We formulate a new concept for computing with quantum cellular automata composed of arrays of nanostructured superconducting devices. The logic states are defined by the position of two trapped flux quanta (vortices) in a 2x2 blind-hole-matrix etched on a mesoscopic superconducting square. Such small computational unit-cells are well within reach of current fabrication technology. In an array of unit-cells, the vortex configuration of one cell influences the penetrating flux lines in the

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... ring cell through the screening currents. Alternatively, in conjoined cells, the information transfer can be strengthened by the interactions between the supercurrents in adjacent cells. Here we present the functioning logic gates based on this fluxonic cellular automata (FCA), where the logic operations are verified through theoretical simulations performed in the framework of the time-dependent Ginzburg-Landau theory. The input signals are defined by current loops placed on top of the two diagonal blind holes of the input cell. For given current-polarization, external flux lines are attracted or repelled by the loops, forming the '0' or '1' configuration. The read-out technology may be chosen from a large variety of modern vortex imaging methods, transport and LDOS measurements.