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Chinese Control Conference (CCC)

Track Control of the States Evolution of Quantum System Based on Bloch Sphere

Lou Yuesheng, Shuang Cong

Published Jul 1, 2006• 2 citations
Publisher

Abstract

The effect of control fields is analyzed based on Bloch sphere representation of a single qubit, the conditions which parameters obey are given under two different requirements in order to achieve the transition from a arbitrary initial state to a target state, and numerical simulation is also be done to get the track of evolution on Bloch sphere. The relationship between the parameters and the track is analyzed.

DOI
10.1109/chicc.2006.4347229

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The effect of control fields is analyzed based on Bloch sphere representation of a single qubit, the conditions which parameters obey are given under two different requirements in order to achieve the transition from a arbitrary initial state to a target state, and numerical simulation is also be done to get the track of evolution on Bloch sphere. The relationship between the parameters and the track is analyzed.
The orbit tracking of free-evolutionary target system in closed quantum systems is studied in this paper. Based on the concept of system control theory, the unitary transformation is applied to change the time-dependent target function into a stationary target state so that the orbit tracking problem is changed into the state transfer one. A Lyapunov function with virtual mechanical quantity P is employed to design a control law for such a state transferring. The target states in density matrix are grouped into two classes: diagonal and non-diagonal. The specific convergent conditions for target state of diagonal mixed-states are derived. In the case that the target state is a non-diagonal superposition state, we propose a non-diagonal P construction method; if the target state is a non-diagonal mixed-state we use a unitary transformation to change it into a diagonal state and design a diagonal P. In such a way, the orbit tracking problem with arbitrary initial state is properly solved. The explicit expressions of P are derived to obtain a convergent control law. At last, the system simulation experiments are performed on a two-level quantum system and the tracking process is illustrated on the Bloch sphere.
This paper is concerned with the tracking control problem for the closed single spin-1/2 quantum ensembles. Based on the Liouville-von Neumann equation, mathematical model of dynamics for single spin-1/2 quantum ensembles are formulated as a bilinear form with states evolving on Bloch sphere. An associated matrix Lie group equation describing transfers from the initial state being an identity matrix is naturally introduced. The tracking problem for the system in question is transformed to a stabilization problem such that the control techniques are able to be used. In order to deal with the tracking problem, the Lyapunov-based feedback design method is employed to obtain the control such that the dynamics of system can converge to the reference trajectory.
This thesis is concerned with the control and filtering problems for quantum systems. Two kinds of control problems are considered for a class of quantum systems: optimal control and trajectory tracking control. Filtering of a class of linear quantum systems is also investigated. \n \nFor the optimal control of quantum systems, the system in question evolves on a manifold inR^3 and is modeled as a bilinear control form whose states are represented as coherence vectors. An associated matrix Lie group system with state space S O(3) is introduced in order to facilitate solving the given problem. The controllability as well as the reachable set of the system is analyzed in detail. The maximum principle is applied to the optimal control for systems evolving on the Lie group of special orthogonal matrices of dimension 3, with cost that is quadratic in the control input. Controlling the evolutions of coherence vector is equivalent to controlling the states of an associated matrix Lie group system. Thus, the optimal control problem for single spin-1/2 systems can be transformed to steer state of the matrix Lie group system from the identity matrix to a final matrix corresponding to a target state of single spin-1/2 systems in an optimal fashion. The obtained result is employed to perform a reversible logic quantum operation NOT on single spin-1/2 systems. Explicit expressions for the optimal control are given which are linked to the initial state of the system. The main contribution of the result is to extend the existing result to a larger class of systems, generalizing existing wave-function-based results to single spin-1/2 systems whose states are represented by density matrices such that it allows the inclusion of mixed states. \n \nFor the tracking control problem of closed single spin-1/2 quantum ensembles, based on the Liouville-von Neumann equation, mathematical model of dynamics for single spin-1/2 quantum ensembles are formulated as a bilinear form with states evolving on Bloch sphere. For a typical setting in NMR, one in general would not cancel the applied electromagnetic field along a specified direction, thus leading to the dynamics in a plane perpendicular to that direction on the Bloch sphere. One natural question arises that how can one design a feedback controller such that given an initial condition for quantum ensembles, the system evolutions can be steered to the pre-determined trajectory. An associated matrix Lie group differential equation describing transfers from the initial state being an identity matrix is naturally introduced. The tracking problem for the system of interest is transformed to a stabilization problem such that the control techniques are able to be used. A distance function as well as a distance vector defined on the Lie group SO(3) are introduced, which can be used to construct a Lyapunov function to deal with the tracking problem. In order to deal with the tracking problem, the Lyapunov-based feedback design method is employed to obtain the feedback control such that the dynamics of system can converge to the reference trajectory. \n \nFor filtering of quantum systems, the filter problem of a class of linear quantum stochastic systems with quantum noise is investigated. The quantum system model is first analyzed, then the necessary conditions of nondemolition are obtained by applying the observation principle. The relations between the input and output noise are necessary for the nondemolition measurement of the quantum output channels. Given prescribed quantum system, the filter of the system is derived using those necessary conditions.
We develop a method for controlling the ensemble states of a two-level quantum system on the Bloch sphere. A spectrum of resonance frequencies parametrizes the ensemble. The evolution of the ensemble states under externally applied controls is uniquely determined by their respective resonance frequencies. Therefore, the control fields must be designed to simultaneously evolve the ensemble states across the domain of resonance frequencies. The objective is to evolve the discretized quantum ensemble states from an arbitrary initial dispersion to a targeted final configuration on the Bloch sphere. The proposed method holds potential for broad applicability in areas such as nuclear magnetic resonance (NMR), electron spin resonance (ESR), quantum computing and information processing, sampling-based learning control (SLC), quantum sensing and metrology, and quantum error correction.
Quantum computing leverages fundamental principles of quantum mechanics to perform computations far exceeding classical architectures. At the core of single-qubit quantum state manipulation is the Bloch sphere, a geometric representation mapping the state space of two-level quantum systems onto a three-dimensional unit sphere. This paper presents a comprehensive theoretical framework of qubits, starting from single-qubit geometric state vectors and expanding to rigorous multi-qubit mathematical formulations. Furthermore, it analyzes single-qubit quantum gates as unitary rotations on the Bloch sphere, evaluates multi-qubit system scaling, and explicitly proves state normalization conditions using Born's rule across multi-dimensional Hilbert spaces.
Quantum state on Bloch sphere for superconducting charge qubit, phase qubit and flux qubit for all time in absence of external drive is stable to initial state. By driving the qubits, approximation of charge and flux Hamiltonian lead to quantum state rotation in Bloch sphere around an axis completely differ from rotation vector of exact Hamiltonian. The trajectory of quantum state for phase qubit for approximated and exact Hamiltonian is the same but the expectation of quantum observable has considerable errors as two other qubits. microwave drive control is designed for approximated Hamiltonian and exerted on actual systems and shows completely different trajectory with respect to desired trajectory. Finally a nonlinear control with external μμμV voltage control and nA current control is designed for general qubit which completely stabilizes quantum state toward a desired state.