Project QASAR Results and hands-on demonstration of a joint … · 2018. 4. 17. · Project QASAR...

Project QASAR Results and hands-on demonstration of a joint

project of Volkswagen and LMU Sebastian Feld, Thomas Gabor

LMU Munich {sebastian.feld, thomas.gabor}@ifi.lmu.de

Qubits Europe 2018 D-Wave Users Conference

12 April 2018, Munich, Germany

The Goals of QASAR 2

based on current D-WAVE platform

Quantum Annealing Systems, Applications and Research



practical challenges…




practical challenges… …that are part of

larger system solutions




practical challenges… …that are part of

larger system solutions

relevant within 2–5 years

derived from other current developments


Quantum Annealing meets Machine Learning 6

Problem

Classification of student success / car selling prices / …

Central Question

Can Quantum Annealing help to automatically construct better classification functions?

Application

Possible interconnections between Quantum Annealing and Machine Learning

use Boosting to build better classifiers from simple basic classifiers

Vote

Vote

Application

Practical QBoost

Choosing a Classifier Set 7

Distribution of Solutions 8

Central Question

Among multiple possible solutions, how does the Quantum Annealer choose?

Application

Solution-Space Illumination

Application

Goal-Biased Sampling

Application

Real Random Numbers

Application

Realiability of Sampling

Problem

Canonical Boolean Formula Satisfaction (3-SAT)

3-SAT

• Given a Boolean formula in CNF (with 3 literals per clause)

• Is this formula satisfiable?

• 𝑥1 ∨ 𝑥2 ∨ 𝑥3 ∧ 𝑥1 ∨ 𝑥4 ∨ 𝑥5

• Yes, for example choose 𝑥1 = 𝑇, 𝑥2 = 𝑇, 𝑥3 = 𝑇, 𝑥4 = 𝐹, 𝑥5 = 𝑇,

9

NP-Hard and “NP-Easy” 10

Problem Criticality for QA 11

non-critical critical

The Importance of Post-Processing 12 non-critical critical

With

D-WAVE

post-processing

Without

D-WAVE

post-processing

Quantum-Classical Hybrid Applications 13

Central Question

How can Quantum Annealing problems and classical algorithms be combined to solve a single problem?

Application

Integration Patterns for Quantum API

Application

Subproblem Construction for Efficiency/Quality/…

1

Project QASAR Results and hands-on demonstration of a joint

project of Volkswagen and LMU Sebastian Feld, Thomas Gabor

LMU Munich {sebastian.feld, thomas.gabor}@ifi.lmu.de

Qubits Europe 2018 D-Wave Users Conference

12 April 2018, Munich, Germany

2

Motivation

Master’s Thesis “A hybrid solution method for the

Capacitated Vehicle Routing Problem on a Quantum Annealer”

3

Motivation

Master’s Thesis “A hybrid solution method for the

Capacitated Vehicle Routing Problem on a Quantum Annealer”

Research Question Solving the CVRP on a QA:

Is it profitable in terms of solution quality and computation time?

Capacitated Vehicle Routing Problem (CVRP)

