Adiabatic-Quantum-Computing-to-solve-the-MaxCut-graph-problem

Adiabatic Quantum Computing to solve the MaxCut graph problem Fenella McAndrew Supervisors: Prof Kate Smith-Miles & Dr Charles Hill The University of Melbourne School of Mathematics and Statistics A thesis submitted in partial fulfilment of the requirements of the degree of Master of Science (Mathematics and Statistics) October 2020 Abstract MaxCut is an NP-hard graph theory problem which has been extensively researched by computer scientists, data scientists and mathematicians As there are no polynomial time algorithms for NP-Hard problems, there has been a great amount of research into approximation algorithms and heuristics This research project focuses on using adiabatic quantum computing as a technique for solving MaxCut Quantum computing in general has been a fast growing field, with adiabatic quantum computing introduced as a useful technique to solve classically hard optimisation problems The main aim of this project is to investigate if there exist any graph features that affect the performance of adiabatic quantum computing By identifying such features it is possible to gain insights into predicting the performance of the MaxCut algorithm based off some prior feature extraction i Acknowledgements First and foremost, I want to thank my supervisors Kate Smith-Miles and Charles Hill, who have been a great support throughout this whole project The combination of Kate’s maths insights and Charles’ physics knowledge as well as your patience and encouragement made this thesis come together A special thank you to Vivek, all those hours you spent with me coding will not soon be forgotten A big thank you to all my friends inside and outside of university, listening to me talk endlessly about MaxCut and quantum computing could not have been easy Special shout out to my proofreaders DJ and Lukas I want to thank my brothers Angus, Edward, Will and Fin and my dad Alasdair You all call me smart when I feel the opposite and always put time aside for me when I need it You’re all amazing and without you I’d be nowhere Finally, I want to thank my mum Felicity, for being there in good times and bad times Your love and our friendship has made this possible and I’m eternally grateful ii Contents Introduction 1.1 Motivation 1.2 Outline Literature Review 2.1 2.2 2.3 MaxCut Graph Problem 2.1.1 Problem Description 2.1.2 Solution Algorithms Quantum Computing 2.2.1 Overview 2.2.2 Adiabatic Quantum Computing 11 Mapping MaxCut to an AQC Problem 13 2.3.1 Defining Initial Hamiltonian HB 13 2.3.2 Defining Problem Hamiltonian HP 15 Methodology 17 3.1 Instance Space Analysis 17 3.2 Instances 21 3.3 Graph Features 23 3.4 Simulating AQC 25 3.5 Performance Metrics 28 3.5.1 29 Probability of Success iii Contents 3.5.2 Minimum Energy Gap 30 3.5.3 Entanglement 31 Results & Discussion 34 4.1 Experimental Results 34 4.2 Analysis of Results 40 4.2.1 Graph Features 40 4.2.2 Quantum Measurements 44 4.2.3 Algorithm Run-time 47 Conclusion 5.1 5.2 49 Potential Future Work 49 5.1.1 Expanding on MaxCut 49 5.1.2 Other NP-Hard Problems 50 Final Thoughts 51 A Assorted Definitions 53 Bibliography 57 iv List of Figures 2.1 MaxCut example The two partitions are S = {1, 4} and S¯ = {2, 3, 5} The size of the cut is f (x) = with x = (1, −1, −1, 1, −1) 2.2 Simple n = node graph, with m = edges 16 3.1 Instance Space Methodology as outlined in [29] 18 3.2 Table of graph features that are measured for each instance 24 3.3 Running AQC on node graph for tstep = 0.1 and T = 27 4.1 Sources Distribution projected onto the instance space 35 4.2 Algorithm Performance based on probability of success Probability of success above 0.9 is considered Good, below 0.9 is considered Bad 4.3 37 Support Vector Machine performance prediction based on the definition of Good and Bad given in Figure 4.2 38 4.4 Distribution of features projected onto the instance space 39 4.5 The two graphs corresponding to the highest and lowest values of the log ratio of the two maximum eigenvalues of the normalized Laplacian 4.6 matrix 42 Probability of Success vs Maximum Entanglement 46 v Chapter Introduction 1.1 Motivation The Maximum Cut (MaxCut) graph problem simply stated is the problem of partitioning the nodes of a graph into two subsets such that the number of edges between the subsets is maximised There also exist generalisations of this problem such as the n-MaxCut problem, where the aim is to partition the nodes into n subsets, and the weighted MaxCut problem, where each edge is assigned a numerical weight and the aim is to maximise the sum of the edge weights between subsets This thesis will focus on the unweighted 2-MaxCut problem MaxCut is a well studied NP-Hard problem, with many researchers attempting to find new and innovative algorithms These algorithms come in many forms from random greedy approaches [35] to approximations through the graph’s spectra [39] Each algorithm offers different advancements in terms of approximation ratios