Quantized contact transformations are Toeplitz operators over a contact manifold $(X,\alpha )$ of the form $U_{\chi }=\Pi A\chi \Pi$, where $\Pi :H^2\left(X\right)\to L^2\left(X\right)$ is a Szegö projector, where $\chi$ is a contact transformation and where $A$ is a pseudodifferential operator over $X$. They provide a flexible alternative to the Kähler quantization of symplectic maps, and encompass many of the examples in the physics literature, e.g. quantized cat maps and kicked rotors. The index problem is to determine $\mathrm{ind}\left(U_{\chi }\right)$ when the principal symbol is unitary, or equivalently to determine...