Quantum annealing of the traveling-salesman problem.

We propose a path-integral Monte Carlo quantum annealing scheme for the symmetric traveling-salesman problem, based on a highly constrained Ising-like representation, and we compare its performance against standard thermal simulated annealing. The Monte Carlo moves implemented are standard, and consist in restructuring a tour by exchanging two links (two-opt moves). The quantum annealing scheme, even with a drastically simple form of kinetic energy, appears definitely superior to the classical one, when tested on a 1002-city instance of the standard TSPLIB.

[1]  M. Suzuki,et al.  Relationship between d-Dimensional Quantal Spin Systems and (d+1)-Dimensional Ising Systems: Equivalence, Critical Exponents and Systematic Approximants of the Partition Function and Spin Correlations , 1976 .

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

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

[4]  J. Hopfield,et al.  Computing with neural circuits: a model. , 1986, Science.

[5]  J. Doll,et al.  Quantum annealing: A new method for minimizing multidimensional functions , 1994, chem-ph/9404003.

[6]  Olivier C. Martin,et al.  Combining simulated annealing with local search heuristics , 1993, Ann. Oper. Res..

[7]  J. K. Lenstra,et al.  Local Search in Combinatorial Optimisation. , 1997 .

[8]  J. Doye,et al.  Global Optimization by Basin-Hopping and the Lowest Energy Structures of Lennard-Jones Clusters Containing up to 110 Atoms , 1997, cond-mat/9803344.

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

[10]  Rosenbaum,et al.  Quantum annealing of a disordered magnet , 1999, Science.

[11]  Bruce J. Berne,et al.  Global Optimization : Quantum Thermal Annealing with Path Integral Monte Carlo , 1999 .

[12]  E. Farhi,et al.  A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem , 2001, Science.

[13]  G. Aeppli,et al.  Tunable quantum tunnelling of magnetic domain walls , 2001, Nature.

[14]  Quantum Thermal Annealing with Renormalization: Application to a Frustrated Model Protein , 2001 .

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

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

[17]  Christian Blum,et al.  Metaheuristics in combinatorial optimization: Overview and conceptual comparison , 2003, CSUR.