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Control theory & applications

Optimal preparation of arbitrary quantum state of electron spins quantum-bit system based on invariant magnetic field

Ming Li

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

In the presentation of single quantum state on Bloch sphere,a strategy of optimal preparation of arbitrary quantum state of electron spin quantum systems is proposed.By using the mechanical logic gate,the manipulation of quantum system is realized by way of unitary-evolution.In the course,a switch idea for the control systems is presented. The models of quantum systems and the control systems using invariant magnetic field are then established.With the help of the Lie group and the classical control theory,a strategy of optimal preparation of arbitrary quantum state is also realized. Finally,the theoretical analysis and the simulation of the preparation indicate the advantage of the idea.

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In the presentation of single quantum state on Bloch sphere,a strategy of optimal preparation of arbitrary quantum state of electron spin quantum systems is proposed.By using the mechanical logic gate,the manipulation of quantum system is realized by way of unitary-evolution.In the course,a switch idea for the control systems is presented. The models of quantum systems and the control systems using invariant magnetic field are then established.With the help of the Lie group and the classical control theory,a strategy of optimal preparation of arbitrary quantum state is also realized. Finally,the theoretical analysis and the simulation of the preparation indicate the advantage of the idea.
Based on analyzing the presentation of the single quantum state on Bloch sphere,a strategy of optimal preparation of arbitrary quantum state of two-level closed quantum systems is proposed.With the help of quantum mechanical logic gate,manipulation of quantum system is realized,on unitary-evolution.The models of two-level quantum systems and its control systems are set up.Then,by using the Lie group,and with classical control theory and dynamics of quantum system,a strategy of optimal preparation of arbitrary quantum state is developed.The theoretical analysis and the simulation of the preparation show the advantage of the idea.
This paper studies the realization of the quantum Hadamard gate of spin with a single electron. First the canonical form of the unitary rotation gate which is feasibly applied in plane control fields is given. Because the Hadamard gate can't be realized directly by applying the plane control fields, so the matrix operator of Hadamard gate is decomposed into two matrices which are feasible in canonical forms. These two canonical matrices correspond to the rotate-operation and reflect-operation of the vector coordinates on the Bloch sphere, respectively. Then the Hadamard gate can be realized in two steps by choosing appropriate control fields, in such a way the realization of Hadamard gate becomes to design external tailored control fields. Finally, the control fields are designed based on Lyapunov method. Numerical simulation experiments and results analysis are given.
With the help of the spin-orbit interaction, we propose a scheme to perform holonomic single-qubit gates on the electron spin confined to a quantum dot. The manipulation is done in the absence (or presence) of an applied magnetic field. By adiabatic changing the position of the confinement potential, one can rotate the spin state of the electron around the Bloch sphere in semiconductor heterostructures. The dynamics of the system is equivalent to employing an effective non-Abelian gauge potential whose structure depends on the type of the spin-orbit interaction. As an example, we find an analytic expression for the electron spin dynamics when the dot is moved around a circular path (with radius RRR) on the two dimensional electron gas (2DEG) and show that all single-qubit gates can be realized by tuning the radius and orientation of the circular paths. Moreover, using the Heisenberg exchange interaction, we demonstrate how one can generate two-qubit gates by bringing two quantum dots near each other, yielding a scalable scheme to perform quantum computing on arbitrary NNN qubits. This proposal shows a way of realizing holonomic quantum computers in solid-state systems.
We study a quantum spin system acting on a single quantum bit. The evolution of this system is governed by the Schrodinger equation which takes the form of a right-invariant system on the special unitary group SU(2) with two control inputs. Using a suitable version of Pontryagin's Principle which is tailor-made for control problems on Lie groups, the optimal controls are derived in two cases: the energy-optimal case (in which the control effort is minimized for a specified end time) and the time-optimal case (in which the control duration is minimized for given constraints on the size of the controls).
We study the manipulation of two-level quantum systems. This research is motivated by the design of quantum mechanical logic gates which perform prescribed logic operations on a two-level quantum system, a quantum bit. We consider the problem of driving the evolution operator to a desired state, while minimizing the energy-type cost. Mathematically, this problem translates into an optimal control problem for systems varying on the Lie group of special unitary matrices, with cost that is quadratic in the control. We develop a comprehensive theory of optimal control for two-level quantum systems. In particular, we prove the 'normality' of the given problem and the 'regularity' of the optimal control functions. The impact of the results of the paper on nuclear magnetic resonance experiments and quantum computation is discussed.
In the last decade, a lot of effort have gone to find the most appropriate physical implementation of quantum computer. Quantum dot based system is one of the most promising system owing to its integrability with classical computation hardware and its versatility in realizing qubits and quantum gates. In this work, we propose a model for quantum dot based realization of qubits and quantum gates. The proposed system makes use of semiconductor hetrostructure based quantum dots and gate voltages to confine electrons and thus defining the qubits. Electron spin resonance is used to manipulate the spin state of the electrons. We implement single, double and triple qubit systems and two most widely used universal sets of quantum gates. We show full control of Bloch sphere in single qubit system, CNOT gate implementation in double qubit system and a novel way to implement Toffoli gate in triple qubit system.