4

Customers with demands

Central depot

Capacity-constrained vehicles

Objective: Find shortest route serving all customers


5

Customers with demands for goods

Central depot


Find shortest route serving all customers


6


Central depot




7


Central depot




8


Central depot




9 Bad solutions


10 Bad solutions Good solutions

Related Work

11

Simulated Annealing

Branch and Bound

Tabu Search

Genetic Algorithm Quantum

Annealing

Ant Colony Optimization

Or-opt Operator

𝜆-opt Operator

Savings Heuristic

Insertion Heuristic

2-Phase Heuristics

Improvement Heuristics Construction Heuristics Meta Heuristics Exact Approaches

Related Work

12

Simulated Annealing

Branch and Bound

Tabu Search

Genetic Algorithm Quantum

Annealing

Ant Colony Optimization

Or-opt Operator

𝜆-opt Operator

Savings Heuristic

Insertion Heuristic

2-Phase Heuristics

Improvement Heuristics Construction Heuristics Meta Heuristics Exact Approaches

2-Phase Heuristics

13

2-Phase Heuristics

14

1. Clustering

– Divide customers into subsets

– Possible route must not exceed vehicle capacity

2. Routing

– Find shortest path within subset

2-Phase Heuristics

1. Clustering

– Divide customers into subsets

– Possible route must not exceed vehicle capacity

2. Routing

– Find shortest path within subset

15

3 Approaches

16

C

R

Classic Quantum

Quantum Two QUBO

3 Approaches

17

C

R

C&R

Classic Quantum Classic Quantum

Quantum Two QUBO Quantum One QUBO

3 Approaches

18

C

R

C&R

C

R

Classic Quantum Classic Quantum Classic Quantum

Quantum Two QUBO Quantum One QUBO Hybrid Solution

Hybrid Solution

19

Classic Quantum

Hybrid Solution

Hybrid Solution 1. Classic Clustering

– Choice of core customer (distance/demand)

– Clustering (Construction/Improvement)

2. Traveling Salesman Problem QUBO – Every customer is visited exactly once

(has got a position)

– Each position is assigned exactly once (to a customer)

– Minimize distances between customers

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C

Classic Quantum

Hybrid Solution








21

C

R

Classic Quantum

Hybrid Solution








22

C

R

Classic Quantum

Hybrid Solution

Traveling Salesman Problem (TSP)

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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?

Source: Wikipedia

The Good, the Bad and the Ugly

24

D E

F

G

H A

B

C

D E

F

G

H A

B

C

D E

F

G

H A

B

C

The Good, the Bad and the Ugly

25 All possible solutions

Qu

alit

y o

f so

luti

on

Formulate TSP as QUBO

26

𝐻 = 𝛼 1− 𝑥𝑣,𝑗

𝑁

𝑗=1

2

+

𝑛

𝑣=1

𝛼 1 − 𝑥𝑣,𝑗

𝑁

𝑣=1

2𝑛

𝑗=1

+ 𝛽 𝑊𝑢𝑣 𝑥𝑢,𝑗𝑥𝑣,𝑗+1

𝑁

𝑗=1𝑢𝑣 ∈𝐸

A1 A2 A3 A4 B1 B2 B3 B4 C1 C2 C3 C4

A1

A2

A3

A4

B1

B2

B3

B4

Given graph 𝐺 = 𝑉, 𝐸 with edge weights 𝑊𝑢𝑣,

find hamiltonian cycle with minimum sum of edge weights

Source: http://arxiv.org/pdf/1302.5843.pdf


27

𝐻 = 𝛼 1− 𝑥𝑣,𝑗

𝑁

𝑗=1

2

+

𝑛

𝑣=1


𝑁

𝑣=1

2𝑛

𝑗=1


𝑁


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 customer can only appear once in a tour


28

𝐻 = 𝛼 1− 𝑥𝑣,𝑗

𝑁

𝑗=1

2

+

𝑛

𝑣=1


𝑁

𝑣=1

2𝑛

𝑗=1


𝑁


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 𝑗𝑡ℎ position in the tour for each 𝑗


29

𝐻 = 𝛼 1− 𝑥𝑣,𝑗

𝑁

𝑗=1

2

+

𝑛

𝑣=1


𝑁

𝑣=1

2𝑛

𝑗=1


𝑁


If the edge is part of the tour, apply the edge weight

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


30

𝐻 = 𝛼 1− 𝑥𝑣,𝑗

𝑁

𝑗=1

2

+

𝑛

𝑣=1


𝑁

𝑣=1

2𝑛

𝑗=1


𝑁


Every customer can only appear once in a tour There must be a 𝑗𝑡ℎ position in the tour for each 𝑗 If the edge is part of the tour, apply the edge weight

31

Hands on…

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)

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33







34

Main Logic

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# 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)

Main Logic

36

# Create QUBO









Main Logic

37

# Create QUBO









D E

F

G

H A

B

C

Main Logic

38

# Create QUBO









D E

F

G

H A

B

C

Main Logic

39

# Create QUBO









Hybrid Solution

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C

R

Classic Quantum

Hybrid Solution

Results (TSP only): Solution Quality • Very good results for small test instances

