Controlled U-door Study, Based On The Josephson Junction

The subject of quantum computation brings together ideas from classical information theory, computer science, and quantum physics.It can solve some problems which cannot be attacked by traditional computation efficiently. The physical realization of quantum computation is done by quantum computer. The basic unit of the quantum computer is quantum gate, so it has great significant to find a suitable method to generate quantum gate for the realization of quantum computer.There are five main physical implementations of quantum logic gate by now. These include nuclear magnetic resonance, cavity-QED, ion traps, quantum dots,and Josephson junction. The generation of quantum gate can be realized by steering those physical systems. The role of a quantum gate is implementing a unitary transformation to a quantum state which can be regarded as a process of steering the quantum state from its initial state to a target state.From these we could conclude that the generation of a quantum gate can be turn into a controlling problem of a quantum system.In this paper, we study the generation of the controlled unitary gate in quantum circuit from perspective of the control theory. By the control theory of quantum mechanical system and the Cartan decomposition of the semi-simple Lie algebra su(4), the generation of arbitrary two-qubit gate can be transformed into a steering problem in a tetrahedron(the Weyl chamber) representing all the local-equivalence classes of two-qubit gate.The local equivalence classes of two-qubit gate are in one-to-one correspondence with the point in a tetrahedron except on the base, and every local equivalence class can be represented by a class vector in the Weyl chamber. First, bypassing all kinds of physical models, we study the unitary evolution of coupled qubits which can be described by exchange Hamiltonian containing Rabi terms, and derive both the Weyl chamber steering trajectory and the accompanying local rotations. Based on those above, we give the conditions which the control parameters in the Hamiltonian should be sufficed to implement a controlled unitary gate in a physical system that can be described by exchange Hamiltonian. Finally, we regard Josephson junction as the physical realization of qubit and propose a method of generating controlled unitary logic gate with coupled superconducting qubits driven by rf Rabi pulse, and then derive all the possible frequencies in the Hamiltonian needed to implement the controlled-unitary gate in the systems with capacitive coupling and inductive coupling. Through the generation of the controlled-phase shift gate and controlled-NOT gate, we demonstrate the efficiency and feasibility of the proposed method.