Part of what's going on here is conflicting definitions of what "Turing complete" means in a formal context and how it's often used among practitioners. A spec

Part of what's going on here is conflicting definitions of what "Turing complete" means in a formal context and how it's often used among practitioners. A specific circuit has a fixed running time, but it can do arbitrary computations in that runtime, including emulating a fixed number of steps of an arbitrary Turing machine. Our physical computers also have limitations (particularly space) on what Turing machines they can simulate, however they are often called "Turing complete" in general parlance. Obviously a laptop though has limitations that are on completely different orders of magnitude though, and are much more efficient. Many "surprisingly Turing complete"[1] systems have similar limitations, but the more interesting question is "can you use this system to simulate a bounded number of steps of a Universal Turing machine?" and the answer for those systems, and FHE is yes. [1]: https://www.gwern.net/Turing-complete https://www.gwern.net/Turing-complete