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Optical Memory and Neural Networks

Gradient-based feedback control of quantum systems

Gerasimos G. Rigatos

Published Apr 1, 2012• 0 citations
Publisher

Abstract

The paper proposes a gradient method for feedback control and stabilization of quantum systems using Schröodinger’s and Lindblad’s descriptions. The eigenstates of the quantum system are defined by the spin model. First, a gradient-based control law is computed using Schröodinger’s description. Next, an estimate of state of the quantum system is obtained using Lindblad’s differential equation. In the latter case, by applying Lyapunov’s stability theory and LaSalle’s invariance principle one can compute a gradient control law which assures that the quantum system’s state will track the desirable state within acceptable accuracy levels. The performance of the control loop is studied through simulation experiments for the case of a two-qubit quantum system.

DOI
10.3103/s1060992x12020087

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The paper proposes a gradient method for feedback control and stabilization of quantum systems using Schröodinger’s and Lindblad’s descriptions. The eigenstates of the quantum system are defined by the spin model. First, a gradient-based control law is computed using Schröodinger’s description. Next, an estimate of state of the quantum system is obtained using Lindblad’s differential equation. In the latter case, by applying Lyapunov’s stability theory and LaSalle’s invariance principle one can compute a gradient control law which assures that the quantum system’s state will track the desirable state within acceptable accuracy levels. The performance of the control loop is studied through simulation experiments for the case of a two-qubit quantum system.
In the design of complex quantum systems like ion traps for quantum computing, it is usually desired to stabilize a particular system state or make the system state track a desired trajectory. Several control theoretical approaches based on feedback seem attractive to solve such problems. But the uncertain dynamics introduced by measurement on quantum systems makes the synthesis of feedback control laws very complicated. Although we have not explicitly modeled the change in system dynamics due to measurement (we have assumed weak measurements), this is a first step towards a more detailed analysis and closed-loop feedback design. Here, we present a Lyapunov-based control approach on the lines of that developed by Mirrahimi, Rouchon, Turnici (2005). The states are assumed to be obtained from weak measurements. The Lyapunov control technique has not been applied to realistic quantum systems so far. We have extended and applied the technique to two realistic physical systems - the quantum harmonic oscillator and the n-qubit system. We also propose to extend this concept to ion traps.
Recently, feedback-based quantum algorithms have been introduced to calculate the ground states of Hamiltonians, inspired by quantum Lyapunov control theory. This paper aims to generalize these algorithms to the problem of calculating an eigenstate of a given Hamiltonian, assuming that the lower energy eigenstates are known. To this aim, we propose a new design methodology that combines the layer wise construction of the quantum circuit in feedback-based quantum algorithms with a new feedback law based on a new Lyapunov function to assign the quantum circuit parameters. We present two approaches for evaluating the circuit parameters: one based on the expectation and overlap estimation of the terms in the feedback law and another based on the gradient of the Lyapunov function. We demonstrate the algorithm through an illustrative example and through an application in quantum chemistry. To assess its performance, we conduct numerical simulations and execution on IBM's superconducting quantum computer.
Control of quantum systems described by the linear Schrödinger equation are considered. Control inputs enter through coupling operators and result in a bilinear control system. Feedback control laws are developed for orbit tracking. The asymptotic properties of the feedback laws are analyzed by the LaSalle-type invariance principle. Numerical integration via time-splitting is also investigated and used to demonstrate the feasibility of the proposed feedback laws and to compare their performance.
We develop a model-free framework for stabilizing quantum states using only empirical finite-difference evaluations of a measurement-derived Lyapunov observable. The controller requires no knowledge of the Hamiltonian, dissipative structure, or generator of the dynamics, and relies solely on discrete measurement data. The approach combines three key elements: sign-based Lyapunov descent, adaptive gain amplification, and a finite-difference analogue of LaSalle's invariance principle. We provide rigorous conditions under which these mechanisms guarantee asymptotic stabilization along the sampling instants in the drift-free case and practical input-to-state stability (ISS) in the presence of unknown drift and noise. The resulting feedback law is simple, derivative-free, and experimentally feasible. A qubit example illustrates the complete closed-loop scheme and the predicted ISS-type behavior. Although demonstrated on a single qubit, the theory applies to arbitrary finite-dimensional quantum systems and offers a foundation for further developments in stochastic, subspace, and multi-qudit model-free quantum control.
By using a quadratic Lyapunov function with un-known parameters and a particle swarm optimization (PSO) algorithm, this paper proposes a new Lyapunov control scheme for quantum systems. This approach can achieve high-probability population transfer to a given target state. For the case when the target state is an eigenstate of the internal Hamiltonian, we introduce a virtual control law into the system model and design corresponding control law. The stability of the system under the action of the Lyapunov control law is analyzed via the LaSalle invariance principle. For the case when the target state is a superposition state, we design a control law by performing a unitary transformation for the quantum system model under consideration. To achieve desired state transfer, we further introduce a PSO algorithm to search for the unknown parameters contained in the control law. Numerical results on a five-level quantum system and a three-qubit system are presented to demonstrate the effectiveness of the proposed approaches.