Quantum Simulation Of Topological Physics

Quantum information technique has developed quickly in the past decades, and people have achieved fruitful results in both theory and experiment, this promotes the development of fundamental physics, information science, material science, et al. As the main research branches in quantum information technique, developing large-scale quantum computing and simulating complex quantum system have attracted re-searchers’most attentions. The parallel nature of quantum computer ensures its power-ful computational ability to solve problems that are far beyond the classical computer ability, but the physical realization of quantum computer must meet rigorous technique requirements. Topological quantum computation, a scheme to realize quantum compu-tation based on topological physics, is fault tolerant, this makes it as one of the most important computing schemes.Quantum simulation was first proposed by Richard Feynman, by constructing a quantum simulator, i.e., an artificial quantum system with more controllable parame-ters, one can simulate and study a realistic physical system which is less controllable and hard to study by current experimental technique, the technique requirements for quantum simulators is much lower than that for quantum computers. A quantum sim-ulator can naturally solve the problems associated with the corresponding quantum many-body system, which can not solved by classical computers. Quantum simulation allow us not only to study the existed physical systems, but also to construct new physi-cal modes with new phenomenons. Currently, there are various experimental platforms used for quantum simulation, such as ultracold neutral atoms, trapped ions, electrons, photons et al. Topological physics play a central role in the study of fundamental and applied physics, and are difficult to investigate because of the stringent experimental conditions required. Quantum simulation offers a powerful tool to overcome this diffi-culty and enable in-depth study of topological physics.In this thesis, we studied the quantum simulation of topological physics and quan-tum computation based on the topological properties, using superconducting circuits and photonic circuits. We mainly concentrate on the simulation of quantum systems with non-trivial topological properties. The details are as follows:1. We give a brief review of some basic concepts in quantum computing, and quantum memories, particularly, we will introduce the basic idea of topological quan- turn computing and the concept of anyon. We also introduce various experimental platforms used for quantum simulation, and we show some important results in quan-tum simulation of topological physics. We give the derivation of the Hamiltonian for photonic circuit, and its application in quantum simulation.2. In quantum simulation, one of the most interesting problem is the simulation of many-body system with topological properties. Especially, the simulation of two di-mensional strong correlated systems with anyon excitations has attracted people’s great concern. On one hand, because anyons have exotic statistic properties, they play an im-portant role in fundamental physics, on the other hand, non-Abelian anyons can be used as the basic qubits for topological protected quantum computing. As the quasi-particle excitation existing in two dimensional system and obeying fractional statistics, anyon was predicted to appear in fractional quantum Hall system and strong correlated lattice system. People have dynamically simulated the Abelian anyon using photonic system, but experimental observation of more interesting non-Abelian anyon is still blank. S3group is the smallest non-Abelian group for universal topological quantum computa-tion. Here we proposed a scheme to dynamically simulate the non-Abelian anyons in quantum double model. According to proper quantum control of the many-body states of the system, we can realize the creation, exchange, detection of non-Abelian anyon based on S3group, and prove the existence of non-Abelian and their statistic property. We give the possible physical realization of our scheme using superconducting circuit, and also give the smallest scale of a system that is sufficient for proof-of-principle demonstration of our scheme.3. Topological physics are at the heart of extraordinary quantum phenomena that arise in2D systems subject to external gauge fields. In two dimensional system, gauge field is key factor for quantum (spin) Hall effect and topological insulator. The simula-tion of gauge field helps us to investigate topological physics, including the Hofstadter’s butterfly spectrum, chiral edge state, topological Chern number, et al. Edge states are very important because of their potential application for dissipationless transport. In cold atomic system, people have done many works, and gauge field has been realized in experiment, but the observation of topological properties require more demanding conditions, such as extremely low temperature and low noise, besides, the limited de-tection technique in cold atomic system also makes the observation of topological state being challenging. Schemes based on superconducting circuit and optical cavities have also been proposed, but experimentally, it’s hard to realize a large system suitable for simulation. Here we proposed a scheme to simulate two dimensional lattice models in gauge field, using a one dimensional optical cavity array. By making use of photon’s inner degrees of freedom, the orbital angular momentum (OAM), we can simulate a large-scale two dimensional lattice with only a small one dimensional coupled cav-ity array, which reduces the experimental resources significantly. OAM number can be very large, this increases the feasible scale of simulation. The tunneling between differ-ent OAM states can be realized by spatial light modulator (SLM). The tunneling phase are introduce by imbalance optical path of the tunneling, which will create the proper gauge field. We show that the Hofstadter’s butterfly band structure, chiral edge state and topological Chern number can be detected by measuring the photon transmission spectroscopy. The topological quantum phase transition is also studied. Our scheme offers a new platform for studying topological physics.