as well as spatial and time complexity This thesis will be concerned with only a single solution method The sole method to be considered is Adiabatic Quantum Computing to solve the MaxCut graph problem Quantum computing is a fast evolving area from its original conception by Benioff in 1980 [4] Since then the size and power of the physical quantum devices are seen to 1.1 Motivation advance every year The specific form of adiabatic quantum computing was introduced by Farhi [16] as a method to solve certain classically difficult optimisation problems The algorithm was first introduced by solving the 3-satisfiability problem There have since been a variety of extensions to further NP-Hard problems and even moving towards machine learning applications [31] Much research has been done around solving MaxCut using adiabatic quantum computing, but there is yet to be any research around the analysis of graph features and how they relate to the performance of the method While certain algorithms indeed perform better than others, we know that by the “no free lunch” (NFL) theorem [40] for NP-Hard problems there really is no single algorithm that performs better on every problem instance than all other algorithms This motivates the idea of a problem instance based investigation through an Instance Space Analysis [29] Instance Space Analysis is a recent methodology introduced to understand algorithm performance based on instances and their measurable features The objective of this approach is concerned with examining what problem instances are hard for some algorithms but easier for others Furthermore, what features make these instances interesting and different from other instances in the space As this thesis is entirely focused on a single algorithm the investigation will be around simply what graph instances are easy and hard for adiabatic quantum computing when solving MaxCut The aim of this project is to run the algorithm on a wide variety of different graph instances and take measurements of particular features of these instances An Instance Space Analysis is a valuable tool that has been used for problems such as Graph Colouring [37] and Machine Learning Classification [28] to understand what measurable features impact algorithm performance This thesis will be concerned with using this methodology to find certain features that greatly affect the performance of adiabatic quantum computing when solving MaxCut Finding such features can offer valuable insight into predicting how the algorithm might perform based on prior feature extraction of the instance 1.2 Outline 1.2 Outline The format of the remainder of this thesis will be as follows; Chapter will be a literature review of the MaxCut problem and quantum computing The chapter will also contain relevant background material important to the remainder of the thesis Chapter will give an introduction to Instance Space Analysis and give an overview of the experimental methodology used in simulating adiabatic quantum computing to solve MaxCut This chapter will also cover the graph features chosen and outline what metrics will be used to measure the success of the algorithm Chapter will show the results and discussion from running experiments Finally, Chapter will conclude the thesis and offer relevant extensions to the project and direction of potential future research 4.2 Analysis of Results 4.2.3 Algorithm Run-time So far we have found that there are indeed graph characteristics that affect the probability of success of the algorithm We have also found that entanglement plays a large part in the algorithm performance However, it is clear to see from the MATILDA plots that the largest factor in the performance of AQC is the total run-time of the algorithm As we increase the total run-time, T , the probability of success increases For our 9-qubit instances we used for this analysis, we can see that the amount of ‘Good’ instances increases very fast from T 20 to T 30 then begins to slow down when we increase from T 50 to T 60 This leads us to think that there must be some sort of critical point where the algorithm is successful enough that we not need to run it any longer We are interested in total run-time, as it is an important factor when performing AQC on an actual quantum device Theoretically, as T → ∞ we have P r(success) → However, this is infeasible as we not want to run an algorithm for an unreasonable amount of time It is important to look for some run-time that is long enough to achieve relatively good results This is where MATILDA offers some nice insights into predictive algorithm properties from feature analysis From our graph feature analysis, we found that graphs with a large value of the log ratio of the two maximum eigenvalues of the normalised Laplacian matrix were easy to solve From the plots we see that these instances tend to need only T = 20 to T = 30 to achieve a probability of success of 0.9 or higher Moreover, we found that for lower values of this measurement