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A possibility of performing the C-NOT gate
operation at the ground and the first excited states of two harmonic oscillators interacting via a two-level system subject to complete control is demonstrated. The system resembles Turing machine, where the result of interaction between oscillators and the two level system is restricted to a certain fixed unitary transformation matrix, while all the control required for the implementation of the gate is provided via manipulations with the two-level system, which remains the only fully-controllable part of the entire system. Each gate operation requires a "Turing programming", - it can be realized as a series of > 63 elementary unitary operations. The result shows a way how one can construct a quantum processor in a multimode microwave cavity equipped with a fully controlled two-level system, such as Josephson junction chip. Parameters of already existing experimental devises could allow one to perform up to 15 gate operations in an ensemble of about 10 qubits.

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