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We present a new type of transmon split-junction qubit which can be tuned by Meissner screening currents in the adjacent superconducting film electrodes. The best detected relaxation time ($T_1$) was of the order of 50 $\mu$s and the dephasing time ($T_2$) about 40 $\mu$s. The achieved period of oscillation with magnetic field was much smaller than in usual SQUID-based transmon qubits, thus a strong effective field amplification has been realized. This Meissner qubit allows an efficient coupling to superconducting vortices. We present a quantitative analysis of the radiation-free energy relaxation in qubits coupled to Abrikosov vortices. The observation of coherent quantum oscillations provides strong evidence that vortices can exist in coherent quantum superpositions of different position states. According to our suggested model, the wave function collapse is defined by Caldeira-Leggett dissipation associated with viscous motion of the vortex cores., Comment: 13 pages and 9 figures; added references in the introduction section