~~By OS to put on it I'm assuming you mean non-android OS~~ (edit: I just reread the post, though FWIW I think android technically counts as a Linux distribution, it's just not GNU/Linux for lack of a better distinction)
The processor in the Pixel 1 is the Snapdragon 821, which has not been officially supported since a decade ago. Qualcomm are notorious for not providing open drivers for their chipsets, so getting it to run anything that isn't binary compatible with the driver blob it comes with, is a huge undertaking. From my reading, no one ever completed that reverse engineering work for the pixel 1 before development effort dried up.
I believe you might find some lineage OS releases got the phone through a few extra versions of Android, but there's very little chance you're going to run anything that isn't android on it, unless you're a particularly talented engineer with a lot of free time