PREPARATION AND CHARACTERIZATION OF TOPOLOGICALLY ORDERED STATE IN NOVEL QUANTUM TECHNOLOGICAL PLATFORM

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2022

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Abstract

With the rapid development of programmable quantum simulators, the quantum states can be controlled with unprecedented precision. Thus, it opens a new opportunity to explore the strongly correlated phase of matter with new quantum technology platforms. In quantum simulators, one can engineer interactions between the microscopic degree of freedom and create exotic phases of matter that presumably are beyond the reach of natural materials. Moreover, quantum states can be directly measured instead of probing physical properties indirectly via optical and electrical responses of material as done in traditional condensed matter. Therefore, it is pressing to develop new approaches to efficiently prepare and characterize desired quantum states in the novel quantum technology platforms.

In this thesis, we discuss the preparation and characterization of the topologically ordered state in nobel quantum technological platforms. First, we show that optically driven monolayer graphene in the quantum Hall regime creates an effective bilayer quantum Hall system. It provides a flexible platform for engineering quantum Hall phases. We use infinite density matrix renormalization group (iDMRG) techniques combined with exact diagonalization (ED) to show that the system exhibits a non-abelian bilayer Fibonacci phase at filling fraction $\nu = 2/3$. Moreover, at integer filling $\nu = 1$, the system exhibits quantum Hall ferromagnetism. Using Hartree-Fock theory and exact diagonalization, we show that excitations of the quantum Hall ferromagnet are topological textures known as skyrmions.

Then we turn our attention to the characterization of the topological invariants from a ground state wave function of the topological order phase and the implementation in noisy intermediate quantum devices. Using topological field theory and tensor network simulations, we demonstrate how to extract the many-body Chern number (MBCN) given a bulk fractional quantum Hall wave function. We further propose an ancilla-free experimental scheme for measuring the MBCN without requiring any knowledge of the Hamiltonian. Specifically, we use the statistical correlations of randomized measurements to infer the MBCN of a wave function.

Finally, we discuss an unbiased numerical optimization scheme to systematically find the Wilson loop operators given a ground state wave function of a gapped, translationally invariant Hamiltonian on a disk. We then show how these Wilson loop operators can be cut and glued through further optimization to give operators that can create, move, and annihilate anyon excitations. We then use these operators to determine the braiding statistics and topological twists of the anyons, yielding a way to fully characterize topological order from the bulk of a ground state wave function.

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