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In this work we propose a novel design for coupling of a superconducting Josephson junction qubit to the fundamental mode of a klystron microwave cavity. The cavity is a copper cylinder on the order of mm size radius with a copper axis-symmetrical post. The peculiarity of the klystron cavity consists in the electric field being concentred in a small gap (~50 micrometers) between the cavity and the post. Outside of the gap it quickly decreases with distance from the gap, whereas the magnetic field has the opposite effect, being localised around the post and weakly depending on the radial coordinate.