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In this thesis we discuss how one can derive the quantum spectral curve forthe $\eta$-deformed AdS$_5 \times S^5$ superstring, an integrable deformationof the AdS$_5 \times $S$^5$ superstring with quantum group symmetry. This modelcan be viewed as a trigonometric version of the AdS$_5 \times $S$^5$superstring, like the Heisenberg xxz spin chain is a trigonometric version ofthe xxx spin chain. Our derivation starts from the ground-state thermodynamicBethe ansatz equations and discusses the construction of both the undeformedand the $\eta$-deformed quantum spectral curve. We reformulate it first as ananalytic $Y$-system, and map this to an analytic $T$-system which upon suitablegauge fixing leads to a $\mathbf{P}\mu$ system, the quantum spectral curve. Wethen discuss constraints on the asymptotics of this system to single outparticular excited states. At the spectral level the $\eta$-deformed string andits quantum spectral curve interpolate between the AdS$_5 \times $S$^5$superstring and a superstring on "mirror" AdS$_5 \times $S$^5$, reflecting amore general relationship between the spectral and thermodynamic data of the$\eta$-deformed string. The thesis is set up such that it simultaneouslyreviews the development of the undeformed AdS$_5 \times $S$^5$ quantum spectralcurve.

Quantum spectral curve for the $\eta$-deformed AdS$_5 \times $S$^5$ superstring