近似琼斯多项式有多难？
How hard is it to approximate the Jones polynomial?
摘要：
Freedman、Kitaev和Wang [arXiv：quant-ph/0001071]，以及后来的Aharonov、Jones和Landau [arXiv：quant-ph/0511096]，建立了一个量子算法来“加性地”近似琼斯多项式V（L，t）在单位t的任何主根处。这种加法近似的强度指数地依赖于链路表示的桥数。Freedman、Larsen和Wang [arXiv：math/0103200]确定了近似对于非格点的量子计算是普适的，即单位主根; Aharonov和Arad [arXiv：quant-ph/0605181]建立了这个结果的统一版本。在这篇文章中，我们证明了琼斯多项式在这些非格单位根上的任何依赖于值的逼近都是P-困难的。如果有权力决定|V（L，t）|> a或|V（L，t）|b< b for fixed constants a >> 0，则存在多项式时间算法来精确计数任意组合方程的解。在我们的论证中，结果相当直接地从普适性结果和Aaronson定理得出，即PostBQP = PP [arXiv：quant-ph/0412187]。
英文摘要：
Freedman, Kitaev, and Wang [arXiv:quant-ph/0001071], and later Aharonov, Jones, and Landau [arXiv:quant-ph/0511096], established a quantum algorithm to "additively" approximate the Jones polynomial V(L,t) at any principal root of unity t. The strength of this additive approximation depends exponentially on the bridge number of the link presentation. Freedman, Larsen, and Wang [arXiv:math/0103200] established that the approximation is universal for quantum computation at a non-lattice, principal root of unity; and Aharonov and Arad [arXiv:quant-ph/0605181] established a uniform version of this result. In this article, we show that any value-dependent approximation of the Jones polynomial at these non-lattice roots of unity is #P-hard. If given the power to decide whether |V(L,t)| > a or |V(L,t)| < b for fixed constants a > b > 0, there is a polynomial-time algorithm to exactly count the solutions to arbitrary combinatorial equations. In our argument, the result follows fairly directly from the universality result and Aaronson's theorem that PostBQP = PP [arXiv:quant-ph/0412187].
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