After I finished playing The Witcher 3 on Steam on my PC, I downloaded The Witcher 3 on Steameck, but after I opened the game and logged into the gog account, I found that there was only a NEW GAME option in the game, and it didn't sync my archive on my computer. But both PC and Steamdeck show that steam's cloud synchronization is up to date. How can I solve this problem?

It's because the old version and the new one have different steam ID and therefore separate cloud storage. And, since on steam deck every game has a separate filesystem, you have to manually copy the saves to the right path.

For Witcher 3 it'd be ~/.local/share/Steam/steamapps/compatdata/[game ID]/pfx/drive_c/users/steamuser/Documents - copy the Witcher 3 folder from Documents on your PC (through google drive or something) in there on the steam deck. The game ID is either 499450 or 292030 depending on which version you have installed.