Eigenstate Entanglement Scaling and Quantum Simulation of Many-body Systems by Entanglement Renormalization

Quantum entanglement lies at the heart of modern physics and pervades various research fields. In the field of quantum many-body physics, celebrated area and log-area laws have been established for the entanglement entropy of ground states. However, there exists a long-standing question regarding the transition of eigenstate entanglement entropy from the ground state to the highly excited states. Our study fills this gap and elucidates a crossover behavior with universal scaling properties. In the study of quantum matters, knowledge about the entanglement structure can be used to guide the design of tensor network state simulations. For example, we may iteratively eliminate short-range entanglement in a so-called entanglement renormalization scheme so that the entangled ground state is mapped to a product state and then resolved exactly. This idea can be adapted to hybrid quantum-classical algorithms and speed up the simulation of strongly correlated quantum many-body systems.

In the first part of this dissertation, we investigate eigenstate entanglement scaling in quantum many-body systems and characterize the crossover from the ground-state entanglement regime at low energies and small subsystem sizes to extensive volume laws at high energies or large subsystem sizes. We first establish a weak eigenstate thermalization hypothesis (ETH) for translation-invariant systems, argue that the entanglement entropies of (almost) all energy eigenstates are described by a single crossover function whenever the (weak) ETH applies, and point out the universal scaling properties in the quantum critical regime. We then comprehensively confirm these scaling properties by analyzing large classes of quantum many-body systems. Particularly, we give the eigenstate entanglement scaling functions in analytical form for critical one-dimensional systems based on conformal field theory and for $d$-dimensional fermionic systems with Fermi surfaces. For $d=1,2,3$ non-interacting fermions, the scaling functions are numerically verified, and for $d=1,2,3$ harmonic lattice models (free scalar field theory), they are numerically determined. ETH is confirmed with Monte Carlo methods by sampling energy eigenstates or squeezed states for fermions or bosons with $d=1,2$. We also probe and confirm the described scaling properties and the applicability of the ETH in integrable and non-integrable interacting spin-1/2 chains by using exact diagonalization. All the evidence appearing here strongly suggests the existence of crossover functions. Their transition from ground-state scaling to extensive scaling, as well as the universal scaling properties in quantum-critical regimes, are generic.

In the second part of this dissertation, we present a quantum-classical tensor network state algorithm for condensed matter systems. First, we describe this algorithm, which is based on the multi-scale entanglement renormalization ansatz (MERA) and gradient-based optimizations. Due to its narrow causal cone, the algorithm can be implemented on noisy intermediate-scale quantum (NISQ) devices and still describe large systems. We show that the number of required qubits is independent of the system size, increasing only to logarithmic scaling when using quantum amplitude estimation to speed up gradient evaluations. Translational invariance can be used to make the computational cost square-logarithmic with respect to the system size and to describe the thermodynamic limit. The method is particularly attractive for ion-trap devices with ion shuttling capabilities. We then demonstrate it numerically for MERA with Trotterized disentanglers and isometries and find that the computational cost of such MERA quantum eigensolvers is substantially lower than that of the corresponding classical algorithms. In particular, numerical results in various strongly-correlated quantum magnet models show that it has a polynomial quantum advantage over the classical approach. In the experimental implementation, small angles in the employed two-qubit quantum gates are advantageous. We find that, by adding an angle penalty term to the energy functional, the average absolute values of the angles can be moderately reduced without significantly affecting the energy accuracies. Finally, we propose that the Trotter-type circuit in each tensor can be replaced by a parallel random circuit. However, this replacement does not seem to result in further gains as long as the tensor-network bond dimensions are small.