The implementation of fault-tolerant quantum gates on encoded logic qubits isconsidered. It is shown that transversal implementation of logic gates based onsimple geometric control ideas is problematic for realistic physical systemssuffering from imperfections such as qubit inhomogeneity or uncontrollableinteractions between qubits. However, this problem can be overcome byformulating the task as an optimal control problem and designing efficientalgorithms to solve it. In particular, we can find solutions that implement allof the elementary logic gates in a fixed amount of time with limited controlresources for the five-qubit stabilizer code. Most importantly, logic gatesthat are extremely difficult to implement using conventional techniques evenfor ideal systems, such as the T-gate for the five-qubit stabilizer code, donot appear to pose a problem for optimal control.
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