A Number Theoretic Interpolation Between Quantum and Classical Complexity Classes

Details A Number Theoretic Interpolation Between Quantum and Classical Complexity Classes A Number Theoretic Interpolation Between Quantum and Classical Complexity Classes

2006-04-12

arxiv.org/abs/quant-ph/0604089v2

Physics

Quantum Physics

14 pages, no figures

Scientific paper

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.

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