the instances needed longer to solve, as they needed around T = 50 or T = 60 achieve a high probability of success This is a valuable and significant realisation as it means one can take this feature measure before performing AQC and estimate the best possible total run-time It is quite simple to calculate eigenvalues of the normalised Laplacian matrix as most algorithms for finding eigenvalues work in O(n2 ) complexity Knowing this easy-to-calculate prior information can lead to more time efficient uses of the physical devices 47 4.2 Analysis of Results It is important to note that all analysis has been done using graphs of n = The time required may vary for graphs of other sizes However, we have shown that for node graphs no more than potentially T = 70 is needed to achieve a good solution for all problems in our instance space Furthermore, this analysis has been conducted from a classically simulated version of AQC When it comes to using an actual quantum device there may need to be some further thought put into how noise may affect total run-time However, for classically simulated cases this is a good predictive measure 48 Chapter Conclusion 5.1 Potential Future Work In this project, we showed that many graph features had no effect on the overall performance of AQC However, we did find that the log ratio of the maximum two eigenvalues of the normalised Laplacian matrix had a great effect on the performance of AQC when solving MaxCut 5.1.1 Expanding on MaxCut We have only been able to test our algorithm on random unweighted small graph instances From this we have been able to gather some insights, but there is certainly much more that can be done in terms of MaxCut and AQC From the discussion in Chapter 4, we noted that all graphs used were randomly generated from five different sources This gave a nice overview of where different graphs sit in the instance space and how different types of graphs work with the algorithm However, there was little control in the creation of these instances An extension to this could be to find further sources to create instances in terms of highly structured graphs This could potentially fill areas of the instance space where no other graphs lie, and offer insights into how AQC performs in other quadrants of the space 49 5.1 Potential Future Work As mentioned in Chapter 3, we were restricted to using rather small graphs All graphs used in the testing of the algorithm were of size n = Using our methods we could test graphs of up to n = 12 but it was difficult to go any larger than that as those graphs took many days to run using a classical simulation To test larger graphs it is possible to employ tensor network methods It would be useful to employ our analysis on the connection between the log ratio of the maximum two eigenvalues of the normalised Laplacian matrix of a graph and the entanglement of a system, to determine what sort of larger graphs may be suited to such methods This would also allow us to expand on our previous discussion of entanglement and tensor networks This thesis was only concerned with testing the algorithm on unweighted MaxCut In Karp’s original paper [24], he proved that general weighted MaxCut is NP-Hard In this project we chose only to focus on the sub-problem of this Extending the methodology to also solve for weighted MaxCut could offer all sorts of new and different graph features to examine Lastly, all of these experiments were done as simulations on a classical computer These simulations have offered us valuable insight into theoretically how the algorithm performs on MaxCut instances It would be useful to further explore these instances by extending the methodology to run experiments on a physical quantum device A physical quantum device has decoherence and noise, this would result in outputs that aren’t perfect as they are in our simulations Testing on a physical device would offer the chance to further investigate how these graph features affect AQC, and how the errors change the performance 5.1.2 Other NP-Hard Problems As mentioned above, this project has purely focused on MaxCut and AQC However, there are a variety of different NP-hard problems which too could be mapped to a quantum computer In Farhi’s original paper on AQC [16] the example he used to explain the func- 50 5.2 Final Thoughts tionality of the algorithm was the 3-satisfiability problem 3-SAT and MaxCut are just a couple of examples that can be solved using AQC Hen [23] presents a review of mapping the graph partition problem and graph colouring to a quantum annealing method, which is closely related to adiabatic quantum computing Childs and Farhi [6] have also presented work on using AQC for finding the largest clique in a random graph It would be a valuable extension to simulate these algorithms using AQC on our set of instances used when testing MaxCut Further use of the ISA methodology would allow us to see if the specific graph feature such of the log ratio of the two largest