  To control and measure the state of a quantum system it must necessarily be
coupled to external degrees of freedom. This inevitably leads to spontaneous
emission via the Purcell effect, photon-induced dephasing from measurement
back-action, and errors caused by unwanted interactions with nearby quantum
systems. To tackle this fundamental challenge, we make use of the design
flexibility of superconducting quantum circuits to form a multi-mode element --
an artificial molecule -- with symmetry-protected modes. The proposed circuit
consists of three superconducting islands coupled to a central island via
Josephson junctions. It exhibits two essential non-linear modes, one of which
is flux-insensitive and used as the protected qubit mode. The second mode is
flux-tunable and serves via a cross-Kerr type coupling as a mediator to control
the dispersive coupling of the qubit mode to the readout resonator. We
demonstrate the Purcell protection of the qubit mode by measuring relaxation
times that are independent of the mediated dispersive coupling. We show that
the coherence of the qubit is not limited by photon-induced dephasing when
detuning the mediator mode from the readout resonator and thereby reducing the
dispersive coupling. The resulting highly protected qubit with tunable
interactions may serve as a basic building block of a scalable quantum
processor architecture, in which qubit decoherence is strongly suppressed.
