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The Jones polynomial: quantum algorithms and applications in quantum complexity theory
"... We analyze relationships between the Jones polynomial and quantum computation. First, we present two polynomialtime quantum algorithms which give additive approximations of the Jones polynomial, in the sense of Bordewich, Freedman, Lovász and Welsh, of any link obtained from a certain general famil ..."
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Cited by 37 (6 self)
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We analyze relationships between the Jones polynomial and quantum computation. First, we present two polynomialtime quantum algorithms which give additive approximations of the Jones polynomial, in the sense of Bordewich, Freedman, Lovász and Welsh, of any link obtained from a certain general family of closures of braids, evaluated at any primitive root of unity. This family encompasses the wellknown plat and trace closures, generalizing results recently obtained by Aharonov, Jones and Landau. We base our algorithms on a local qubit implementation of the unitary JonesWenzl representations of the braid group which makes the underlying representation theory apparent, allowing us to provide an algorithm for approximating the HOMFLYPT twovariable polynomial of the trace closure of a braid at certain pairs of values as well. Next, we provide a selfcontained proof that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. This theorem was originally proved by Freedman, Larsen and Wang in the context of topological quantum computation, and the necessary notion of approximation was later provided by Bordewich et al. Our proof is simpler as it uses a more natural encoding of twoqubit unitaries into the rectangular representation of the eightstrand braid group. We then give QCMAcomplete and PSPACEcomplete problems which are based on braids. Finally, we conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #Phard problem, while learning its most significant bit is PPhard, without taking the usual route through the Tutte polynomial and graph coloring. 1
qdeformed spin networks, knot polynomials and anyonic topological . . .
, 2006
"... We review the qdeformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results ..."
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Cited by 22 (9 self)
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We review the qdeformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and selfcontained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the WittenReshetikhinTuraev invariant of three manifolds.
Topological quantum computing and the Jones polynomial
, 2006
"... In this paper, we give a description of a recent quantum algorithm created by Aharonov, Jones, and Landau for approximating the values of the Jones polynomial at roots of unity of the form e 2πi/k. This description is given with two objectives in mind. The first is to describe the algorithm in such ..."
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Cited by 6 (2 self)
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In this paper, we give a description of a recent quantum algorithm created by Aharonov, Jones, and Landau for approximating the values of the Jones polynomial at roots of unity of the form e 2πi/k. This description is given with two objectives in mind. The first is to describe the algorithm in such a way as to make explicit the underlying and inherent control structure. The second is to make this algorithm accessible to a larger audience.
Microscopic description of 2d topological phases, duality and 3d state sums
, 907
"... Doubled topological phases introduced by Kitaev, Levin and Wen supported on two dimensional lattices are Hamiltonian versions of three dimensional topological quantum field theories described by the TuraevViro state sum models. We introduce the latter with an emphasis on obtaining them from theorie ..."
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Cited by 4 (0 self)
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Doubled topological phases introduced by Kitaev, Levin and Wen supported on two dimensional lattices are Hamiltonian versions of three dimensional topological quantum field theories described by the TuraevViro state sum models. We introduce the latter with an emphasis on obtaining them from theories in the continuum. Equivalence of the previous models in the ground state are shown in case of the honeycomb lattice and the gauge group being a finite group by means of the wellknown duality transformation between the group algebra and the spin network basis of lattice gauge theory. An analysis of the ribbon operators describing excitations in both types of models and the three dimensional geometrical interpretation are given. 1
Quantum knitting
, 2006
"... We analyze the connections between the mathematical theory of knots and quantum physics by addressing a number of algorithmic questions related to both knots and braid groups. Knots can be distinguished by means of ‘knot invariants’, among which the Jones polynomial plays a prominent role, since it ..."
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We analyze the connections between the mathematical theory of knots and quantum physics by addressing a number of algorithmic questions related to both knots and braid groups. Knots can be distinguished by means of ‘knot invariants’, among which the Jones polynomial plays a prominent role, since it can be associated with observables in topological quantum field theory. Although the problem of computing the Jones polynomial is intractable in the framework of classical complexity theory, it has been recently recognized that a quantum computer is capable of approximating it in an efficient way. The quantum algorithms discussed here represent a breakthrough for quantum computation, since approximating the Jones polynomial is actually a ‘universal problem’, namely the hardest problem that a quantum computer can efficiently handle.
Topological Quantum Information Theory
"... This paper is an introduction to relationships between quantum topology and quantum computing. In this paper we discuss unitary solutions to the YangBaxter equation that are universal quantum gates, quantum entanglement and topological entanglement, and we give an exposition of knottheoretic recou ..."
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This paper is an introduction to relationships between quantum topology and quantum computing. In this paper we discuss unitary solutions to the YangBaxter equation that are universal quantum gates, quantum entanglement and topological entanglement, and we give an exposition of knottheoretic recoupling theory, its relationship with topological quantum field theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and selfcontained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the
An efficient quantum algorithm for colored Jones polynomials
, 2006
"... 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) ChernSimons topological quantum field theory (and associated WessZuminoWitten conformal field the ..."
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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) ChernSimons topological quantum field theory (and associated WessZuminoWitten conformal field theory) and exploits the qdeformed 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. 1 Introduction. The present paper is the natural completion of previous work [1, 2, 3] which focused