On the computational complexity of the Jones and Tutte polynomials

Jaeger, François; Vertigan, Dirk; Welsh, Dominic

doi:10.1017/s0305004100068936

Cited by 406 publications

(423 citation statements)

References 37 publications

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“…This was shown by Jaeger, Vertigan and Welsh [JVW90]. They proved that for any primitive root of unity e 2πi/n , n > 4 and n = 6, the evaluation of the Jones polynomial at this value is #P -hard.…”

Section: Complexity Theorymentioning

confidence: 83%

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Quantum Morphing and the Jones Polynomial

Dasbach¹,

Le²,

Lin³

2001

Communications in Mathematical Physics

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We will explore the experimental observation that on the set of knots with bounded crossing number, algebraically independent Vassiliev invariants become correlated, as noticed first by S. Willerton. We will see this through the value distribution of the Jones polynomial at roots of unit. As the degree of the roots of unit is getting larger, the higher order fluctuation is diminishing and a more organized shape will emerge from a rather random value distribution of the Jones polynomial. We call such a phenomenon "quantum morphing". Evaluations of the Jones polynomial at roots of unity play a crucial role, for example in the volume conjecture. When

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Section: Complexity Theorymentioning

confidence: 83%

“…The Alexander polynomial is computable in polynomial time in the number of crossings of a knot while evaluations of the Jones polynomial at all but eight points are known to be #P -hard [JVW90].…”

Section: Introductionmentioning

confidence: 99%

Quantum Morphing and the Jones Polynomial

Dasbach¹,

Le²,

Lin³

2001

Communications in Mathematical Physics

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“…The case where q = 3 is more interesting. A classical algorithm to compute this invariant exists which works in a time that scales polynomially with the number of crossings [26]. In turn, using a generalisation to three-level systems of the scheme presented in Section II allows to estimate the quantum invariant Z L in constant time with an additive error that scales like 1/ √ R where R is the number of repetitions of the experiment, now independent of the number of crossings.…”

Section: Link Invariantsmentioning

confidence: 99%

Low depth quantum circuits for Ising models

et al. 2014

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A scheme for measuring complex temperature partition functions of Ising models is introduced. In the context of ordered qubit registers this scheme finds a natural translation in terms of global operations, and single particle measurements on the edge of the array. Two applications of this scheme are presented. First, through appropriate Wick rotations, those amplitudes can be analytically continued to yield estimates for partition functions of Ising models. Bounds on the estimation error, valid with high confidence, are provided through a central-limit theorem, which validity extends beyond the present context. It holds for example for estimations of the Jones polynomial. Interestingly, the kind of state preparations and measurements involved in this application can in principle be made "instantaneous", i.e. independent of the system size or the parameters being simulated. Second, the scheme allows to accurately estimate some non-trivial invariants of links. A third result concerns the computational power of estimations of partition functions for real temperature classical ferromagnetic Ising models on a square lattice. We provide conditions under which estimating such partition functions allows one to reconstruct scattering amplitudes of quantum circuits making the problem BQP-hard. Using this mapping, we show that fidelity overlaps for ground states of quantum Hamiltonians, which serve as a witness to quantum phase transitions, can be estimated from classical Ising model partition functions. Finally, we show that the ability to accurately measure corner magnetizations on thermal states of two-dimensional Ising models with magnetic field leads to fully polynomial random approximation schemes (FPRAS) for the partition function. Each of these results corresponds to a section of the text that can be essentially read independently.

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“…It is well-known that the computation of the Tutte polynomial is NP-hard [16]. Also, evaluating TðG; x; yÞ for specific points ðx; yÞ is NP-hard except for the points on the hyperbola ðx À 1Þðy À 1Þ¼ 1 or when ðx; yÞ equals ð1; 1Þ; ðÀ1; À1Þ; ð0; À1Þ; ðÀ1; 0Þ; ði; ÀiÞ; ðÀi; iÞ; ðj; j 2 Þ; ðj 2 ; jÞ where j ¼ e ð2pi=3Þ for which it can be done in polynomial time [16,26].…”

Section: Introductionmentioning

confidence: 99%

“…Also, evaluating TðG; x; yÞ for specific points ðx; yÞ is NP-hard except for the points on the hyperbola ðx À 1Þðy À 1Þ¼ 1 or when ðx; yÞ equals ð1; 1Þ; ðÀ1; À1Þ; ð0; À1Þ; ðÀ1; 0Þ; ði; ÀiÞ; ðÀi; iÞ; ðj; j 2 Þ; ðj 2 ; jÞ where j ¼ e ð2pi=3Þ for which it can be done in polynomial time [16,26]. Vertigan [29] has extended this by showing that a similar result holds for planar graphs except that, in this case, for the additional points lying on the hyperbola ðx À 1Þðy À 1Þ¼ 2, the problem can be solved again in polynomial time.…”

Section: Introductionmentioning

confidence: 99%

Computing the Tutte polynomial of Archimedean tilings

Garijo

Gegúndez

Márquez

et al. 2014

Applied Mathematics and Computation

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We describe an algorithm to compute the Tutte polynomial of large fragments of Archimedean tilings by squares, triangles, hexagons and combinations thereof. Our algorithm improves a well known method for computing the Tutte polynomial of square lattices. We also address the problem of obtaining Tutte polynomial evaluations from the symbolic expressions generated by our algorithm, improving the best known lower bound for the asymptotics of the number of spanning forests, and the lower and upper bounds for the asymptotics of the number of acyclic orientations of the square lattice.

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On the computational complexity of the Jones and Tutte polynomials

Cited by 406 publications

References 37 publications

Quantum Morphing and the Jones Polynomial

Quantum Morphing and the Jones Polynomial

Low depth quantum circuits for Ising models

Computing the Tutte polynomial of Archimedean tilings

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