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• W85898295 abstract "The quantum algorithm of AJL [3] (following the work of Freedman et al. [10]) to approximate the Jones polynomial, is of a new type: rather than using the quantum Fourier transform, it encodes the local combinatorial structure of the problem by the relations of an algebra, called the Temperley-Lieb algebra, whose matrix representation is then applied, using a quantum computer. By the results of [11], the problems this algorithm solves are the first non trivial BQP-complete problems. A natural question is whether it is possible to generalize this algorithm to other points of the famous Tutte polynomial, of which the Jones polynomial is a special case. However, the only points for which unitary representations of the Temperley-Lieb algebra exist are exactly those points to which the AJL algorithms apply. It seems counter-intuitive to be able to apply non-unitary representations in a quantum algorithm, and even more so to be able to show that these non-unitary algorithms are BQP-complete. In the first part of this work we present two approaches that aim to generalize the results of AJL to other points of the Tutte polynomial. The first approach uses interpolation techniques in order to approximate the polynomial on an interval. The second approach uses a transformation of graphs, namely graph tensor, in order to approximate a discrete set of points of the Tutte polynomial that were not accessible using the algorithm of AJL. Both approaches achieve an approximation scale which is exponential in the number of edges of the graph. This scale is of the same order as the trivial bound and we aimed to improve it. The main result of the present work is a polynomial quantum additive approximation of the Tutte polynomial, at any point, for any planar graph. This is done by first generalizing the Temperley-Lieb algebra so that any planar graph can be dealt with and then observing that non-unitary operators can in fact be applied. We note that by the result of Aharonov, Arad, Landau and the author [2], the approximation of many points of the Tutte polynomial is BQP-hard. However, this result is not described in this work. An implication is an additive approximation of the partition function of the Potts model with any set of couplings at any temperature. However, the question of the complexity of this approximation is not fully settled; as the techniques used in [2] to prove BQP-hardness are not known to apply to the parameters of the Potts model. Thus, the exact quality of our algorithm regarding the Potts model is yet to be determined. In addition, our result implies approximations to many other combinatorial graph properties captured by the (multivariate or not) Tutte polynomial." @default.
• W85898295 created "2016-06-24" @default.
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• W85898295 date "2007-01-01" @default.
• W85898295 modified "2023-09-27" @default.
• W85898295 title "Quantum Algorithms Beyond the Jones Polynomial" @default.
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