We derive that sufficient and necessary conditions for existence of a quantum channel φ and a generalized unitary operation ε sending Ai to Bi(1≤ i≤ k) for two given families {Ai}i=1k,{Bi}i=1k of matrices,respectively.As an application,a sufficient and necessary condition for existence of a unitary duality quantum computer with given input-output states is obtained.
In this paper, we derive an upper bound for the adiabatic approximation error, which is the distance between the exact solution to a Schrodinger equation and the adiabatic approximation solution. As an application, we obtain an upper bound for 1 minus the fidelity of the exact solution and the adiabatic approximation solution to a SchrOdinger equation.
The classical adiabatic approximation theory gives an adiabatic approximate solution to the Schr6dinger equation （SE） by choosing a single eigenstate of the Hamiltonian as the initial state. The superposition principle of quantum states enables us to mathematically discuss the exact solution to the SE starting from a superposition of two different eigenstates of the time-dependent Hamiltonian H（0）. Also, we can construct an approximate solution to the SE in terms of the corresponding instantaneous eigenstates of H（t）. On the other hand, any physical experiment may bring errors so that the initial state （input state） may be a superposition of different eigenstates, not just at the desired eigenstate. In this paper, we consider the generalized adiabatic evolution of a quantum system starting from a superposition of two different eigenstates of the Hamiltonian at t = 0. A generalized adiabatic approximate solution （GAAS） is constructed and an upper bound for the generalized adiabatic approximation error is given. As an application, the fidelity of the exact solution and the GAAS is estimated.
Two linear In this letter, we prove the following conclusions by introducing a function Fn（t）： （1） If a quantum system S with a time-dependent non-degenerate Hamiltonian H（t） is initially in the n-th eigenstate of H（0）, then the state of the system at time t will remain in the n-th eigenstate of H（t） up to a multiplicative phase factor if and only if the values Fn（t） for all t are always on the circle centered at 1 with radius 1; （2） If a quantum system S with a time-dependent Hamiltonian H（t） is initially in the n-th eigenstate of H（0）, then the state of the system at time t will remain c-uniformly approximately in the n-th eigenstate of H（t） up to a multiplicative phase factor if and only if the values F,（t） for all t are always outside of the circle centered at 1 with radius 1-ε. Moreover, some quantitative sufficient conditions for the state of the system at time t to remain ε-uniformly approximately in the n-th eigenstate of H（t） up to a multiplicative phase factor are established. Lastly, our results are illustrated by a spin-half particle in a rotating magnetic field.
