The spectral problem for the AdS_5× 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 η-deformed AdS_5× S^5 superstring, an integrable deformation of the AdS_5× S^5 superstring with quantum group symmetry. This model can be viewed as a trigonometric version of the AdS_5× S^5 superstring, like the relation between the XXZ

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... nd XXX spin chains, or the sausage and the S^2 sigma models for instance. We derive the quantum spectral curve for the η-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 Pμ 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 η-deformed string and its quantum spectral curve interpolate between the AdS_5× S^5 superstring and a superstring on "mirror" AdS_5× S^5, reflecting a more general relationship between the spectral and thermodynamic data of the η-deformed string. In particular, the spectral problem of the mirror AdS_5× S^5 string, and the thermodynamics of the undeformed AdS_5× S^5 string, are described by a second rational limit of our trigonometric quantum spectral curve, distinct from the regular undeformed limit.