  The spectral problem for the ${\rm AdS}_5\times {\rm S}^5$ superstring and
its dual planar maximally supersymmetric Yang-Mills theory can be efficiently
solved through a set of functional equations known as the quantum spectral
curve. We discuss how the same concepts apply to the $\eta$-deformed ${\rm
AdS}_5\times {\rm S}^5$ superstring, an integrable deformation of the ${\rm
AdS}_5\times {\rm S}^5$ superstring with quantum group symmetry. This model can
be viewed as a trigonometric version of the ${\rm AdS}_5\times {\rm S}^5$
superstring, like the relation between the XXZ and XXX spin chains, or the
sausage and the ${\rm S}^2$ sigma models for instance. We derive the quantum
spectral curve for the $\eta$-deformed string by reformulating the
corresponding ground-state thermodynamic Bethe ansatz equations as an analytic
$Y$ system, and map this to an analytic $T$ system which upon suitable gauge
fixing leads to a $\mathbf{P} \mu$ system -- the quantum spectral curve. We
then discuss constraints on the asymptotics of this system to single out
particular excited states. At the spectral level the $\eta$-deformed string and
its quantum spectral curve interpolate between the ${\rm AdS}_5\times {\rm
S}^5$ superstring and a superstring on "mirror" ${\rm AdS}_5\times {\rm S}^5$,
reflecting a more general relationship between the spectral and thermodynamic
data of the $\eta$-deformed string. In particular, the spectral problem of the
mirror ${\rm AdS}_5\times {\rm S}^5$ string, and the thermodynamics of the
undeformed ${\rm AdS}_5\times {\rm S}^5$ string, are described by a second
rational limit of our trigonometric quantum spectral curve, distinct from the
regular undeformed limit.
