  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 \subseteq BPP. This type of quantum/classical interpolation phenomenon
appears to new.
