What they actually built was a 2-bit wide machine with one instruction. No, they can't run Doom, which they mention a lot.
There's a lot of hand-waving about memory, around page 10. They seem to have used a delay line, which is very slow; you have to wait for the bits you want to come around. That's been a classic problem with photonics. You can build gates, which is nice for switching packets, but how do you store data?
Much of the architectural discussion is about what you can do if memory is mostly ROM. They talk about fast-read, really slow write memory. Here's an article about building something like that.[2] It's a clunky technology. Writing involves on-chip heaters and switching memory cells back and forth from amorphous to crystalline. There's a long history of forgotten devices like that - photochromic memory, UV-erasable EEPROMS, rewritable DVDs, Ovonics, etc. All were superseded by something with better read-write properties.
The underlying device technology is not theirs. It's from the Cornerstone project.[1]