(BQ) Part 2 book "Introduction to quantum computers" has contents: Unitary transformations and quantum dynamics, quantum dynamics at finite temperature, physical realization of quantum computations, linear chains of nuclear spins, experimental logic gates in quantum systems, error correction for quantum computers,.and other contents. | Chapter 15 Unitary Transformations and Quantum Dynamics We can wonder what the connection is between the quantum dynamics described by the Schrodinger equation and the unitary transformations which describe the quantum logic gates. In this chapter, we shall describe their relation. Let us suppose, for simplicity, that the Hamiltonian of the system is time-independent. Then, the Schrodinger equation, ih+ = WP, () ~ ( t=)e - i x r / A \ I , ( 0 ) , () has the solution, where for any operator F it is assumed, =E +iF (iF)* ( i F ) 3 + -+-+. () 2! 3! Equation () defines the unitary transformation of the initial state e ( O ) into the final state Q ( t ) , ,iF Q ( t ) = U(t)Q(O), U ( t ) = e- i x t / h . () Consider, as an example, a spin 1/2 in a permanent magnetic field, under the action of a resonant electromagnetic pulse. The Hamiltonian 85 86 INTRODUCTION TO QUANTUM COMPUTERS of the system is given by Eq. (). We can get the time-independent Hamiltonian using the transformation to the rotating system of coordinates. This transformation can be performed using the formulas, W = UJQ, Ft = UJFU,., () where Ur is the unitary matrix of the transformation in (), ur -- e i ~ I z t 9 () W is the wave function in the rotating frame; F is an arbitrary operator in the initial reference frame; Ft is the same operator in the rotating frame; and w = 00 is the frequency of the rotating magnetic field. In our case, we make the substitution in (), This gives, () From () we get, after simplifications, the Scrodinger equation in the rotating frame, ih+' = WP', () The right side in Eq. () describes the interaction of the spin with the electromagnetic field, in the rotating frame. To simplify the right side of Eq. (), let us find the time-dependent operator, 1- - e - i y ~ I Z t ~ - e i y ) I Z t () t - 87 15 Unitarv Transformations and Quantum Dynamics For this purpose we consider the .
