Implementation Of A Toffoli Gate Using An Array Of Coupled Cavities In A Single Step And The Physical Study Of Topological Basis

Quantum computers provide the possibility of solving certain computational tasks much faster than any classical counterpart using the best currently known algorithms.Any quantum algorithm can be decomposed into a sequence of single-qubit rotations and entangling two-qubit logical gates,which form a universal set of quantum operations.However,if only single-qubit and two-qubit gates are available,the qubits scale up so that the approach becomes very complicated and it may be hard to implement.Using multiqubit gates can simplify this process.However,if we still require two-qubit or qubit-qutrit gates to implement the Toffoli gate so that the fidelity would be significantly deteriorated by decoherence.Therefore,it is important to implement the quantum Toffoli gate in one-step without using any two-qubit or qubit-qutrit gate.We present an efficient method implement one typical three-qubit gate the Toffoli gate(controlled-controlled-NOT gate)using interaction between atom and cavity.The quantum information is encoded into the low-lying states of identical atoms and it's convenient to address qubit individually.The large detuning between atoms and classical(quantum)fields plays an important role.Based on the Markovian master equation,we numerically show how the decay of cavities and the spontaneous emission of atoms affect the fidelity of the resulting gate.It is found that the Toffoli gate implemented by one step is robust to the decoherence.On the other hand,various spin chain models have been widely used in the simulation of quantum computing.The Heisenberg spin chain model is relatively simple.The spin chain model to be constructed in this paper is the Heisenberg XY model.In the second part of this paper,it is explained that the XY model can be constructed by linear superposition of the U matrix generators of the Temperly-Lieb algebra.Since there is a certain relationship between Temperly-Lieb algebra and topological figure,we can use topological figure to reduce the Hamiltonian of XY model to find the eigen values and eigen states of XY model.In this section,we show a graphing approach to solve the problem of XY spin chains and investigate the properties of topological ground states in this system.