Google is looking into Android 8.1 beta installation failures, as reports have been coming in from users on the Pixel User Community over at the Google Product Forums about the update not completing. According to various users, notifications came in that the update was available for download, and after proceeding with the download and installation, upon hitting the restart button to complete the update process, the phones would just reboot and they would get an update failed notification.
Based on the forum comments this appears to be happening to just Pixel users as there doesn't seem to be any mention from users about this problem with the Pixel XL, but it is happening to both Pixels purchased from Google and from Verizon so where the phone was purchased likely doesn't matter. Google says they're currently investigating the problem by gathering as much information as they can to allow them to pinpoint the cause, then correct it so users can properly install the beta software update. Google is also urging users who are having the problem to post in the Pixel User Community thread (which you can find linked at the source below) so they might reach out to individual users about bug reports or try to grab more detail about specific instances.
As of right now the most recent post from a Google employee about this issue was yesterday and that was the initial response given stating that the issue would be investigated. That means for now it doesn't appear that there is a way to fix this problem, though one user does offer up a potential workaround of doing a factory reset instead of tapping the "restart to complete update." According to the user who suggested this method, they downloaded and installed the update, then instead of tapping the button that says restart to complete, they navigated to the factory reset option in settings and after the factory reset was finished the update was installed properly and they were then on the Android 8.1 beta. Users can try this option if they wish, but it's worth keeping in mind that a factory reset is going to wipe your phone's data completely, meaning you'll lose everything that isn't backed up. Expert members in the thread are recommending users avoid doing the reset, so it might just be worth waiting for Google to figure out the problem and implement the fix.