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Lyapunov Control of Quantum Systems with Applications to Quantum Computing

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.