Quantum Simulation Of Topological Phases And Topological Quantum Computation Based On Nuclear Magnetic Resonance

A quantum computer based on the principles of quantum mechanics has in-tense superiority over a classical computer in computational power. However, one of the major obstacles to realize a quantum computer is the noise and decoher-ence. The noise arises from the imperfection of the operations acting on qubits and quantum decoherence arises from the inevitable interactions between system and environment. The noise and quantum decoherence might make the encoded information loss or even completely wrong. One of solutions is quantum error correction. But it needs to introduce some auxiliary qubits and more noises will added in the error correction process itself. An alternative strategy is topological quantum computation. Unlike the active behaviors of quantum error correction, topological quantum computing does not try to make the system noiseless, but instead makes it deaf, that is. immune to usual sources. So it is passive, due to the globally robust topological nature of the computation. Topological quantum computing has the highest known tolerable error rate.The scheme of topological quantum computation depends on the existence of the topological phases. Topological phases are exotic phases of matter which are beyond the usual symmetry description. These phases have some interesting properties, such as robust ground state degeneracy that depends on the surface topology, quasiparticle fractional statistics, topological entanglement entropy, and so on. A special kind of topology phases of matter with energy gap is called topo-logical order, which corresponds to the pattern of long-range entanglement. A well-known example is the fractional quantum hall states, which is the first obser-vation of topological order in the natural systems. Besides, some two-dimensional lattice models exhibit topological orders as well. Topological order not only pro-vides a natural medium for fault-tolerant quantum computation, but also plays a significant role in the basic scientific research of condensed matter physics, such as topological properties, topological phase transitions and the discovery of new materials, etc.It is an extremely challenge to observe topological phases and implemen-t topological quantum computation in a natural system, mainly limited to the technical level of the experimental control. For instance, two-dimensional lattice models with topological orders usually involve many-body interactions (e.g., toric code model has four-body interactions), and multi qubits are required to build a lattice. Therefore, it is not an easy thing to engineer and control such complicated multi-qubit systems. Due to this difficulty, there is no related experimental report on topological quantum computation besides the theoretical works.Quantum simulation suggests that the complicated or inaccessible physical phenomena can be simulated by a controlled system. Many applications have successfully demonstrated in condensed matter physics, high energy physics and quantum chemistry. Quantum simulation will provide a powerful means to explore topological phases and their application in topological quantum computation. On the other hand, nuclear magnetic resonance, as one physical implementation of quantum simulation, is a good test platform due to its sophisticated control and precise measurement in multi-qubit experiments.In this thesis, I introduce a series of theoretical and experimental researches on quantum simulation of topological phases and topological quantum computa-tion based on nuclear magnetic resonance systems. The main achievements are as follows:1. The experimental researches on quantum simulation of topological phases: (ⅰ) We first report an experimental implementation of adiabatic passage be-tween different topological orders in the Wen-plaquette spin model. This is a novel quantum phase transition that cannot be described by Landau’s symmetry-breaking theory. Our work presents an important new step in the development of quantum simulation, corresponding to a type of system with topological order of great current interest. See section 3.1 in detail, (ⅱ) We experimentally observe a series of dynamical quantum Hall effects in two to four spin one-dimensional Heisenberg chains. In experiments, Berry curva-ture in the parameter space of Hamiltonian is probed by means of dynamical response and then the first Chern number is extracted by integrating the curvature over the closed surface. From the resulting Chern number, we can visualize the geometric structure of Hamiltonian. The precise quanti-zation of first Chern number may be applied to parameter estimation of Hamiltonian. See section 3.2 in detail, (ⅲ) We show a practical measuring method to identify topological order for a given lattice model using a nuclear magnetic resonance simulator. One of most important tasks in condensed matter theory is to characterize the emergent topological orders in interact-ing many-body quantum systems. However, there is no such experiment so far, due to the intrinsic difficulty in measurement. Central to our scheme is to extract the nonabelian geometric phases from the degenerate ground-state subspace on a torus. The information of topological order, including quantum dimension of quasiparticles, fusion rules and braiding statistics, can be obtained from the resulting modular matrices. See section 3.3 in detail.2. The researches on topological quantum computation:(i) We engineer the time-dependent Hamiltonian of such models and prepare a topologically ordered state through the adiabatic evolution. The other sectors in degen-erate ground-state space are realized by performing nontrivial closed string operations. Each sector is highly entangled, as shown from the complete-ly reconstructed density matrices. This paves the way towards exploring topological properties and constructing a robust quantum memory based on topological degeneracy and long-range entanglement of its ground state. See section 4.1 in detail, (ⅱ) We theoretically analyze two key properties of topological quantum memory from its robustness against local perturba-tion and thermal noise, and further consider the possibility of experimental demonstration with a concrete example. See section 4.2 in detail.3. The development of a new four-qubit sample and the automatic pulse gener-ating program. It not only improve the precision of the experiment control, but also reduce the complexity of pulse design. The sample and related pro-gram has been used widely in laboratory and play a important significance in the experiment. See appendix A and B in detail.