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arXiv:quant-ph/0503019 · Journal of Optics B Quantum and Semiclassical Optics

Control of non-controllable quantum systems: a quantum control algorithm based on Grover iteration

Chen-Bin Zhang, Dao-Yi Dong, Zong-Hai Chen

Published Sep 21, 2005• 32 citations
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Abstract

A new notion of controllability, eigenstate controllability, is defined for finite-dimensional bilinear quantum mechanical systems which are neither strongly completely controllable nor completely controllable. Moreover, a quantum control algorithm based on Grover iteration is designed to perform a quantum control task of steering a system, which is eigenstate controllable but may not be (strongly) completely controllable, from an arbitrary state to a target state.

DOI
10.1088/1464-4266/7/10/015

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A new notion of controllability, eigenstate controllability, is defined for finite-dimensional bilinear quantum mechanical systems which are neither strongly completely controllable nor completely controllable. Moreover, a quantum control algorithm based on Grover iteration is designed to perform a quantum control task of steering a system, which is eigenstate controllable but may not be (strongly) completely controllable, from an arbitrary state to a target state.
In this paper, a new quantum hybrid controller for controlling the strongly eigenstate controllable systems, is designed.For this purpose, a Lyapunov control law is implemented when the target state is in reachable set of the initial state.On the other hand, if the target state is not in the reachable set of the given initial state, based on Grover algorithm, a new interface state that the target state is in its reachable set, will be found and the given initial state is transferred to the new interface state.Then, the new interface state is transferred to the target state, based on a designed Lyapunov control law.A theorem and a new algorithm, for implementing the presented method, is designed.Finally, a simulated example illustrates the applications and advantages of presented method.
This paper proposes a new numerical quantum optimal control method with Grover Iteration for two-level quantum systems. We show that our treatment of using the piecewise constant controls with desired control structure is implementable for laboratory laser experiments. And then, the robustness of the algorithm is also considered when parameter perturbations of quantum systems exist. The performance of the system is enhanced using the combination of the quantum optimal control and the Grover Iteration. In particular, the feasibility, effectiveness and priority of the new control scheme are presented by a comparison between our algorithm and the traditional quantum optimal control.
Quantum phenomena of interest in connection with applications to computation and communication often involve generating specific transfers between eigenstates, and their linear superpositions. For some quantum systems, such as spin systems, the quantum evolution equation (the Schrödinger equation) is finite-dimensional and old results on controllability of systems defined on on Lie groups and quotient spaces provide most of what is needed insofar as controllability of non-dissipative systems is concerned. However, in an infinite-dimensional setting, controlling the evolution of quantum systems often presents difficulties, both conceptual and technical. In this paper we present a systematic approach to a class of such problems for which it is possible to avoid some of the technical issues. In particular, we analyze controllability for infinite-dimensional bilinear systems under assumptions that make controllability possible using trajectories lying in a nested family of pre-defined subspaces. This result, which we call the Finite Controllability Theorem, provides a set of sufficient conditions for controllability in an infinite-dimensional setting. We consider specific physical systems that are of interest for quantum computing, and provide insights into the types of quantum operations (gates) that may be developed.
Realizing coherent quantum computation requires precise and robust manipulation of quantum systems through quantum control protocols. Most quantum control techniques rely on heuristic methods for designing the driving pulses that steer the system towards a target state. Such methods are often based on brute-force optimization and offer limited understanding of the solution landscape. In contrast, quantum algorithms offer a rich body of analytical methods with rigorous error guarantees for implementing unitary and non-unitary transformations, which suggests a promising direction for developing new approaches to quantum control. Among various such algorithms, quantum signal processing (QSP) has emerged as a powerful framework for quantum algorithm design, implementation, and optimization. However, its potential for quantum control remains largely unexplored. In this work, we establish QSP-Control, an analytical framework for quantum control of qubit-oscillator dynamics. We focus on dispersively coupled qubit-oscillator systems and employ the QSP formalism to mitigate unwanted nonlinear effects arising from cross-Kerr interactions. In addition, we develop constructions for precise manipulation of Fock states by designing Fock-state-selective operators, based on structural parallels between the Jaynes-Cummings interaction and QSP. These findings demonstrate how several practically relevant problems in quantum control can be mapped to forms amenable to QSP, offering both a systematic design framework and an interpretable perspective on quantum control.