Interpolating Between Quantum and Classical Complexity Classes

We reveal a natural algebraic problem whose complexity appears to interpolate between the well-known complexity classes BQP and NP: ⋆ Decide whether a univariate polynomial with exactly m monomial terms has a p-adic rational root. In particular, we show that while (⋆) is doable in quantum randomized polynomial time when m=2 (and no classical randomized polynomial time algorithm is known), (⋆) is nearly NP-hard for general m: Under a plausible hypothesis involving primes in arithmetic progression (implied by the Generalized Riemann Hypothesis for certain cyclotomic fields), a randomized polynomial time algorithm for (⋆) would imply the widely disbelieved inclusion NP⊆BPP. This type of quantum/classical interpolation phenomenon appears to new.