Optimization and Quantum Annealing

Optimization deals with problem of finding the minimum of a given cost function (the relationship between the total cost of production and the quantity of a product produced) and in the combinatorial optimisation problems the cost function depends on a large number of variables and hard problems are those for which the computational time is not bound by any polynomial in the problem size [1]. To solve hard optimization problems is a challenging task. Several techniques have been developed to get to the solution(s) of such problems. Here we will discuss about some of the problems and also the techniques that have already been implemented.

[1]  A. Kolen.  Combinatorial optimization algorithm and complexity: Prentice-Hall, Englewood Cliffs, 1982, 496 pages, $49.50 , 1983 .

[2]  Fisher,et al.  Residual energies after slow cooling of disordered systems. , 1986, Physical review letters.

[3]  B. Derrida.  Random-energy model: An exactly solvable model of disordered systems , 1981 .

[4]  H. Nishimori.  Statistical Physics of Spin Glasses and Information Processing , 2001 .

[5]  J. Wheeler,et al.  Physicist’s version of traveling salesman problem: statistical analysis , 1983 .

[6]  Ground-state entropy of the infinite-range model of a spin glass , 1980 .

[7]  Levin,et al.  Cooling-rate dependence for the spin-glass ground-state energy: Implications for optimization by simulated annealing. , 1986, Physical review letters.

[8]  F. Barahona.  On the computational complexity of Ising spin glass models , 1982 .

[9]  J. Bouchaud,et al.  Energy exponents and corrections to scaling in Ising spin glasses , 2002, cond-mat/0212070.

[10]  G. Rinaldi,et al.  Exact ground states of Ising spin glasses: New experimental results with a branch-and-cut algorithm , 1995 .

[11]  B. Chakrabarti,et al.  Reaching the ground state of a quantum spin glass using a zero-temperature quantum Monte Carlo method. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  J. Joanny.  Adsorption of a polyampholyte chain , 1994 .

[13]  Stefan Boettcher,et al.  Optimization with Extremal Dynamics , 2000, Complex..

[14]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[15]  Relaxation to spin glass ground state in one to six dimensions , 1995, cond-mat/9503070.

[16]  E. Tosatti,et al.  Optimization using quantum mechanics: quantum annealing through adiabatic evolution , 2006 .

[17]  S. Boettcher.  Extremal optimization for Sherrington-Kirkpatrick spin glasses , 2004, cond-mat/0407130.

[18]  A. Lüchow.  Quantum Monte Carlo methods , 2011 .

[19]  K. F. Pál,et al.  Hysteretic optimization for the Sherrington Kirkpatrick spin glass , 2006 .

[20]  Károly F. Pál,et al.  The ground state energy of the Edwards-Anderson Ising spin glass with a hybrid genetic algorithm , 1996 .

[21]  Giorgio Parisi,et al.  Infinite Number of Order Parameters for Spin-Glasses , 1979 .

[22]  Bikas K. Chakrabarti,et al.  The travelling salesman problem on a dilute lattice: a simulated annealing study , 1988 .

[23]  Erio Tosatti,et al.  Quantum annealing by the path-integral Monte Carlo method: The two-dimensional random Ising model , 2002 .

[24]  B. Chakrabarti,et al.  Colloquium : Quantum annealing and analog quantum computation , 2008, 0801.2193.

[25]  V. Cerný.  Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm , 1985 .

[26]  Brian W. Kernighan,et al.  An Effective Heuristic Algorithm for the Traveling-Salesman Problem , 1973, Oper. Res..

[27]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[28]  Masatoshi Kitaoka,et al.  Traveling Salesman Problem and Statistical Physics , 1997 .

[29]  B. Chakrabarti,et al.  The travelling salesman problem on a randomly diluted lattice , 1987 .

[30]  I. Morgenstern,et al.  Magnetic correlations in two-dimensional spin-glasses , 1980 .

[31]  B. Chakrabarti,et al.  Quantum Annealing and Related Optimization Methods , 2008 .

[32]  B. Chakrabarti.  Directed travelling salesman problem , 1986 .

[33]  Edward W. Felten,et al.  Large-step markov chains for the TSP incorporating local search heuristics , 1992, Oper. Res. Lett..

[34]  K. Binder,et al.  Spin glasses: Experimental facts, theoretical concepts, and open questions , 1986 .

[35]  I. Morgenstern,et al.  Magnetic correlations in three-dimensional ising spin glasses , 1980 .

[36]  H. Nishimori,et al.  Quantum annealing in the transverse Ising model , 1998, cond-mat/9804280.

[37]  Dietrich Stauffer,et al.  Evolution by damage spreading in Kauffman model , 1994 .

[38]  Daniel A. Lidar,et al.  Consistency of the Adiabatic Theorem , 2004, Quantum Inf. Process..

[39]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[40]  Monte Carlo method for obtaining the ground-state properties of quantum spin systems. , 1996, Physical review. B, Condensed matter.

[41]  Uwe Gropengiesser.  The ground-state energy of the ±J sping glass. A comparison of various biologically motivated algorithms , 1995 .

[42]  E. Tosatti,et al.  Quantum annealing of the traveling-salesman problem. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  Uwe Gropengiesser.  SUPERLINEAR SPEEDUP FOR PARALLEL IMPLEMENTATION OF BIOLOGICALLY MOTIVATED SPIN GLASS OPTIMIZATION ALGORITHM , 1995 .

[44]  J. Beardwood,et al.  The shortest path through many points , 1959, Mathematical Proceedings of the Cambridge Philosophical Society.

[45]  S. Kirkpatrick.  Frustration and ground-state degeneracy in spin glasses , 1977 .

[46]  K. F. Pál.  The ground state of the cubic spin glass with short-range interactions of Gaussian distribution , 1996 .

[47]  Hansmann,et al.  Ground-state properties of the three-dimensional Ising spin glass. , 1994, Physical review. B, Condensed matter.

[48]  Gary S. Grest,et al.  Irreversibility and metastability in spin-glasses. I. Ising model , 1983 .

[49]  Ulrich H. E. Hansmann,et al.  Multicanonical Study of the 3D Ising Spin Glass , 1992 .

[50]  高山 一,et al.  D. Chowdhury: Spin Glasses and Other Frustrated Systems, World Scientific, Singapore, 1986, xiv+386ページ, 22.5×16cm, 5,023円. , 1987 .

[51]  Masuo Suzuki,et al.  Relationship among Exactly Soluble Models of Critical Phenomena. I ---2D Ising Model, Dimer Problem and the Generalized XY-Model--- , 1971 .

[52]  Jooyoung Lee,et al.  Ground-state energy and energy landscape of the Sherrington-Kirkpatrick spin glass , 2007 .

[53]  Spin-glass energy landscape , 1994 .

[54]  Myriam Preissmann,et al.  Optimal cuts in graphs and statistical mechanics , 1997 .

[55]  S. Kirkpatrick,et al.  Infinite-ranged models of spin-glasses , 1978 .

[56]  Martin,et al.  Finite size and dimensional dependence in the Euclidean traveling salesman problem. , 1996, Physical review letters.

[57]  R. Car,et al.  Theory of Quantum Annealing of an Ising Spin Glass , 2002, Science.