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

Dorit Aharonov and Vaughan Jones and Zeph Landau
2006 arXiv   pre-print
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^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.  ... 
arXiv:quant-ph/0511096v2 fatcat:7fb7yys7dva57io2vk4btove5m

A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

Dorit Aharonov, Vaughan Jones, Zeph Landau
2008 Algorithmica  
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,  ...  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.  ... 
doi:10.1007/s00453-008-9168-0 fatcat:xtf6twrrurhkzgm2wwbwg2vd7m

A polynomial quantum algorithm for approximating the Jones polynomial

Dorit Aharonov, Vaughan Jones, Zeph Landau
2006 Proceedings of the thirty-eighth annual ACM symposium on Theory of computing - STOC '06  
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,  ...  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.  ... 
doi:10.1145/1132516.1132579 dblp:conf/stoc/AharonovJL06 fatcat:ylv2j2uxvjgazdjjlwgnc7lq7y

On the Complexity of Random Quantum Computations and the Jones Polynomial [article]

Ryan L. Mann, Michael J. Bremner
2017 arXiv   pre-print
Our results provide 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  ... 
arXiv:1711.00686v1 fatcat:i7xvfw4rejdrnpzbgcooubj3li

Experimental Approximation of the Jones Polynomial with One Quantum Bit

G. Passante, O. Moussa, C. A. Ryan, R. Laflamme
2009 Physical Review Letters  
We present experimental results 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.  ... 
doi:10.1103/physrevlett.103.250501 pmid:20366244 fatcat:5oedkj5oaram3iefcgo2pt2zoy

AN EFFICIENT QUANTUM ALGORITHM FOR COLORED JONES POLYNOMIALS

MARIO RASETTI, SILVANO GARNERONE, ANNALISA MARZUOLI
2008 International Journal of Quantum Information  
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.  ...  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.  ... 
doi:10.1142/s0219749908004092 fatcat:zzwxqfkxtjdtxosvmm5luqs7wu

An efficient quantum algorithm for colored Jones polynomials [article]

S. Garnerone, A. Marzuoli, M. Rasetti
2006 arXiv   pre-print
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.  ...  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.  ... 
arXiv:quant-ph/0606167v1 fatcat:tkihefvnxnco7lf3yo5733ddmi

Nuclear-magnetic-resonance quantum calculations of the Jones polynomial

Raimund Marx, Amr Fahmy, Louis Kauffman, Samuel Lomonaco, Andreas Spörl, Nikolas Pomplun, Thomas Schulte-Herbrüggen, John M. Myers, Steffen J. Glaser
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.  ... 
doi:10.1103/physreva.81.032319 pmid:21461143 pmcid:PMC3069923 fatcat:ftcohfw42rhotfsdk3g6qr76wa

Knot theory and quantum computing [article]

Robin Gaudreau, David Ledvinka
2019 arXiv   pre-print
On one side, knot theory has been used to create models of 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  ... 
arXiv:1901.03186v1 fatcat:2q2pc2r47ngtvm6dkfywidhgie

The Jones polynomial: quantum algorithms and applications in quantum complexity theory [article]

Pawel Wocjan, Jon Yard
2007 arXiv   pre-print
We also give 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.  ... 
arXiv:quant-ph/0603069v3 fatcat:njscgc56org53pcq46dbf5ug6q

Approximate Counting and Quantum Computation

M. BORDEWICH, M. FREEDMAN, L. LOVÁSZ, D. WELSH
2005 Combinatorics, probability & computing  
Motivated by 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.  ... 
doi:10.1017/s0963548305007005 fatcat:2s5kpzlx2zfbvc7lxyc2j5bbkm

Approximate Counting and Quantum Computation [article]

M. Bordewich, M. Freedman, L. Lovász, D. Welsh
2009 arXiv   pre-print
Motivated by 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.  ... 
arXiv:0908.2122v1 fatcat:2q5ccnskevhcfk4p3ciltfaxp4

Quantum knitting

S. Garnerone, A. Marzuoli, M. Rasetti
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  ... 
doi:10.1134/s1054660x06110120 fatcat:e5lrh3cvzzbz3hg3jkr6ktjnde

Two paradigms for topological quantum computation [article]

Eric C. Rowell
2008 arXiv   pre-print
We present two paradigms relating algebraic, topological and 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.  ... 
arXiv:0803.1258v1 fatcat:xajxm6kyfvhbrgwkuhwvtubsvi

Quantum geometry and quantum algorithms

S Garnerone, A Marzuoli, M Rasetti
2007 Journal of Physics A: Mathematical and Theoretical  
Motivated by 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.  ... 
doi:10.1088/1751-8113/40/12/s10 fatcat:npd6gcbtdrchlkjhs3km4fyuaa
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