IBM Research has published a write-up, dated 15 September, of a mathematical result separating what shallow quantum circuits can do from what shallow classical language models can do. The underlying paper went up on arXiv in August.
There are two results. The first is functional: the authors exhibit a function computable by a constant-depth quantum circuit followed by a single classical AND gate, which a constant-depth transformer can only compute if its width grows polynomially with the length of the input. The task is an iterated index problem, where each lookup returns the position of the next. The second is distributional: a probability distribution that constant-depth quantum circuits can sample from, but which shallow diffusion language models cannot reproduce within a constant statistical distance, even given chain-of-thought and token remasking.
The caveat is one the authors state plainly. The work is entirely theoretical and no models were run. These are statements about constant-depth architectures, a strong restriction that deployed systems do not obey.
It is still worth noting. Most public claims about what language models cannot do are extrapolations from prompts that failed. This is a rare case where the limit is proved and its scope stated precisely.