How hard is it to approximate the Jones polynomial?

TL;DR

This paper proves that approximating the Jones polynomial at certain roots of unity is #P-hard, linking quantum computational universality with classical complexity, and showing the difficulty of value-dependent approximation.

Contribution

It establishes the #P-hardness of any value-dependent approximation of the Jones polynomial at non-lattice roots of unity, connecting quantum universality with classical complexity classes.

Findings

Any value-dependent approximation is #P-hard.

Deciding whether |V(L,t)| > a or < b is computationally hard.

The result follows from universality and Aaronson's theorem.

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 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].