Entanglement complexity of quantum states, dynamics and quantum computation

Quantum entanglement has become the key notion bridging originally distinct fields of research over the last decade, namely, quantum information and computation, condensed matter physics, and quantum gravity. Previous studies on quantum entanglement have largely focused on the entanglement entropy, which quantifies the amount of entanglement. However, a natural question arises: is there additional information of a quantum state that is not captured by the entanglement entropy alone? For ground states of gapped Hamiltonians, this question has been answered in the affirmative. In this dissertation, I extend this idea to study highly entangled states typically having volume law entropy, and demonstrate that there is indeed much richer information on the complexity of a quantum state beyond the entanglement entropy. In the first part, I study the entanglement spectrum of highly entangled states corresponding to highly excited eigenstates of non-integrable Hamiltonians, time-evolved states after a quantum quench with Hamiltonians exhibiting different dynamical phases, and random unitary circuits consisting of random braids of non-Abelian anyons. I demonstrate that the entanglement spectrum is able to capture the degree of randomness of a quantum state, which we call the entanglement complexity. In the context of scrambling, this quantifies the degree of randomness produced by scrambling beyond entropic diagnostics. Our understanding of quantum entanglement in condensed matter systems and high energy physics have largely benefited from the field of quantum computation. In the second part of the dissertation, I present two examples of novel platforms for quantum computation using state-of-the-art experimental technologies. I demonstrate how one can use hybrid quantum-classical architecture to solve computational problems based on an optimal variational ansatz of the evolution protocol. I also present a hierarchical architecture of constructing logical Majorana zero modes which can be used for demonstrating non-Abelian braiding statistics experimentally