eigenvalues of the normalised Laplacian matrix is simply a feature that affects MaxCut, or if it is a feature that for some reason has quite an effect on all graph problems when applying AQC 5.2 Final Thoughts In this project we managed to use an ISA effectively to determine graph characteristics of interest However, this project has mostly introduced a variety of new questions We have discussed extensions to testing MaxCut and potential NP-Hard problems that can be tested While these are natural extensions to experimentally testing AQC an important next step would be to step away from the experimental side and look more into the theoretical arguments A necessary follow up to this project would be to undertake an in depth investigation into the area of Spectral Graph Theory and the potential connection to Quantum Computing We have shown that a spectral graph property does indeed have a great effect on the probability of success, but what we still not know is why it has such a great effect A future direction is to mathematically investigate the connections between our Hamiltonian H(s) and graph matrices such as Adjacency, Laplacian and Normalised Laplacian to name a few When creating the problem Hamiltonian the structure is based on the number of nodes and connecting edges in the graph, which may well 51 5.2 Final Thoughts have some correlation with graph matrices Even though the Hamiltonian is dimension 2n × 2n and the graph matrices are dimension n × n, there may be some sort of similarities hidden within their construction This project has been a good initial survey of AQC and MaxCut, but there is still so much more that is left to explore 52 Appendix A Assorted Definitions Definition A.1 (Tensor product) If A is an m × n matrix and B is a p × q matrix, then the Kronecker product A ⊗ B is the pm × qn block matrix:  a11 B a1n B  A ⊗ B =  am1 B amn B  (A.1) Example A.1.1 Given the computational bases 1 |+ = √ (|0 + |1 ) = √ 2 1 and 1 |− = √ (|0 − |1 ) = √ 2 −1 (A.2) The Kronecker product of the two is  1 −1 |+ ⊗ |− =   = (|00 − |01 + |10 − |11 ) 2 −1  (A.3) Definition A.2 (Graph Density) Given a graph G = (V, E) with n = |V | and m = |E| the density, d, for a simple undirected graph is d= 2m n(n − 1) (A.4) 53 Definition A.3 (Average Clustering) For unweighted graphs, the clustering of a node u is the fraction of possible triangles through that node that exist, cu = 2T (u) , deg(u)(deg(u) − 1) (A.5) where T(u) is the number of triangles through node u and deg(u) is the degree of u The clustering coefficient for the graph is the average, C= cv , n v∈G (A.6) where n is the number of nodes in G Definition A.4 (Average Shortest Path) The average shortest path length is a= s,t∈V d(s, t) , n(n − 1) (A.7) where V is the set of nodes in G, d(s, t) is the shortest path from s to t, and n is the number of nodes in G Definition A.5 (Adjacency Matrix) For a simple graph with vertex set V, the adjacency matrix is a square n × n matrix A such that its element Aij is one when there is an edge from vertex i to vertex j, and zero when there is no edge Definition A.6 (Laplacian Matrix) The graph Laplacian is the matrix L = D − A, where A is the adjacency matrix and D is the diagonal matrix of node degrees Definition A.7 (Normalised Laplacian Matrix) The Normalised graph Laplacian ˆ = D−1/2 LD1/2 is the Laplacian matrix normalised by the degree matrix L Definition A.8 (Algebraic Connectivity) The algebraic connectivity of a connected undirected graph is the second smallest eigenvalue of its Laplacian matrix 54 Definition A.9 (Eccentricity) The eccentricity of a node v is the maximum distance from v to any other nodes in G Definition A.10 (Radius) The radius is the minimum eccentricity in a graph Definition A.11 (Diameter) The diameter is the maximum eccentricity in a graph Definition A.12 (Laplacian Energy) Given a graph G = (V, E) with a Laplacian matrix L and spectra λ1 , λ2 , , λn , the Laplacian energy is n LE(G) = λi (A.8) i=1 Definition A.13 (Harmonic Mean) Given a set of positive real numbers x1 , x2 , · · · , xn the Harmonic mean, H, is given as  H= x1 + x2 n + ··· + = xn n n i=1 xi n −1 −1 xi   i=1 =  n   (A.9) Definition A.14 (Geometric Mean) Given a set of positive real numbers x1 , x2 , · · · , xn the Geometric Mean, G, is given as n n G= xi = √ n x1 x2 · · · xn (A.10) i=1 Definition A.15 (Skewness) Skewness is the distortion or asymmetry in a symmetrical bell curve, or normal distribution, in a set of data The skewness of a dataset is computed as the Fisher-Pearson coefficient of skewness Moreover, given a set of positive real numbers x1 , x2 , · · · , xn the Fisher-Pearson coefficient is given as g1 = m3 2/3 (A.11) m2 55 where mi = n n (xk − x¯)i (A.12) k=1 and x¯ is the mean value of the data set Definition A.16 (Kurtosis) Kurtosis is the measure of the "tailedness" of set of data Calculated much like the coefficient of skewness of a dataset, the coefficient of kurtosis is given 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