noisy quantum computing hardware is to reduce noise.It is also shown that the output states of noisy random quantum circuits are not highly entangled states,making it possible to efficiently simulate these systems using tensor networks.