Quantum Circuits for Chiral Topological Order

Abstract

Quantum simulation stands as an important application of quantum computing, offering insights into quantum many-body systems that are beyond the reach of classical computational methods. For many quantum simulation applications, accurate initial state preparation is typically the first step for subsequent computational processes. This dissertation specifically focuses on state preparation procedures for quantum states with chiral topological order, states that are notable for their robust edge modes and topological properties. These states are interesting due to their profound connections to the behavior of electrons and spins in real-world solid-state materials. In this dissertation, we explore a type of state preparation procedure known as entanglement renormalization circuits. This class of quantum circuits is characterized by its hierarchical arrangement of quantum gates (or quantum operations in general), which systematically organize and prepare the entanglement of the target states across various length scales.

In the first part of the dissertation, we present an entanglement renormalization circuit for a non-interacting chiral topological system. The non-interacting chiral topological system we consider is a continuous Chern insulator model, which can serve as a toy model for the integer quantum Hall effect. The entanglement renormalization circuit for the continuous Chern insulator is the continuous multiscale entanglement renormalization ansatz (cMERA). The cMERA circuit, adapted for field theories, provides a natural framework for quantum systems that are continuous in momentum space. One of the key findings of this work is that we find a scale-invariant cMERA for which the continuous Chern insulator is a fixed-point wavefunction, a property that is believed to be impossible within the traditional lattice multiscale entanglement renormalization ansatz (MERA) framework. Furthermore, we provide an experimental proposal to realize the cMERA circuit using cold atoms.

In the second part of this dissertation, we shift our focus to entanglement renormalization circuits for interacting chiral topologically ordered states. We analytically derive a class of exactly solvable chiral spin liquids, classified under Kitaev's 16-fold way. Some of these chiral spin liquids share universal properties with certain fractional quantum Hall states. We then construct entanglement renormalization circuits for these chiral spin liquids by combining traditional MERA circuits with time-dependent quasi-local Hamiltonians. We refer to this class of circuits as the multiscale entanglement renormalization ansatz with quasi-local evolution (MERAQLE).