  A phase-slip flux qubit, exactly dual to a charge qubit, is composed of a
superconducting loop interrupted by a phase-slip junction. Here we propose a
tunable phase-slip flux qubit by replacing the phase-slip junction with a
charge-related superconducting quantum interference device (SQUID) consisting
of two phase-slip junctions connected in series with a superconducting island.
This charge-SQUID acts as an effective phase-slip junction controlled by the
applied gate voltage and can be used to tune the energy-level splitting of the
qubit. Also, we show that a large inductance inserted in the loop can reduce
the inductance energy and consequently suppress the dominating flux noise of
the phase-slip flux qubit. This enhanced phase-slip flux qubit is exactly dual
to a transmon qubit.
