Frobenius light cone and the shift unitary

We bound the time necessary to implement the shift unitary on a one-dimensional ring, both using local Hamiltonians and those with power-law interactions. This time is constrained by the Frobenius light cone; hence we prove that shift unitaries are parametrically harder to implement than preparing long-range Bell pairs for sufficiently small power-law exponent. We note an intriguing similarity between the proof of our results, and the hardness of preparing symmetry-protected topological states with symmetry-preserving Hamiltonians.
Comment: 17 pages, 1 figure