  Superposition states of circular currents of exciton-polaritons mimic the
superconducting flux qubits. The phase of a polariton fluid must change by an
integer number of $2\pi$, when going around the ring. If one introduces a
${\pi}$-phase delay line in the ring, the fluid is obliged to propagate a
clockwise or anticlockwise circular current to reduce the total phase gained
over one round-trip to zero or to build it up to $2\pi$. We show that such a
$\pi$-delay line can be provided by a dark soliton pinned to a potential well
created by a C-shape non-resonant pump-spot. The resulting split-ring polariton
condensates exhibit pronounced coherent oscillations passing periodically
through clockwise and anticlockwise current states. These oscillations may
persist far beyond the coherence time of polariton condensates. The qubits
based on split-ring polariton condensates are expected to possess very high
figures of merit that makes them a valuable alternative to superconducting
qubits. The use of the dipole-polarized polaritons allows to control coherently
the state of the qubit with the external electric field. This is shown to be
one of the tools for realization of single-qubit logic operations. We propose
the design of an $i$SWAP gate based on a pair of coupled polariton qubits. To
demonstrate the capacity of the polariton platform for quantum computations, we
propose a protocol for the realization of the Deutsch's algorithm with
polariton qubit networks.
