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arXiv:math/0701056 · IEEE Transactions on Automatic Control

Flatness-Based Control of a Single Qubit Gate

Paulo Sergio Pereira da Silva, Pierre Rouchon

Published Apr 1, 2008• 18 citations
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Abstract

This work considers the open-loop control problem of steering a two-level quantum system from any initial to any final condition. The model of this system evolves on the state space , having two inputs that correspond to the complex amplitude of a resonant laser field. A symmetry preserving flat output is constructed using a fully geometric construction and quaternion computations. Simulation results of this flatness-based open-loop control are provided.

DOI
10.1109/tac.2008.917650

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This work considers the open-loop control problem of steering a two-level quantum system from any initial to any final condition. The model of this system evolves on the state space , having two inputs that correspond to the complex amplitude of a resonant laser field. A symmetry preserving flat output is constructed using a fully geometric construction and quaternion computations. Simulation results of this flatness-based open-loop control are provided.
Abstract—This work considers the open-loop control problem of steering a two-level quantum system from any initial to any final condition. The model of this system evolves on the state space X = SU(2), havingtwo inputs that correspond to the complex amplitude of a resonant laser field. A symmetry preserving flat output is constructed using a fully geometric construction and quaternion computations. Simulation results of this flatness-based open-loop control are provided. Index Terms—Flatness, geometric control, nonlinear systems, quantum control, qubit gate. I.
In the design of quantum computing devices of the future the basic element is the qubit. It is a two-level quantum system which may describe population transfer from one steady-state to another controlled by a coherent laser field. A four-dimensional real-variable differential equation model is constructed from the complex-valued two-level model describing the wave function of the system. The state transition matrix of the model is constructed via the Wei-Norman technique and Lie algebraic methodology. The idea of parametrization using flatness-based control, is applied to construct feasible input-output pairs of the model. This input drives the state of the system from the given initial state to the given final state in a finite time producing the corresponding output of the pair. The population transfer is obtained by nullifying part of the state vector via careful selection of the parameter functions.
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.
The coherence-preserving of two-level open quantum systems is recognized as a key to preserve the striking feature of the parallel character of the quantum information processing,which is crucial for implementing the quantum theory into reality. Aiming at the problem,in order to reduce this unexpected decoherence effect,the quantum master equation was mainly investigated. The orthogonal basis of geometric algebra was used to convert the quantum master equation into the state-space model; in which the useful coherent vector was attained. Then the corresponding recursive optimal control algorithm was designed to achieve both the maximal suppression of decoherence and the control energy minimizing based on a typical two-level quantum system. Afterwards,some tips of how to choose the parameters were discussed. Simulation results verify its effectiveness and superiority to coherence-preserving.