  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].
