Solving the Travelling Salesman Problem Using Quantum Computing

Transcription

1 Solving the Travelling Salesman Problem Using Quantum Computing Sebastian Feld, Christoph Roch, Thomas Gabor Ludwig-Maximilians-Universität München OpenMunich , Munich

2 Agenda I. Quantum Computing and Quantum Annealing II. TSP and Ising Model III. qbsolv IV. Conclusion 2

3 Source: D-Wave I. QUANTUM COMPUTING AND QUANTUM ANNEALING

4 WHAT CAN I USE QUANTUM COMPUTING FOR? 4

5 Source: Neukart et al., 2017 Traffic Flow Optimization Find (near) optimal solution from possibilities in several 20μs 5

6 HOW DOES QUANTUM COMPUTING WORK? 6

7 Source: Wikipedia; Elena Natasha Liston Wave-particle duality Quantum Physics Non-determinism Influence by observation A B C 7

8 Quantum Computing Quantum Gate Model Quantum Annealing 8

9 Quantum Bits 0 1 9

10 Superposition 01 10

11 Entanglement

12 Quality of solution Quantum Annealing Best solution All possible solutions 12

13 Portfolio Optimization?????? Spend budget, minimize risk, maximize outcome 13

14 Portfolio Optimization Buy Buy Buy Spend budget, minimize risk, maximize outcome 14

15 Qubits represent stocks 15

16 Initialize qubits in superposition???????????????? 16

17 Formulate constraints???????????????? 17

18 Anneal to optimal solution Y Y Y Y Y N N Y Y Y N Y N Y Y N 18

19 Quality of solution Quantum Annealing Best solution All possible solutions 19

20 ARE THERE REAL-WORLD APPLICATIONS FOR QUANTUM COMPUTING? 20

21 Source: Neukart et al., 2017 Traffic Flow Optimization Intelligent Mobility Self-driving cars 21

22 Source: Christine Mumford Vehicle Routing Problem Logistic Problems E-Mobility 22

23 Source: Robert R. McCormick School of Engineering and Applied Science Multi-Criteria Optimization Financial Risk Opt Portfolio Mgmt 23

24 Source: Chao-Yang Lu, Xin-Dong Cai Probabilistic Sampling Artificial Intelligence Machine Learning 24

25 Source: Eric.Ray; University of Warwick, Department of Computer Science Search unsorted database in O n Search Engines Social Media Lov Grover 25

26 Source: Universität Innsbruck, Instiut für Experimentalphysik; MIT, Department of Mathematics Prime Factorization in polynomial time RSA Cryptography Peter Shor 26

27 Source: D-Wave II. TSP AND ISING MODEL

28 Source: Wikipedia Travelling Salesman Problem Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city? 28

29 The Good, the Bad and the Ugly D E C F B G A H 29

30 The Good, the Bad and the Ugly D E C F B G A H 30

31 The Good, the Bad and the Ugly D E C F B G A H 31

32 Combinatorial Optimization n cities n 1! combinations ,

33 Quality of solution Quantum Annealing (QA) All possible solutions 33

34 QA using Ising Model H s = i h i s i + J ij s i s j i<j Mathematical model for studying properties of physical systems that evolve in time 34

35 Source: McGeoch QA using Ising Model H s = i h i s i + i<j J ij s i s j Particles can be in state +1 or 1 35

36 Source: McGeoch QA using Ising Model H s = i h i s i + i<j J ij s i s j External forces on individual particles 36

37 Source: McGeoch QA using Ising Model H s = i h i s i + i<j J ij s i s j Interaction forces between neighbors 37

38 Source: McGeoch QA using Ising Model H s = i h i s i + i<j J ij s i s j Energy of spin configuration 38

39 Source: TSP as Ising Formulation n N 2 n N 2 N H = α 1 x v,j + α 1 x v,j + β W uv x u,j x v,j+1 v=1 j=1 j=1 v=1 uv E j=1 D E C F B G A H Given graph G = V, E with edge weights W uv, find hamiltonian cycle with minimum sum of edge weights 39

40 TSP as Ising Formulation n N 2 n N 2 N H = α 1 x v,j + α 1 x v,j + β W uv x u,j x v,j+1 v=1 j=1 j=1 v=1 uv E j=1 D E A1 A1 A2 A3 A4 B1 B2 B3 B4 C1 C2 C3 C4 C B F G A2 A3 A4 B1 B2 A H B3 B4 40

41 TSP as Ising Formulation n N 2 n N 2 N H = α 1 x v,j + α 1 x v,j + β W uv x u,j x v,j+1 v=1 j=1 j=1 v=1 uv E j=1 A1 A2 A3 A4 B1 B2 B3 B4 C1 C2 C3 C4 A1 α α α A2 α α A3 α A4 B1 α α α B2 α α B3 α B4 Every vertex can only appear once in a circle 41

42 TSP as Ising Formulation n N 2 n N 2 N H = α 1 x v,j + α 1 x v,j + β W uv x u,j x v,j+1 v=1 j=1 j=1 v=1 uv E j=1 A1 A2 A3 A4 B1 B2 B3 B4 C1 C2 C3 C4 A1 α α α α α A2 α α α α A3 α α α A4 α α B1 α α α α B2 α α α B3 α α B4 α There must be a j th node in the cycle for each j 42

