Sony QX1 has a Sony E lens mount and currently there are 363 native lenses available for this mount.
The latest released lenses that are compatible with the QX1 are the Tamron 12-20mm f2.8, the Viltrox 26mm f2.8 FE and the SG-image AF 35mm f2.2.
Sony QX1 doesn't have a sensor based image stabilization system but
54 of these lenses already comes with optical image stabilization.
Since Sony QX1 has a APS-C sensor, it has a focal length multiplier of 1.5× so you have to multiply lenses original focal length with this multiplier to find the Full Frame equivalent focal length of the lens when mounted on QX1.
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Sony QX1 Lenses with Image Stabilization
Sony QX1 Lenses with Weather Sealings