Open Sys. & Information Dyn. (2006) 13: 373–382 Quantum Computation of Universal Link Invariants Silvano Garnerone Dipartimento di Fisica, Politecnico di Torino corso Duca degli Abruzzi 24, 10129 Torino, Italy e-mail: silvano.garnerone@polito.it Annalisa Marzuoli Dipartimento di Fisica Nucleare e Teorica, Universit`a di Pavia and Sezione INFN Pavia via Bassi 6, 27100 Pavia, Italy e-mail: annalisa.marzuoli@pv.infn.it Mario Rasetti Dipartimento di Fisica, Politecnico di Torino corso Duca degli Abruzzi 24, 10129 Torino, Italy e-mail: mario.rasetti@polito.it (Received: June 7, 2006) Abstract. In the framework of the spin-network simulator based on the SUq (2) tensor algebra, we implement families of finite state quantum automata capable of accepting the language gen- erated by the braid group, and whose transition amplitudes are coloured Jones polynomials. The automaton calculation of the polynomial of a link L on n strands at any fixed root of unity q is bounded from above by a linear function of the number of crossings of the link, on the one hand, and polynomially bounded in terms of the braid index n, on the other. 1. Introduction Since the beginning of quantum computation the understanding of the capabilities and limitations of quantum computers with respect to classical machines has been one of the most important problem in the area. Untill recently, all known quantum algorithms endowed with an exponential speed-up with respect to classical ones have been based on the quantum Fourier transform. In [1, 2] radically different techniques have been introduced in handling quantum algorithms aimed to evaluate (or approximate) the Jones polynomial, a topological invariant which characterizes knots (collections of closed circles in the 3-space). Actually, the connection between the topology of knots, physics and computer science dates back to the papers by Jones [3], Witten [4] and Jaeger et al. [5], while the possibility of performing quantum simulations on observables of topological nature arising in non-abelian topological quantum field theories has been first addressed in [6]. c  Springer 2006 DOI: 10.1007/s11080-006- 9019-x