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Advances in Theoretical and Mathematical Physics

Efficient quantum processing of three-manifold topological invariants

Silvano Garnerone, Annalisa Marzuoli, Mario Rasetti

Published Jan 1, 2009• 14 citations
Publisher

Abstract

A quantum algorithm for approximating efficiently three-manifold topological invariants in the framework of SU(2) Chern-Simons-Witten (CSW) topological quantum field theory at finite values of the coupling constant k is provided.The model of computation adopted is the q-deformed spin network model viewed as a quantum recognizer in the sense of [1], where each basic unitary transition function can be efficiently processed by a standard quantum circuit.

DOI
10.4310/atmp.2009.v13.n6.a1

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A quantum algorithm for approximating efficiently three-manifold topological invariants in the framework of SU(2) Chern-Simons-Witten (CSW) topological quantum field theory at finite values of the coupling constant k is provided.The model of computation adopted is the q-deformed spin network model viewed as a quantum recognizer in the sense of [1], where each basic unitary transition function can be efficiently processed by a standard quantum circuit.
A quantum algorithm for approximating efficiently three–manifold topological invariants in the framework of SU(2)SU(2)SU(2) Chern–Simons–Witten (CSW) topological quantum field theory at finite values of the coupling constant kkk is provided. The model of computation adopted is the qqq-deformed spin network model viewed as a quantum recognizer in the sense of [1], where each basic unitary transition function can be efficiently processed by a standard quantum circuit. This achievement is an extension of the algorithm for approximating polynomial invariants of colored oriented links found in "Spin networks, quantum automata and link invariants" and "An efficient quantum algorithm for colored Jones polynomials." Thus all the significant quantities — partition functions and observables — of quantum CSW theory can be processed efficiently on a quantum computer, reflecting the intrinsic, field-theoretic solvability of such theory at finite kkk. The paper is supplemented by a critical overview of the basic conceptual tools underlying the construction of quantum invariants of links and three–manifolds and connections with algorithmic questions that arise in geometry and quantum gravity models are discussed.
A quantum algorithm for approximating efficiently 3--manifold topological invariants in the framework of SU(2) Chern-Simons-Witten (CSW) topological quantum field theory at finite values of the coupling constant k is provided. The model of computation adopted is the q-deformed spin network model viewed as a quantum recognizer in the sense of Wiesner and Crutchfield, where each basic unitary transition function can be efficiently processed by a standard quantum circuit. This achievement is an extension of the algorithm for approximating polynomial invariants of colored oriented links found in Refs 2,3. Thus all the significant quantities - partition functions and observables - of quantum CSW theory can be processed efficiently on a quantum computer, reflecting the intrinsic, field-theoretic solvability of such theory at finite k. The paper is supplemented by a critical overview of the basic conceptual tools underlying the construction of quantum invariants of links and 3-manifolds and connections with algorithmic questions that arise in geometry and quantum gravity models are discussed.
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.
Motivated by algorithmic problems arising in quantum field theories whose dynamical variables are geometric in nature, we provide a quantum algorithm that efficiently approximates the coloured Jones polynomial. The construction is based on the complete solution of the Chern–Simons topological quantum field theory and its connection to Wess–Zumino–Witten conformal field theory. The coloured Jones polynomial is expressed as the expectation value of the evolution of the q -deformed spin-network quantum automaton. A quantum circuit is constructed capable of simulating the automaton and hence of computing such an expectation value. The latter is efficiently approximated using a standard sampling procedure in quantum computation.
Abstract. Motivated by algorithmic problems arising in quantum field theories whose dynamical variables are geometric in nature, we provide a quantum algorithm that efficiently approximates the colored Jones polynomial. The construction is based on the complete solution of Chern-Simons topological quantum field theory and its connection to Wess-Zumino-Witten conformal field theory. The colored Jones polynomial is expressed as the expectation value of the evolution of the q-deformed spin-network quantum automaton. A quantum circuit is constructed capable of simulating the automaton and hence of computing such expectation value. The latter is efficiently approximated using a standard sampling procedure in quantum computation. PACS numbers: 03.67.Lx, 02.10.Kn, 04.60.Kz, 04.60.NcQuantum geometry and quantum algorithms 2 1.
Certain quantum topological invariants of three manifolds can be written in the form of the Gaussian sum. It is shown that such topological invariants can be approximated efficiently by a quantum computer. The invariants discussed here are obtained as a partition function of the gauge theory on three manifolds with various gauge groups. Our algorithms are applicable to Abelian and finite gauge groups and to some classes of non-Abelian gauge groups. These invariants can be directly estimated by the nuclear magnetic resonance (NMR) technique used for evaluating the Gaussian sum.