• Increasing problem size higher average deviation & variance

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Problem Best Known Solution

TSP-QUBO Best of 10

TSP-QUBO Avg. Dev. of 10

Burma14 3323 3323 0%

Ulysses16 6859 6859 0.21%

Ulysses22 7013 7019 2.01%

WesternSa29 27603 28564 7.63%

Djibouti38 6656 7480 21.31%

Results (TSP only): Solution Quality • Very good results for small test instances

• Increasing problem size higher average deviation & variance

42

Problem Best Known Solution

TSP-QUBO Best of 10

TSP-QUBO Avg. Dev. of 10

Burma14 3323 3323 0%

Ulysses16 6859 6859 0.21%

Ulysses22 7013 7019 2.01%

WesternSa29 27603 28564 7.63%

Djibouti38 6656 7480 21.31%

Results (CVRP): Solution Quality • HS competitive with comparable construction heuristics

• However, BKS never found

43

Clarke-Wright Fisher-Jaikumar Christofides et al. Sweep Hybrid Solution

Problem n BKS Best Avg. Dev. Best Avg. Dev. Best Avg. Dev. Best Avg. Dev. Best Avg. Dev.

CMT1 50 524.61 585 11.5% 524 0.12% 550 4.84% 532 1.41% 556 5.98%

CMT2 75 835.26 900 7.75% 857 2.60% 883 5.72% 874 4.64% 926 10.86%

CMT3 100 826.14 886 7.25% 833 0.83% 851 3.01% 851 3.01% 905 9.55%

CMT4 150 1028.42 1204 17.07% - - 1093 6.28% 1079 4.92% 1148 11.63%

CMT5 199 1291.29 1540 19.26% 1420 9.97% 1418 9.81% 1389 7.57% 1429 10.66%

CMT11 120 1042.12 - - - - - - - - 1084 4.02%

CMT12 100 819.56 877 7.01% 848 3.47% 876 6.89% 949 15.79% 828 1.03%

Results (CVRP): Computation Time • Using QBSolv leads to latency and waiting times

• Embedding requires several seconds per SubQUBO

• Pure solution time of SubQUBOS on QA in 𝜇𝑠 range

44

Classic QBSolv Hybrid QBSolv (D-Wave)

Total time 3.672 sec 484.907 sec

QUBO solution time 1.086 sec 0.737 sec

Including network latency, queueing, programming, cool-down, annealing, and readout

Results (CVRP): Computation Time • Using QBSolv leads to latency and waiting times

• Embedding requires several seconds per SubQUBO

• Pure solution time of SubQUBOS on QA in 𝝁𝒔 range

45

Classic QBSolv Hybrid QBSolv (D-Wave)

Total time 3.672 sec 484.907 sec

QUBO solution time 1.086 sec 0.737 sec

Including just programming, cool-down, annealing, and readout

Summary Hybrid Approach

• Splitting problems into subproblems is helpful

• Direct mapping of problem on hardware is possible

Computation time

• Large optimization problems require QBSolv (Latency, waiting times and embedding)

• But: pure solution time of single SubQUBOs in range of 𝜇𝑠

Solution quality

• Comparable with fundamental construction heuristics

• Clustering phase makes finding BKS difficult

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Computation time



Solution quality



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Computation time



Solution quality



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Summary Publication

• Currently in preparation…

Acknowledgment

• Dr. Christian Seidel, Dr. Gabriele Compostella, Isabella Galter (VW)

• Andreas Hessenberger, Christoph Roch, Sebastian Zielinski (LMU)

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Summary Publication

• Currently in preparation…

Acknowledgment

• Dr. Christian Seidel, Isabella Galter (Volkswagen AG Data:Lab Munich)

• Prof. Dr. Wolfgang Mauerer (OTH Regensburg)

• Christoph Roch, Sebastian Zielinski, Andreas Hessenberger (LMU Munich)

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Project QASAR Quantum Annealing Systems, Applications and Research

Project QASAR Results and hands-on demonstration of a joint … · 2018. 4. 17. · Project QASAR...

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