Abstract The challenge is to create an efficient quantum algorithm for the bosonic model capable of calculating the Jones polynomials for a knot resulting from interweaving or interlacing n-vertices. This weave is the construction of braid group representations from nineteen-vertex model. We present eigenbases and eigenvalues for lattice generators and their usefulness for the direct computation of Jones polynomials. The calculation shows that the Temperley-Lieb operators can be used for any braid word. Therefore, we propose a quantum sequence using these singular operators as quantum gates operating on the state of n qubits. We show that quantum calculations give the Jones polynomial for achiral knots and links.
Abstract The challenge is to create an efficient quantum algorithm for the bosonic model capable of calculating the Jones polynomials for a knot resulting from interweaving or interlacing n-vertices. This weave is the construction of braid group representations from nineteen-vertex model. We present eigenbases and eigenvalues for lattice generators and their usefulness for the direct computation of Jones polynomials. The calculation shows that the Temperley-Lieb operators can be used for any braid word. Therefore, we propose a quantum sequence using these singular operators as quantum gates operating on the state of n qubits. We show that quantum calculations give the Jones polynomial for achiral knots and links.
It is a challenging problem to construct an efficient quantum algorithm which can compute the Jones' polynomial for any knot or link obtained from platting or capping of a 2n-strand braid. We recapitulate the construction of braid-group representations from vertex models. We present the eigenbases and eigenvalues for the braiding generators and its usefulness in direct evaluation of Jones' polynomial. The calculation suggests that it is possible to associate a series of unitary operators for any braid word. Hence we propose a quantum algorithm using these unitary operators as quantum gates acting on a 2n qubit state. We show that the quantum computation gives Jones' polynomial for achiral knots and links.
This paper is an exploration of relationships between the Jones polynomial and quantum computing. We discuss the structure of the Jones polynomial in relation to representations of the Temperley Lieb algebra, and give an example of a unitary representation of the braid group. We discuss the evaluation of the polynomial as a generalized quantum amplitude and show how the braiding part of the evaluation can be construed as a quantum computation when the braiding representation is unitary. The question of an efficient quantum algorithm for computing the whole polynomial remains open.
The Jones polynomial, discovered in 1984 [18], is an important knot invariant in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten [32]) to be intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang [13, 14] provide an efficient simulation of TQFT by a quantum computer, and vice versa. These results implicitly imply the existence of an efficient quantum algorithm that provides a certain additive approximation of the Jones polynomial at the fifth root of unity, e2π i/5, and moreover, that this problem is BQP-complete. Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results in [13, 14] are heavily based on TQFT, which makes the algorithm essentially inaccessible to computer scientists.We provide an explicit and simple polynomial quantum algorithm to approximate the Jones polynomial of an n strands braid with m crossings at any primitive root of unity e2π i/k, where the running time of the algorithm is polynomial in m,n and k. Our algorithm is based, rather than on TQFT, on well known mathematical results (specifically, the path model representation of the braid group and the uniqueness of the Markov trace for the Temperly Lieb algebra). By the results of [14], our algorithm solves a BQP complete problem.The algorithm we provide exhibits a structure which we hope is generalizable to other quantum algorithmic problems. Candidates of particular interest are the approximations of other downwards self-reducible P-hard problems, most notably, the Potts model.
FINDING: The Jones polynomial is a knot invariant constructible from quantum R-matrices (braid group representations), and its evaluation at roots of unity yields the computational resource for topological quantum computation — the same algebraic structure underlies both knot theory and TQFT. | MATH: Jones polynomial V_K(t) ∈ ℤ[t^{±1/2}]; skein relation: t⁻¹V(L₊) − tV(L₋) = (t^{1/2} − t^{−1/2})V(L₀). Braid group representation: ρ: B_n → GL(V^⊗n) via R-matrix satisfying Yang-Baxter equation (R⊗I)(I⊗R)(R⊗I) = (I⊗R)(R⊗I)(I⊗R). At t = e^{2πi/k}, the representation factors through Temperley-Lieb algebra TL_n(q) with q + q⁻¹ = −(t^{1/2} + t^{−1/2}). Quantum dimension d = −(q + q⁻¹) = [2]_q. Modular tensor category (MTC) data: fusion rules N_{ij}^k, S-matrix, T-matrix (topological spin), satisfying (ST)³ = e^{2πic/8}·S² (modularity). | CONNECTION: Root-of-unity evaluation t = e^{2πi/k} connects to: (1) quantum group U_q(sl₂) at q = e^{iπ/k} — deformation parameter is a root of unity, truncati Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Abstract This paper is an introduction to relationships between quantum topology and quantum computing. We discuss unitary solutions to the YangBaxter equation that are universal quantum gates, quantum entanglement and topological entanglement, and we give an exposition of knot-theoretic 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 self-contained 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 Witten-Reshetikhin-Turaev invariant of three manifolds.
FINDING: Jones polynomial knot invariants are directly realizable as quantum computing circuits, linking A_n root system representations to topological quantum gates. | MATH: Jones polynomial VL(t) evaluated at roots of unity (e.g., t=e2πi/r) yields unitary representations of braid groups; A_n root system has Cartan matrix Aij=2(αi,αj)/(αj,αj), with simple roots αi in Rn, Weyl group order ∣W(An)∣=(n+1)!. | CONNECTION: A_n root lattice is the n-dimensional hypercubic lattice Zn with all permutations; the golden ratio φ=1.618 appears in the quantum dimension d=[2]q=q+q−1 for q=eiπ/5 (Fibonacci anyons), related to A1 at level 3; base-60 emerges in the cyclic structure of Zn actions on braid groups. | DEPTH: 8 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com