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- W1625398148 abstract "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 > 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]." @default.
- W1625398148 created "2016-06-24" @default.
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- W1625398148 date "2009-08-04" @default.
- W1625398148 modified "2023-09-27" @default.
- W1625398148 title "How hard is it to approximate the Jones polynomial" @default.
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