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Zbl 1152.81718
Garnerone, S.; Marzuoli, A.; Rasetti, M.
Quantum automata, braid group and link polynomials. (English)
[J] Quantum Inf. Comput. 7, No. 5-6, 479-503 (2007). ISSN 1533-7146

Summary: The spin-network quantum simulator model, which essentially encodes the (quantum deformed) $SU(2)$ Racah--Wigner tensor algebra, is particularly suitable to address problems arising in low dimensional topology and group theory. In this combinatorial framework we implement families of finite-states and discrete-time quantum automata capable of accepting the language generated by the braid group, and whose transition amplitudes are colored Jones polynomials. The automaton calculation of the polynomial of (the plat closure of) a link $L$ on $2N$ strands at any fixed root of unity is shown to be 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 $2N$, on the other. The growth rate of the time complexity function in terms of the integer $k$ appearing in the root of unity $q$ can be estimated to be (polynomially) bounded by resorting to the field theoretical background given by the Chern--Simons theory.

MSC 2000:: *81P68 Quantum computation and quantum cryptography
20M35 Semigroups in automata theory, linguistics, etc.
81R10 Repres. of infinite-dim. groups and algebras from quantum theory

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Zentralblatt MATH Berlin [Germany]