A Number Theoretic Interpolation Between Quantum and Classical Complexity Classes [article]

J. Maurice Rojas
2006 arXiv   pre-print
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
more » ... on (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.
arXiv:quant-ph/0604089v2 fatcat:bqi2g2lnfrd6zeao4ziwq5owu4