  Dissipation in open systems enriches the possible symmetries of the
Hamiltonians beyond the Hermitian framework allowing the possibility of novel
non-Hermitian topological phases, which exhibit long-living end states that are
protected against disorder. So far, non-Hermitian topology has been explored
only in settings where probing genuine quantum effects has been challenging. We
theoretically show that a non-Hermitian topological quantum phase can be
realized in a reservoir-engineered transmon chain. The spatial modulation of
dissipation is obtained by coupling each transmon to a quantum circuit
refrigerator allowing in-situ tuning of dissipation strength in a wide range.
By solving the many-body Lindblad master equation using a combination of the
density matrix renormalization group and third quantization approaches, we show
that the topological end modes and the associated phase transition are visible
in simple reflection measurements with experimentally realistic parameters.
Finally, we demonstrate that genuine quantum effects are observable in this
system via robust and slowly decaying long-range quantum entanglement of the
topological end modes, which can be generated passively starting from a locally
excited transmon.
