A PS1 emulator with the ability to upscale games to 4K has recently been made available to manually install on Xbox Series X and S.
The emulator, which is called DuckStation, is able to not only emulate PS1 games but also internally upscale them up to native 4K at 60FPS, although the emulator's default is to cap games at their original framerate. However, as this is a clearly unofficial program, if you were to give it a go, you'll need to put your Xbox Series X and Xbox Series S into developer mode to basically use the console as an Xbox development kit.
DuckStation was developed at the end of last year and was created with a focus on “playability, speed, and long-term maintainability.” The developer of the project; Stenzek, has also added on the emulator’s Github page that “the goal is to be as accurate as possible while maintaining performance suitable for low-end devices.”
In a video by Modern Vintage Gamer (via VGC), we get to see the PS1 emulator in action on an Xbox Series S as well as get an explanation of how this emulating software works. As the video explains, DuckStation recently got a UWP (Universal Windows Platform) Xbox Port which allows the software to run on the likes of the Xbox One, and next-gen Xbox consoles and therefore also emulate PS1 games on the consoles too.
It should be noted however that emulation is a legally grey area that has seen certain sites offering emulated games having lawsuits filed against them. Nintendo recently demonstrated this when it issued an injunction against a ROM website owner and demanded that all illegal games should be "permanently destroyed." This is after Nintendo managed to successfully win its case in court against the owner of the ROM site who also has to compensate Nintendo $2.1 million due to violating trademarks and copyrights.
For a trip down memory lane, take a look at our best PS1 games list.