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arXiv:0801.0721 · Physical Review A

Global controllability with a single local actuator

S. G. Schirmer, I. C. H. Pullen, Peter J. Pemberton-Ross

Published Dec 31, 2008• 36 citations
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Abstract

We show that we can achieve global density-operator controllability for most $N$-dimensional bilinear Hamiltonian control systems with general fixed couplings using a single, locally acting actuator that modulates one energy-level transition. Controllability depends upon the position of the actuator and relies on the absence of either decompositions into noninteracting subgroups or symmetries restricting the dynamics to a subgroup of $\mathrm{SU}(N)$. These results are applied to spin-chain systems and used to explicitly construct control sequences for a single binary-valued switch actuator.

DOI
10.1103/physreva.78.062339

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We show that we can achieve global density-operator controllability for most NNN-dimensional bilinear Hamiltonian control systems with general fixed couplings using a single, locally acting actuator that modulates one energy-level transition. Controllability depends upon the position of the actuator and relies on the absence of either decompositions into noninteracting subgroups or symmetries restricting the dynamics to a subgroup of SU(N)\mathrm{SU}(N)SU(N). These results are applied to spin-chain systems and used to explicitly construct control sequences for a single binary-valued switch actuator.
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Abstract. We provide sufficient conditions for the approximate controllability of infinite-dimensional quantum control systems corresponding to form perturbations of the drift Hamiltonian modulated by a control function. We rely on previous results on controllability of quantum bilinear control systems and obtain a priori [Formula: see text]-bounds of the controls for generic initial and target states. We apply a stability result for the nonautonomous Schrödinger equation to extend the results to systems defined by form perturbations, including singular perturbations. As an application of our results, we prove approximate controllability of a quantum particle in a one-dimensional box with a point-interaction with tuneable strength at the center of the box.
Abstract Quantum control has achieved remarkable success in state transfer, trajectory tracking, and optimal control, yet a fundamental gap remains between trajectory generation and dynamics shaping. Unlike robotic and mechanical systems, where feedback design can directly prescribe transient behaviors such as convergence rates, damping, and robustness, most quantum control methods focus on identifying control fields that achieve a desired evolution without explicitly shaping the dynamics along that evolution. Here we bridge this gap by extending the fully actuated system (FAS) paradigm from classical control to finite-dimensional bilinear quantum systems. We show that controllability does not imply full actuation in quantum dynamics: the Lie-group structure of quantum evolution generally prevents a controllable quantum system from being represented as a FAS. To overcome this limitation, we introduce virtual quantum measurements and construct a local partial FAS representation in the output space, where the highest-order output derivatives are directly assignable by the control Hamiltonians. This representation enables explicit cancellation of nonlinear input–output dynamics and provides a constructive framework for quantum dynamics shaping. The resulting closed-loop behavior can be prescribed through arbitrary pole placement or fixed-time convergence with preassigned settling-time bounds. We further develop an offline closed-loop emulation procedure that converts the feedback law into implementable open-loop control waveforms. Numerical studies on single-qubit and two-qubit systems demonstrate accurate trajectory tracking, fixed-time convergence, prescribed gate-error decay quantified by average gate fidelity, and tunable trade-offs among convergence speed, robustness, and control effort.