Communications in Mathematical Physics

, Volume 279, Issue 3, pp 735–768 | Cite as

On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers

  • Joseph GeraciEmail author
  • Daniel A. Lidar


We present an efficient quantum algorithm for the exact evaluation of either the fully ferromagnetic or anti-ferromagnetic q-state Potts partition function Z for a family of graphs related to irreducible cyclic codes. This problem is related to the evaluation of the Jones and Tutte polynomials. We consider the connection between the weight enumerator polynomial from coding theory and Z and exploit the fact that there exists a quantum algorithm for efficiently estimating Gauss sums in order to obtain the weight enumerator for a certain class of linear codes. In this way we demonstrate that for a certain class of sparse graphs, which we call Irreducible Cyclic Cocycle Code (ICCCε) graphs, quantum computers provide a polynomial speed up in the difference between the number of edges and vertices of the graph, and an exponential speed up in q, over the best classical algorithms known to date.


Partition Function Lidar Linear Code Quantum Algorithm Code Word 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Shor P.W. (1997). Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. on Comp. 26: 1484 zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on the Theory of Computing, New York: ACM, 1996, p. 212Google Scholar
  3. 3.
    Lidar D.A. and Biham O. (1997). Simulating Ising spin glasses on a quantum computer. Phys. Rev. E 56: 3661 CrossRefADSGoogle Scholar
  4. 4.
    Swendsen R.H. and Wang J.-S. (1987). Phys. Rev. Lett. 58: 86 CrossRefADSGoogle Scholar
  5. 5.
    Lidar D.A. (2004). On the Quantum Computational Complexity of the Ising Spin Glass Partition Function and of Knot Invariants. New J. Phys. 6: 167 CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Welsh, D.J.A.: Complexity: Knots, Colourings and Counting. Volume 1 of London Mathematical Society Lecture Note Series 186. London: Cambridge University Press, 1993Google Scholar
  7. 7.
    Reichl L.E. (1998). A Modern Course in Statistical Physics. John Wiley & Sons, New York zbMATHGoogle Scholar
  8. 8.
    Jones V.F.R. (1989). On Knot Invariants Related to Some Statistical Mechanical Models. Pacific J. Math. 137: 311 zbMATHMathSciNetGoogle Scholar
  9. 9.
    Jones V.F.R. (1985). A Polynomial Invariant for Knots via von Neumann Algebras. Bull. Amer. Math. Soc. 12: 103 zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jaeger F., Vertigen D. and Welsh D. (1990). On the Computational Complexity of the Jones’ and Tutte polynomials. Math. Proc. Cambridge Philos. Soc. 108: 35 zbMATHMathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Kauffman, L.H.: Knots and Physics. Volume 1 of Knots and Everything. Singapore: World Scientific, 2001Google Scholar
  12. 12.
    Alon, N., Frieze, A.M., Welsh, D.: Polynomial Time Randomised Approximation Schemes for Tutte-Gröthendieck Invariants: The Dense Case. Electronic Colloquium on Computational Complexity, 1(5) (1994), available at, 1994
  13. 13.
    Nechaev, S.: Statistics of knots and entangled random walks., 1998
  14. 14.
    Witten E. (1988). Topological quantum field theory. Commun. Math. Phys. 117: 353 zbMATHCrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Freedman M.H., Kitaev A. and Wang Z. (2002). Simulation of topological field theories by quantum computers. Commun. Math. Phys. 227: 587 zbMATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Freedman, M.H., Kitaev, A., Larsen, M.J., Wang, Z.: Topological Quantum Computation., 2001
  17. 17.
    Aharonov, D., Jones, V., Landau, Z.: A Polynomial Quantum Algorithm for Approximating the Jones Polynomial., 2005
  18. 18.
    Wocjan, P., Yard, J.: The Jones polynomial: quantum algorithms and applications in quantum complexity theory., 2006
  19. 19.
