Polynomial Quantum algorithms for additive approximations of the Potts model and other points of the Tutte plane Preliminary Version
Dorit Aharonov, Itai Arad, Elad Eban, Zeph A. Landau · 2008
In the first part of this paper, we provide polynomial quantum algorithms for additive approximations of the Tutte polynomial, at any point in the Tutte plane, for any planar graph. This includes an additive approximation of the partition function of the Potts model for any weighted planer graph at any temperature, as well as approximations to many other combinatorial graph properties described by the (multivariate or not) Tutte polynomial. To achieve these algorithms, we generalize the Temperley Lieb algebra representations, used in [6], to apply for any graph (not necessarily coming from a braid). Moreover, our representations are non-unitary, as are all representations of the Temperley Lieb algebra not corresponding to Jones polynomial related parameters. It might seem at first sight that this makes it impossible to apply them by a quantum circuit. We show how to do this nevertheless. The approximation window size turns out to be inverse polynomial in |G| times the product of the norms of the operators we apply. Additive approximations are tricky; the range of the possible outcomes, might be smaller than the size of the approximation window, in which case the outcome is meaningless. Unfortunately, ruling out this possibility is difficult: If we want to argue that our a are meaningful, we have to provide an estimate of the scale of the problem, which is difficult here exactly because no efficient algorithm for the problem exists! In the second part of the paper we provide an indirect but very convincing proof that our approximation is meaningful for a large range of parameters, by showing that in those cases, the problems