ON THE QUANTUM COMPLEXITY OF EVALUATING THE TUTTE POLYNOMIAL
Abbreviated Journal Title
J. Knot Theory Ramifications
Quantum algorithm; quantum complexity; Tutte polynomial; link invariants; JONES; COMPUTATION; Mathematics
We show that the problem of approximately evaluating the Tutte polynomial of triangular graphs at the points (q, 1/q) of the Tutte plane is BQP-complete for (most) roots of unity q. We also consider circular graphs and show that the problem of approximately evaluating the Tutte polynomial of these graphs at the point (e(2 pi i/5), e(-2 pi i/5)) is DQC1-complete and at points (q(k), 1 + 1-q(-k)/(q(1/2)-q(-1/2))(2)) for some integer k is in BQP. To show that these problems can be solved by a quantum computer, we rely on the relation of the Tutte polynomial of a planar G graph with the Jones and HOMFLY polynomial of the alternating link D(G) given by the medial graph of G. In the case of our graphs the corresponding links are equal to the plat and trace closures of braids. It is known how to evaluate the Jones and HOMFLY polynomial for closures of braids. To establish the hardness results, we use the property that the images of the generators of the braid group under the irreducible Jones-Wenzl representations of the Hecke algebra have finite order. We show that for each braid b we can efficiently construct a braid (b) over tilde such that the evaluation of the Jones and HOMFLY polynomials of their closures at a fixed root of unity leads to the same value and that the closures of (b) over tilde are alternating links.
Journal of Knot Theory and Its Ramifications
"ON THE QUANTUM COMPLEXITY OF EVALUATING THE TUTTE POLYNOMIAL" (2010). Faculty Bibliography 2010s. 6932.