Sony QX1 has a Sony E (NEX) lens mount and currently there are 218 native lenses available for this mount.
The latest released lenses that are compatible with the QX1 are the Samyang 14mm F2.8, the Samyang 8mm F2.8 Fisheye and the Samyang 85mm F1.4.
Sony QX1 doesn't have a sensor based image stabilization system but
43 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