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arXiv:quant-ph/0511096

A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

Dorit Aharonov, Vaughan F. R. Jones, Zeph A. Landau

Published Nov 9, 2005• 19 citations
arXivHTMLPDF

Abstract

The Jones polynomial, discovered in 1984, is an important knot invariant in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten) to be intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang 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, e^{2πi/5}, and moreover, that this problem is BQP-complete. Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results mentioned 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 e^{2πi/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. By the results of Freedman et. al., 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 #P-hard problems, most notably, the partition function of the Potts model, a model which is known to be tightly connected to the Jones polynomial.

DOI
10.48550/arxiv.quant-ph/0511096

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A polynomial quantum algorithm for approximating the Jones polynomial

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FINDING: The Jones polynomial V_K(t) is a knot invariant constructible from quantum R-matrices (quantum groups U_q(sl₂)) and is #P-hard to compute exactly at generic t, but admits efficient quantum approximation at specific roots of unity t = e^{2πi/k} via TQFT (Witten–Reshetikhin–Turaev). | MATH: V_K(t) = Σ_i a_i t^i; skein relation t⁻¹V(L₊) − tV(L₋) = (t^{1/2} − t^{−1/2})V(L₀). At t = e^{2πi/k}, V_K is the trace of a braid-group representation ρ: B_n → U(H_k) where H_k is the level-k WZW conformal block space. Quantum algorithm (Freedman–Kitaev–Larsen–Wang) runs in poly(n, k) time. | CONNECTION: The golden ratio φ = (1+√5)/2 enters via the quantum group U_q(sl₂) at q = e^{2πi/5}, where the representation category is the modular tensor category associated to the icosahedral A₅ fusion rules. The A₅ root system (H₃ ⊂ H₄ Coxeter groups) has eigenvalues {2cos(π/5), 2cos(2π/5)} = {φ, φ⁻¹} = {1.618, 0.618}. The Jones polynomial at t = e^{2πi/5} encodes the A₅ representation theory: the 5-fo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al.\ that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a \#P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.

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The Jones polynomial, discovered in 1984, is an important knot invariant in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten) to be intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang 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, e^{2πi/5}, and moreover, that this problem is BQP-complete. Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results mentioned 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 e^{2πi/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. By the results of Freedman et. al., 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 #P-hard problems, most notably, the partition function of the Potts model, a model which is known to be tightly connected to the Jones polynomial.
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.
Freedman, Kitaev, and Wang proved the equivalence between quantum field theory and quantum computation, and consequently showed that the problem of approximating the Jones polynomial (a knot invariant) at the fifth root of unity is BQP-complete. Recently, Aharonov, Jones, and Landau proposed a concrete quantum algorithm, called the AJL algorithm, that approximates the Jones polynomial at the kkkth root of unity in polynomial time. In this paper, we propose a new method for implementing the AJL algorithm, which improves the performance from O(mnlog⁡2k)O(mn\log^2k)O(mnlog2k) to O(mn)O(mn)O(mn). Here, nnn is the number of strands, mmm is the number of the crossings in a braid. Since, in the AJL algorithm, mmm and kkk are assumed to be given as polynomials in nnn, the difference in the performance between the original implementation and our design is significant if kkk is a large-degree polynomial.
FINDING: The Jones polynomial V_K(t) is a knot invariant constructible from quantum R-matrices (quantum groups U_q(sl₂)) and is #P-hard to compute exactly at generic t, but admits efficient quantum approximation at specific roots of unity t = e^{2πi/k} via TQFT (Witten–Reshetikhin–Turaev). | MATH: V_K(t) = Σ_i a_i t^i; skein relation t⁻¹V(L₊) − tV(L₋) = (t^{1/2} − t^{−1/2})V(L₀). At t = e^{2πi/k}, V_K is the trace of a braid-group representation ρ: B_n → U(H_k) where H_k is the level-k WZW conformal block space. Quantum algorithm (Freedman–Kitaev–Larsen–Wang) runs in poly(n, k) time. | CONNECTION: The golden ratio φ = (1+√5)/2 enters via the quantum group U_q(sl₂) at q = e^{2πi/5}, where the representation category is the modular tensor category associated to the icosahedral A₅ fusion rules. The A₅ root system (H₃ ⊂ H₄ Coxeter groups) has eigenvalues {2cos(π/5), 2cos(2π/5)} = {φ, φ⁻¹} = {1.618, 0.618}. The Jones polynomial at t = e^{2πi/5} encodes the A₅ representation theory: the 5-fo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al. that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.
We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al. that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-cornplete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.
We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al.\ that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a \#P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.