UNIVERSITY OF VERONA DEPARTMENT OF COMPUTER SCIENCE GRADUATE SCHOOL OF NATURAL SCIENCES AND ENGINEERING DOCTORAL PROGRAM IN COMPUTER SCIENCE CYCLE 32 Quantum Approaches to Data Science and Data Analytics S.S.D. INF/01 Coordinator: Massimo Merro Tutor: Alessandra Di Pierro Doctoral Student: Riccardo Mengoni This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License, Italy. To read a copy of the licence, visit the web page: http://creativecommons.org/licenses/by-nc-nd/3.0/ b Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. e NonCommercial — You may not use the material for commercial purposes. d NoDerivatives — If you remix, transform, or build upon the material, you may not distribute the modified material. Algorithms and data structures for indexing, mining, and coding of sequential data — RICCARDO MENGONI or Combinatorics and Algorithmics of Sequential Data — Prefix normal words, colored strings, and space-aware encodings — RICCARDO MENGONI PhD Thesis Verona, May 14, 2020 ISBN <ISBN> Contents 1 The Classical Setting ............................................................ 1 1.1 Classical Machine Learning . .1 1.1.1 Supervised Learning . .2 1.1.2 Unsupervised Learning . .5 1.2 Topological Data Analysis . .7 1.2.1 Algebraic Topology Background . .7 1.2.2 Persistent Homology . 10 1.2.3 TDA as a Resource for ML . 12 2 Quantum Entanglement ......................................................... 13 2.1 Postulates of Quantum Mechanics. 13 2.2 Entanglement for Pure States . 14 2.2.1 Schmidt Decomposition. 15 2.3 Entanglement for Mixed States . 16 2.3.1 PPT Criterion for Separability . 18 2.4 Entanglement Monotones . 18 2.4.1 Measures of Entanglement . 19 2.5 Entanglement Classification . 19 3 Quantum Computing ............................................................ 21 3.1 Quantum Circuit Model . 21 3.1.1 Quantum Gates . 21 3.1.2 Quantum Parallelism . 22 3.1.3 Quantum Algorithms . 23 3.2 Adiabatic Quantum Computation and Quantum Annealing . 26 3.2.1 Adiabatic Quantum Computation . 26 3.2.2 Quantum Annealing . 27 3.3 Topological Quantum Computation . 29 3.3.1 Mathematical Background. 29 3.3.2 Kauffman Bracket . 30 3.3.3 Braids and Links . 31 3.3.4 Computing with Anyons . 31 3.3.5 Topological Quantum Computation of the Jones Polynomials . 32 4 Quantum Machine Learning ..................................................... 35 4.1 QC: Machine Learning for Quantum Physics . 36 4.2 CQ: Quantum Computing for Machine Learning . 37 4.2.1 Essential Tools in QML . 37 4.2.2 Quantum K-NN and K-Means . 39 4.2.3 Quantum SVM . 40 4.2.4 Quantum Computation of Hard Kernels . 41 4.2.5 ML with a Quantum Annealer . 44 5 Homological Analysis of Multi-qubit Entanglement ................................. 47 5.1 Creating the Qubit Data Cloud. 47 5.1.1 Random State Generation . 48 5.1.2 Entangled States Selection . 48 5.1.3 Distances Calculation. 48 5.2 Entanglement Classification . 49 5.2.1 Classification of Three Qubits States . 49 5.2.2 Classification of Four Qubits States . 51 5.2.3 Classification of Five Qubits States . 55 5.2.4 Classification of Six Qubits States . 62 5.3 Class Frequencies . 66 5.4 Comments . 68 6 Quantum Kernel Methods ....................................................... 71 6.1 Quantum Error Correcting Output Codes . 71 6.1.1 Error Correcting Output Codes (ECOC) in Classical Machine Learning . 72 6.1.2 Strategy for Quantum Error Correcting Output Codes . 73 6.1.3 Comments . 76 6.2 Hamming Distance Kernelization via TQC . 76 6.2.1 Topological Quantum Calculation of Hamming Distance Between Binary Strings 77 6.2.2 Hamming Distance Based Kernel . 79 6.2.3 Comments . 80 7 Machine Learning with the D-Wave ............................................... 81 7.1 Quantum Annealing Approach to the Minimum Spanning Tree Problem with ∆-Degree Constraint . 81 7.1.1 QUBO Formulation . 81 7.1.2 Resources . 84 7.1.3 Embedding . 87 7.1.4 Experimental Results and Comments . 89 7.2 Quantum Annealing Approach to Edit Distance Calculation . 91 7.2.1 QUBO Formulation . 91 7.2.2 Resources . 92 7.2.3 Embedding . 93 7.2.4 Experimental Results and Comments . 94 8 Machine Learning with IBM Quantum Computer .................................. 97 8.1 Dataset and Preprocessing . 97 8.2 Encoding Graphs into Quantum States . 99 8.2.1 Example. ..