43 TSP as Ising Formulation n N 2 n N 2 N H = α 1 x v,j + α 1 x v,j + β W uv x u,j x v,j+1 v=1 j=1 j=1 v=1 uv E j=1 A1 A2 A3 A4 B1 B2 B3 B4 C1 C2 C3 C4 A1 α α α α (ab) (ab) α (ac) (ac) A2 α α (ab) α (ab) (ac) α (ac) A3 α (ab) α (ab) (ac) α (ac) A4 (ab) (ab) α (ac) (ac) α B1 α α α α (bc) (bc) B2 α α (bc) α (bc) B3 α (bc) α (bc) B4 (bc) (bc) α If the edge is part of the cycle, apply the edge weight 43

44 TSP as Ising Formulation n N 2 n N 2 N H = α 1 x v,j + α 1 x v,j + β W uv x u,j x v,j+1 v=1 j=1 j=1 v=1 uv E j=1 Every vertex can only appear once in a circle There must be a j th node in the cycle for each j If the edge is part of the cycle, apply the edge weight 44

45 Source: D-Wave III. QBSOLV

46 Source: qbsolv on github 46

47 Preprocessing # Import nodes with x/y-coordinates file_parser = FileParser("./datasets/TSP_Testdata.xml") # Get list of nodes (index, x, y) nodelist = file_parser.parse_file_tsp() # Get list of undirected edges (index1, index2, length) edges = file_parser.generate_edge_list(nodelist) 47

48 Preprocessing # Import nodes with x/y-coordinates file_parser = FileParser("./datasets/TSP_Testdata.xml") # Get list of nodes (index, x, y) nodelist = file_parser.parse_file_tsp() # Get list of undirected edges (index1, index2, length) edges = file_parser.generate_edge_list(nodelist) 48

49 Preprocessing # Import nodes with x/y-coordinates file_parser = FileParser("./datasets/TSP_Testdata.xml") # Get list of nodes (index, x, y) nodelist = file_parser.parse_file_tsp() # Get list of undirected edges (index1, index2, length) edges = file_parser.generate_edge_list(nodelist) 49

50 Preprocessing # Import nodes with x/y-coordinates file_parser = FileParser("./datasets/TSP_Testdata.xml") # Get list of nodes (index, x, y) nodelist = file_parser.parse_file_tsp() # Get list of undirected edges (index1, index2, length) edges = file_parser.generate_edge_list(nodelist) 50

51 Main Logic # Create QUBO Q = tsp_solver.generate_tsp_qubo(len(nodelist), edges) # Solve QUBO with qbsolv answer = QBSolv().sample_qubo(Q, 50) # Returns the result distance, used_edges = tsp_solver.get_distance_and_edges_of_tour_qbsolv(adj_ matrix, response, edges, len(nodelist)) # Plot dataset and result plot_drawer = PlotDrawer() plot_drawer.plot_tsp(used_edges, nodelist, node_annotation) 51

52 Main Logic # Create QUBO Q = tsp_solver.generate_tsp_qubo(len(nodelist), edges) # Solve QUBO with qbsolv answer = QBSolv().sample_qubo(Q, 50) # Returns the result distance, used_edges = tsp_solver.get_distance_and_edges_of_tour_qbsolv(adj_ matrix, response, edges, len(nodelist)) # Plot dataset and result plot_drawer = PlotDrawer() plot_drawer.plot_tsp(used_edges, nodelist, node_annotation) 52

53 Main Logic # Create QUBO Q = tsp_solver.generate_tsp_qubo(len(nodelist), edges) # Solve QUBO with qbsolv answer = QBSolv().sample_qubo(Q, 50) # Returns the result distance, used_edges = tsp_solver.get_distance_and_edges_of_tour_qbsolv(adj_ matrix, response, edges, len(nodelist)) # Plot dataset and result plot_drawer = PlotDrawer() plot_drawer.plot_tsp(used_edges, nodelist, node_annotation) 53

54 Main Logic # Create QUBO Q = tsp_solver.generate_tsp_qubo(len(nodelist), edges) # Solve QUBO with qbsolv answer = QBSolv().sample_qubo(Q, 50) # Returns the result distance, used_edges = tsp_solver.get_distance_and_edges_of_tour_qbsolv(adj_ matrix, response, edges, len(nodelist)) # Plot dataset and result plot_drawer = PlotDrawer() plot_drawer.plot_tsp(used_edges, nodelist, node_annotation) C B D E F G A H 54

55 Main Logic # Create QUBO Q = tsp_solver.generate_tsp_qubo(len(nodelist), edges) # Solve QUBO with qbsolv answer = QBSolv().sample_qubo(Q, 50) # Returns the result distance, used_edges = tsp_solver.get_distance_and_edges_of_tour_qbsolv(adj_ matrix, response, edges, len(nodelist)) # Plot dataset and result plot_drawer = PlotDrawer() plot_drawer.plot_tsp(used_edges, nodelist, node_annotation) C B D E F G A H 55

56 Main Logic # Create QUBO Q = tsp_solver.generate_tsp_qubo(len(nodelist), edges) # Solve QUBO with qbsolv answer = QBSolv().sample_qubo(Q, 50) # Returns the result distance, used_edges = tsp_solver.get_distance_and_edges_of_tour_qbsolv(adj_ matrix, response, edges, len(nodelist)) # Plot dataset and result plot_drawer = PlotDrawer() plot_drawer.plot_tsp(used_edges, nodelist, node_annotation) 56

57 Source: D-Wave IV. CONCLUSION

58 Source: MCMBUZZ Complex production and usage, but completely new possibilities 58

59 Source: Gartner Bring talents together and try it now! 59

60 Solving the Travelling Salesman Problem Using Quantum Computing Sebastian Feld, Christoph Roch, Thomas Gabor Ludwig-Maximilians-Universität München OpenMunich , Munich