Quantum Internal Model Principle: Decoherence Control

In this article, we study the problem of designing a Decoherence Control for quantum systems with the help of a scalable ancillary quantum control and techniques from geometric control theory, in order to successfully and completely decouple an open quantum system from its environment. We re-formulate the problem of decoherence control as a disturbance rejection scheme which also leads us to the idea of Internal Model Principle for quantum control systems which is first of its kind in the literature. It is shown that decoupling a quantum disturbance from an open quantum system, is possible only with the help of a quantum controller which takes into account the model of the environmental interaction. This is demonstrated for a simple 2-qubit system wherein the effects of decoherence are completely eliminated. The theory provides conditions to be imposed on the controller to ensure perfect decoupling. Hence the problem of decoherence control naturally gives rise to the quantum internal model principle which relates the disturbance rejecting control to the model of the environmental interaction. Classical internal model principle and disturbance decoupling focus on different aspects viz. perfect output tracking and complete decoupling of output from external disturbances respectively. However for quantum systems, the two problems come together and merge in order to produce an effective platform for decoherence control. In this article we introduce a seminal connection between disturbance decoupling and the corresponding analog for internal model principle for quantum systems.Comment: Submitted to IEEE Transactions on Automatic Control, Mar 15 2010. A basic introduction appeared in 46th IEEE CDC 2007. Acknowledgements: The authors would like to thank the Center for Quantum Information Science and Technology at Tsinghua University, R.-B. Wu, J. Zhang, J.-W. Wu, M. Jiang, C.-W. Li and G.-L. Long for their valuable comments and suggestion

Strong Analytic Controllability for Hydrogen Control Systems

The realization and representation of so(4,2) associated with the hydrogen atom Hamiltonian are derived. By choosing operators from the realization of so(4,2) as interacting Hamiltonians, a hydrogen atom control system is constructed, and it is proved that this control system is strongly analytically controllable based on a time-dependent strong analytic controllability theorem.Comment: 6 pages; corrected typo; added equations in section III for representation states of so(4,2). accepted by CDC 200

Optimal Control of Gene Mutation in DNA Replication

We propose a molecular-level control system view of the gene mutations in DNA replication from the finite field concept. By treating DNA sequences as state variables, chemical mutagens and radiation as control inputs, one cell cycle as a step increment, and the measurements of the resulting DNA sequence as outputs, we derive system equations for both deterministic and stochastic discrete-time, finite-state systems of different scales. Defining the cost function as a summation of the costs of applying mutagens and the off-trajectory penalty, we solve the deterministic and stochastic optimal control problems by dynamic programming algorithm. In addition, given that the system is completely controllable, we find that the global optimum of both base-to-base and codon-to-codon deterministic mutations can always be achieved within a finite number of steps

Genetic Code Based Coding and Mathematical Formulation for DNA Computation

DNA computation is to use DNA molecules for information storing and processing. Challenges currently faced by DNA computation are (1) lack of theoretical computational models for applications, and (2) high error rate for implementation. This paper attempts to address these problems from genetic coding and mathematical modeling aspects. The proposed genetic coding approach provides a promising alternative to reduce high error rate. The mathematical formulation lays down groundwork for studying theoretical aspects of DNA computation