Quantum Computing

QUANTUM DOTS

QUANTUM COMPUTING

The paper that I have chosen is Daniel Loss, David P.DiVincenzo, Quantum computation with quantum dots, Physical Review 1998 57 1. The reason of my choice is because I think is a very interesting field and is completely new for me.

A quantum dot is a system of electrons fully confined in 3D with a discrete charge and electronic states. One of the reasons that the quantum dots are used for the quantum computing is because the time evolution of an arbitrary quantum state is intrinsically more powerful computationally than the evolution of a digital logic state.

The goal of this paper is to develop a detailed scenario for how quantum computing may be achieved in a coupled quantum dot system. It is provide an advance in the study of controlled nonequilibrium spin dynamics of magnetic nanosystems.

In their model the qubit is realized as the spin of the excess electron on a single-electron quantum dot .They have introduced a mechanism for twoqubit quantum-gate operation that operates by a purely electrical gating of the tunnelling barrier between neighbouring quantum dots rather than by spectroscopic manipulation as in other models. The tunnel barrier between dots can be high or low by setting a gate voltage. In the case of the high barrier potential the tunnelling is forbidden between dots (no evolution in time). In the low barrier potential spins will be subject to a transient Heisenberg coupling (Hubbard model). The equation obtained with this model gives a good description of a quantum dot system if several conditions are met: i) Single particle states of the dots can be ignored if ∆E>>KT ii) The time scale for pulsing the gate potential low longer than ħ/∆E in order to prevent transitions to higher orbital levels. iii) Charging energy of a single dot high than the tunnelling matrix elements to(t) for all t iv) Decoherence time >> time scale. If the decoherence time is long, the ideal of quantum computing may be achieved. In their model the transverse and longitudinal relaxation or decoherence rates of the system spins are the same and given by decoherence time.

Their goal is to find the linear map superoperator that connects the input state of the gate with the output state. They consider the swap operation when the two spins are coupled to a magnetic environment. They assume that it acts isotropically and is equal and independent on both dots, maybe they can have mistakes with this approximation.

The level of this paper is very high for me. I think that I have not got the necessary skills to understand all the mathematical models. I can not follow all the mathematical developments. First of all I do not know what is the Hubbard model, I have searched information about it in N. W. Ashcroft and N. D. Mermin, Solid State Physics,1976, Chap. 32, but I could not understand anything. Secondly I do not understand why they apply a particular unitary time evolution operator to the initial state of the two spins as a result of the effect of the pulsed Hamiltonian, and I do not know what the “swap operator” means. Why is not enough by itself to perform quantum computations? It is very difficult for me to know which the things that the authors do not understand are because I have not got the necessary knowledge.