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An efficient quantum algorithm for colored Jones polynomials

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arxiv quant-ph/0606167 v1 pith:T6RTRQQF submitted 2006-06-20 quant-ph

An efficient quantum algorithm for colored Jones polynomials

classification quant-ph
keywords coloredquantumjoneslinknumberalgorithmboundedefficiently
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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We construct a quantum algorithm to approximate efficiently the colored Jones polynomial of the plat presentation of any oriented link L at a fixed root of unity q. Our construction is based on SU(2) Chern-Simons topological quantum field theory (and associated Wess-Zumino-Witten conformal field theory) and exploits the q-deformed spin network as computational background. As proved in (S. Garnerone, A. Marzuoli, M. Rasetti, quant-ph/0601169), the colored Jones polynomial can be evaluated in a number of elementary steps, bounded from above by a linear function of the number of crossings of the link, and polynomially bounded with respect to the number of link strands. Here we show that the Kaul unitary representation of colored oriented braids used there can be efficiently approximated on a standard quantum circuit.

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Cited by 2 Pith papers

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    New combinatorial proofs and circuit designs for quantum error correction reduce physical qubit overhead by up to 10x and time overhead by 2-6x for codes including Steane, Golay, and surface codes.