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A Polynomial Quantum Algorithm for Approximating the Jones Polynomial
[article]

2006
*
arXiv
*
pre-print

These results implicitly imply

arXiv:quant-ph/0511096v2
fatcat:7fb7yys7dva57io2vk4btove5m
*the*existence of an efficient*quantum**algorithm*that provides*a*certain additive*approximation*of*the**Jones**polynomial*at*the*fifth root of unity, e^2π i/5, and moreover, ... 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^2π i/k, where*the*running time ... We are grateful to Umesh Vazirani*for*helpful remarks regarding*the*presentation. ...##
###
A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

2008
*
Algorithmica
*

These results implicitly imply

doi:10.1007/s00453-008-9168-0
fatcat:xtf6twrrurhkzgm2wwbwg2vd7m
*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, ... of*the*Potts model,*a*model which is known to be tightly connected to*the**Jones**polynomial*[34]. ... Thus we obtain*a**polynomial**quantum**algorithm**for**the*BQP-complete problem of*approximating**the**Jones**polynomial*of*a*plat closure of*a*braid. ...##
###
A polynomial quantum algorithm for approximating the Jones polynomial

2006
*
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing - STOC '06
*

These results implicitly imply

doi:10.1145/1132516.1132579
dblp:conf/stoc/AharonovJL06
fatcat:ylv2j2uxvjgazdjjlwgnc7lq7y
*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, ... of*the*Potts model,*a*model which is known to be tightly connected to*the**Jones**polynomial*[34]. ... Thus we obtain*a**polynomial**quantum**algorithm**for**the*BQP-complete problem of*approximating**the**Jones**polynomial*of*a*plat closure of*a*braid. ...##
###
On the Complexity of Random Quantum Computations and the Jones Polynomial
[article]

2017
*
arXiv
*
pre-print

Our results provide

arXiv:1711.00686v1
fatcat:i7xvfw4rejdrnpzbgcooubj3li
*a*straightforward relationship between*the**approximation*of*Jones**polynomials*and*the*complexity of random*quantum*computations. ... There is*a*natural relationship between*Jones**polynomials*and*quantum*computation. ... MJB acknowledges support from*the*Australian Research Council via*the*Future Fellowship scheme (grant FT110101044) and as*a*member of*the*ARC Centre of Excellence*for**Quantum*Computation and Communication ...##
###
Experimental Approximation of the Jones Polynomial with One Quantum Bit

2009
*
Physical Review Letters
*

We present experimental results

doi:10.1103/physrevlett.103.250501
pmid:20366244
fatcat:5oedkj5oaram3iefcgo2pt2zoy
*approximating**the**Jones**polynomial*using 4 qubits in*a*liquid state nuclear magnetic resonance*quantum*information processor. ...*The*implemented*algorithm*is*a*modification of*the**algorithm*developed by Shor and Jordan suitable*for*implementation in NMR. ...*The**algorithm*developed by Shor and Jordan shows that*approximating**the**Jones**polynomial**for**the*trace closure of*a*braid at*the*fifth root of unity is*a*complete problem*for*DQC1. ...##
###
AN EFFICIENT QUANTUM ALGORITHM FOR COLORED JONES POLYNOMIALS

2008
*
International Journal of Quantum Information
*

We construct

doi:10.1142/s0219749908004092
fatcat:zzwxqfkxtjdtxosvmm5luqs7wu
*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. ... 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*... We have provided*a**quantum**algorithm*that efficiently*approximates**the*colored*Jones**polynomial*. ...##
###
An efficient quantum algorithm for colored Jones polynomials
[article]

2006
*
arXiv
*
pre-print

We construct

arXiv:quant-ph/0606167v1
fatcat:tkihefvnxnco7lf3yo5733ddmi
*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. ... 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*... We have provided*a**quantum**algorithm*that efficiently*approximates**the*colored*Jones**polynomial*. ...##
###
Nuclear-magnetic-resonance quantum calculations of the Jones polynomial

2010
*
Physical Review A. Atomic, Molecular, and Optical Physics
*

*The*repertoire of problems theoretically solvable by

*a*

*quantum*computer recently expanded to include

*the*

*approximate*evaluation of knot invariants, specifically

*the*

*Jones*

*polynomial*. ... Here we present experimental results

*for*

*a*small-scale

*approximate*evaluation of

*the*

*Jones*

*Polynomial*by nuclear-magnetic resonance (NMR), in addition we show how to escape from

*the*limitations of NMR ... Pictures

*for*knots and links were created using KnotPlot, http://knotplot.com. ...

