量子复杂性类与经典复杂性类之间的数论插值
A Number Theoretic Interpolation Between Quantum and Classical Complexity Classes
摘要：
我们揭示了一个自然的代数问题，其复杂性似乎介于著名的复杂性类BQP和NP之间：（*）决定一个具有正好m个单项项的一元多项式是否有一个p-adic有理根。特别地，我们证明了当m=2时，while（*）在量子随机多项式时间内是可行的（并且没有已知的经典随机多项式时间算法），（*）对于一般m几乎是NP困难的：在一个合理的假设下，（由某些分圆域的广义黎曼假设所暗示），（*）的随机多项式时间算法将意味着广泛不被相信的包含NP \subseteq BPP。这种类型的量子/经典插值现象似乎是新的。
英文摘要：
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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