    Kauffman L.H. and Lomonaco S.J. (2007). q-Deformed spin networks, knot polynomials and anyonic topological computation. J. of Knot Theory and its Ramifications 16: 267 zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Bonca, J., Gubernatis, J.E.: Real-Time Dynamics from Imaginary-Time Quantum Monte Carlo Simulations: Test on Oscillator Chains., 1995
  21. 21.
    Aharonov, D., Arad, I., Eban, E., Landau, Z.: Polynomial Quantum Algorithms for Additive approximations of the Potts model and other Points of the Tutte Plane., 2007
  22. 22.
    Van den Nest M., Dur W. and Briegel H.J. (2007). Classical spin models and the quantum stabilizer formalism. Phys. Rev. Lett. 98: 117207 CrossRefADSGoogle Scholar
  23. 23.
    Hartmann A.K. (2005). Calculation of partition functions by measuring component distributions. Phys. Rev. Lett. 94: 050601 CrossRefADSGoogle Scholar
  24. 24.
    Denef, J., Vercauteren, F.: Counting Points on C ab Curves using Monsky-Washnitzer Cohomology., 2004
  25. 25.
    Kedlaya K. (2006). Quantum Computation of zeta functions of curves. Comput. Complex. 15: 1–19 zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    van Dam, W., Seroussi, G.: Efficient Quantum Algorithms for Estimating Gauss Sums., 2002
  27. 27.
    Barg A. (2002). On some polynomials related to Weight Enumerators of Linear Codes. SIAM J. Discrete Math. 15: 155 zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Baumert L. and McEliece R. (1972). Weights of Irreducible Cyclic Codes. Inform. and Control 20: 158 CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Welsh D.J.A. (1976). Matroid Theory. Academic Press Inc, London zbMATHGoogle Scholar
  30. 30.
    Gross J. and Yellen J. (1999). Graph theory and its applications. Discrete mathematics and its applications. CRC Press, Boca Raton, FL Google Scholar
  31. 31.
    Lint J.H. (1982). Introduction to Coding Theory. Springer-Verlag, Berlin-Heidelberg-New York zbMATHGoogle Scholar
  32. 32.
    Jaeger F. (1998). The Tutte Polynomial and Link Polynomials. Proc. Amer. Math. Soc. 103: 647 CrossRefMathSciNetGoogle Scholar
  33. 33.
    Evans J., Berndt B.C. and Williams K.S. (1998). Gauss and Jacobi Sums. Wiley-Interscience, New York zbMATHGoogle Scholar
  34. 34.
    Moisio, M.: Exponential Sums, Gauss Sums and Cyclic Codes. 1997. Available at, 1997
  35. 35.
    Aubry, Y., Langevin, P.: On the weights of irreducible cyclic codes. 2005. Available at, 2005
  36. 36.
    Andrews G.E. (1994). Number Theory. Dover Publications Inc., New York Google Scholar
  37. 37.
    Nielsen M.A. and Chuang I.L. (2000). Quantum Computation and Quantum Information. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  38. 38.
    van Dam, W., Seroussi, G.: Quantum algorithms for estimating Gauss sums and calculating discrete logarithms. Available at
  39. 39.
    Brassard, G., Hoyer, P., Tapp, A.: Quantum Counting., 1998
  40. 40.
    van Bussel, F., Geraci, J.: A Note on Cyclotomic Cosets and an Algorithm for finding Coset Representatives and Size and a theorem on the quantum evaluation of weight enumerators for a certain class of cycliccades., 2007
  41. 41.
    Lidl R. and Niederreiter H. (1997). Finite Fields, Volume 20 of Encyclopedia of Mathematics. Cambridge University Press, Cambridge Google Scholar
  42. 42.
    Van Der Glugt M. (1995). Hasse-Davenport Curves, Gauss Sums and Weight Distributions of Irreducible Cyclic Codes. J. Number Theory 55: 145 CrossRefMathSciNetGoogle Scholar
  43. 43.
    Vyalyi, M.N.: Hardness of approximating the weight enumerator of a binary linear code., 2003
  44. 44.
    Khovanov M. (2000). A categorification of the Jones polynomial. Duke Math. J. 101: 359 zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Martinez-Perez C. and Willems W. (2006). Is the Class of Cyclic Codes Asymptotically Good?. IEEE Trans. Inf. Theory 52(2): 696 CrossRefMathSciNetGoogle Scholar
  46. 46.
    Shrock R. (2000). Exact Potts model partition functions on ladder graphs. Physica A 283: 388 CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Departments of Chemistry, Electrical Engineering, and Physics, Center for Quantum Information Science & TechnologyUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations