The Quantum Fourier Transform (QFT) subroutine seems ubiquitous in most quantum algorithms that are conjectured to give an exponential (or at least superpolynomial) speedup over the best classical algorithms for the same classical (non-oracle, non-promise) problem, such as Shor's algorithm, the HHL algorithm, and using the phase estimation algorithm for simulating quantum systems.
Are there any classical non-oracle non-promise problems for which a known quantum algorithm is believed to give an exponential (or superpolynomial) advantage over the best classical algorithm, which doesn't use the QFT as a subroutine? Ideally, I'd prefer a problem of practical real-world interest, but I'd also accept a problem that was constructed solely to prove a complexity-theoretic result. Oracle problems and promise problems are out of scope.
The answer to this question may be somewhat subjective, in that it depends on whether a close variant of the traditional QFT "counts". I'll say that for the purpose of this question, a valid algorithm can't use anything "like" the QFT as a subroutine, although I admit that there may be somewhat some subjectivity in just what quantum subroutines are considered to be "like" the QFT. (The Hadamard transform is perhaps a borderline case; I'll accept an algorithm that uses the Hadamard transformation, but would prefer one that doesn't.)