2006/04/12 by Rojas, J. Maurice
#FOS: Mathematics #FOS: Physical sciences #Number Theory (math.NT) #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.quant-ph/0604089
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.