##
###
Knot theory and quantum computing
[article]

2019
*
arXiv
*
pre-print

On one side, knot theory has been used to create models of

arXiv:1901.03186v1
fatcat:2q2pc2r47ngtvm6dkfywidhgie
*quantum*computing, and on*the*other, it is*a*source of computational problems. ... This paper explores*the*interactions between knot theory and*quantum*computing. ... This work was financed by*the*National Science and Engineering Research Council of Canada, and completed as part of course work*for**Quantum*Computing: Foundations to Frontier given by Henry Yuen at*the*...##
###
The Jones polynomial: quantum algorithms and applications in quantum complexity theory
[article]

2007
*
arXiv
*
pre-print

We also give

arXiv:quant-ph/0603069v3
fatcat:njscgc56org53pcq46dbf5ug6q
*algorithms**for**approximating**the**Jones**polynomial*of*a*general class of closures of braids at roots of unity. ... 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. ... JY is thankful*for*support from*the*National Science Foundation under grant PHY-0456720 through*the*Institute*for**Quantum*Information at*the*California Institute of Technology. ...##
###
Approximate Counting and Quantum Computation

2005
*
Combinatorics, probability & computing
*

Motivated by

doi:10.1017/s0963548305007005
fatcat:2s5kpzlx2zfbvc7lxyc2j5bbkm
*the*result that an '*approximate*' evaluation of*the**Jones**polynomial*of*a*braid at*a*5 th root of unity can be used to simulate*the**quantum*part of any*algorithm*in*the**quantum*complexity class ... However we are unable to determine whether*the*particular functions we are motivated by, such as*the*above evaluation of*the**Jones**polynomial*, can be*approximated*in this way. ... Acknowledgements*The*authors would like to thank*the*referee*for*many helpful comments and suggestions, in particular*for*suggesting*a*simpler proof of Theorem 4.3. ...##
###
Approximate Counting and Quantum Computation
[article]

2009
*
arXiv
*
pre-print

Motivated by

arXiv:0908.2122v1
fatcat:2q5ccnskevhcfk4p3ciltfaxp4
*the*result that an '*approximate*' evaluation of*the**Jones**polynomial*of*a*braid at*a*5^th root of unity can be used to simulate*the**quantum*part of any*algorithm*in*the**quantum*complexity class ... However we are unable to determine whether*the*particular functions we are motivated by, such as*the*above evaluation of*the**Jones**polynomial*, can be*approximated*in this way. ... Acknowledgements*The*authors would like to thank*the*referee*for*many helpful comments and suggestions, in particular*for*suggesting*a*simpler proof of Theorem 4.3. ...##
###
Quantum knitting

2006
*
Laser physics
*

*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*... 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*... Indeed, in

*the*late 2005 Dorit Aharonov, Vaughan

*Jones*and Zeph Landau [6] found an efficient

*quantum*

*algorithm*

*for*

*approximating*

*the*

*Jones*

*polynomial*(

*a*difficult problem to be addressed in

*the*classical ...

##
###
Two paradigms for topological quantum computation
[article]

2008
*
arXiv
*
pre-print

We present two paradigms relating algebraic, topological and

arXiv:0803.1258v1
fatcat:xajxm6kyfvhbrgwkuhwvtubsvi
*quantum*computational statistics*for**the*topological model*for**quantum*computation. ... We give*a*fairly comprehensive list of known examples and formulate two conjectures that would further support*the*paradigms. ... Since*the**Jones**polynomial*can be obtained as*a*specialization of*the*HOM-FLYPT*polynomial*, FPRASability of*the*HOMFLYPT*polynomial*would imply*the*same*for**the**Jones**polynomial*. 3.3.3. ...##
###
Quantum geometry and quantum algorithms

2007
*
Journal of Physics A: Mathematical and Theoretical
*

Motivated by

doi:10.1088/1751-8113/40/12/s10
fatcat:npd6gcbtdrchlkjhs3km4fyuaa
*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*colored*Jones**polynomial*is expressed as*the*expectation value of*the*evolution of*the*q-deformed spin-network*quantum*automaton. ... Acknowledgments*Quantum*geometry and*quantum**algorithms*We are in debt with Romesh Kaul*for*clarifying remarks on his work on colored*polynomials*. ...
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