Quantum Algorithmic Techniques for Fault-Tolerant Quantum Computers by Maria´ Kieferova´ A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Physics (Quantum Information) Waterloo, Ontario, Canada, 2019 c Maria´ Kieferova´ 2019 Supervisor: Michele Mosca, Professor, Dept. of Combinatorics & Optimization, University of Waterloo Supervisor (cotutelle): Dominic Berry, Assoc. Professor, Dept. of Physics & Astronomy, Macquarie University Committee Member: Christine Muschik, Assist. Professor, Dept. of Physics & Astronomy, University of Waterloo Committee Member: Daniel Gottesman, Professor, Dept. of Physics & Astronomy, (Perimeter Institute) University of Waterloo Internal-External Examiner: Ashwin Nayak, Professor, Dept. of Combinatorics & Optimization, University of Waterloo External Examiner: Ashley Montanaro, Reader, School of Mathematics, University of Bristol ii Author’s Declaration This thesis is being submitted to Macquarie University and the University of Waterloo in accordance with the Cotutelle agreement dated February 10, 2017. This thesis consists of material all of which I authored or co-authored: see Statement of Contributions included in the thesis. I understand that my thesis may be made electronically available to the public. Maria´ Kieferova´ iii Statement of Contributions Chapter 1 MK wrote this chapter as an introduction to the thesis. Chapter 2 Section 2.3 is based on part 4.1 of my publication [1] but has been extended to cover Hamiltonian simulation techniques in more detail. The rest of the chapter was written by MK solely for this thesis. MK wrote the majority of Chapter 4 of [1] with inputs and revisions • from other authors, in particular Peter Johnson, Yudong Cao and Ian Kivlichan. Chapter 3 Chapter 3 is based on [2]. This chapter was edited and Sections 3.1 and 3.5 added for completeness. The project was proposed by Dominic Berry and Ryan Babbush. MK contributed to the antisymmetrization procedure in [2] with ideas • and writing. MK designed the parallelized quantum comparator with inputs from • Artur Scherer, Dominic Berry and Craig Gidney. Chapter 4 Chapter 4 is based on the publication [3]. This chapter was edited, and Fig. 4.1 added for clarity. The project was proposed by Dominic Berry. MK contributed to all parts of the project. • MK wrote the majority of the paper together with Artur Scherer and • Dominic Berry. iv Chapter 5 Chapter 5 is based on [4]. This chapter was extensively edited, updated for new developments, and an introduction to Boltzmann machines was added. We also added multiple figures for clarity. The project was proposed by Nathan Wiebe. MK contributed with ideas for defining a quantum training set. • MK contributed to the development of POVM and relative entropy • training. MK performed the simulations in Sections 5.6.3 and 5.6.4 and the • comparison between Golden-Thompson and relative entropy training in 5.6.5. MK proved Theorem 8. • MK contributed to the writing of the paper. • Chapter 7 MK wrote this chapter as a conclusion for this thesis. Appendix A MK wrote this appendix to provide additional background information for Chapter 5. v Abstract Quantum computers have the potential to push the limits of computation in areas such as quantum chemistry, cryptography, optimization, and ma- chine learning. Even though many quantum algorithms show asymptotic improvement compared to classical ones, the overhead of running quan- tum computers limits when quantum computing becomes useful. Thus, by optimizing components of quantum algorithms, we can bring the regime of quantum advantage closer. My work focuses on developing efficient subroutines for quantum computation. I focus specifically on algorithms for scalable, fault-tolerant quantum computers. While it is possible that even noisy quantum computers can outperform classical ones for specific tasks, high-depth and therefore fault-tolerance is likely required for most applications. In this thesis, I introduce three sets of techniques that can be used by themselves or as subroutines in other algorithms. The first components are coherent versions of classical sort and shuffle. We require that a quantum shuffle prepares a uniform superposition over all permutations of a sequence. The quantum sort is used within the shuffle and as well as in the next algorithm in this thesis. The quantum shuffle is an essential part of state preparation for quantum chemistry computation in first quantization. Second, I review the progress of Hamiltonian simulations and give a new algorithm for simulating time-dependent Hamiltonians. This algo- rithm scales polylogarithmic in the inverse error, and the query complexity does not depend on the derivatives of the Hamiltonian. A time-dependent Hamiltonian simulation was recently used for interaction picture simula- tion with applications to quantum chemistry. Next, I present a fully quantum Boltzmann machine. I show that our algorithm can train on quantum data and learn a classical description of vi quantum states. This type of machine learning can be used for tomography, Hamiltonian learning, and approximate quantum cloning. vii Acknowledgement This thesis would not be possible without a number of people supporting me throughout my PhD studies. I would first like to thank my supervisors Michele Mosca, Dominic Berry and Gavin Brennen for all their time, advice and encouragement. I cannot thank Michele enough for his guidance since my first internship at IQC in 2012. I would also like to thank Dominic for everything I learned from him and his many comments on earlier drafts of this thesis. I am very grateful to my advisory committee consisting of Daniel Gottes- man, Ashwin Nayak, and Roger Melko as well as additional examiners Christine Muschik, Ashley Montanaro, David Poulin and Jingbo Wang for taking the time to read my thesis and for their helpful suggestions. I want to thank Microsoft Research and Zapata Computing for giving me the opportunities to join them as an intern during my studies. Spending time in a corporate environment enriched my academic experience, and I thoroughly enjoyed both of my internships. I would also like to thank everyone involved in the FKS competition for opening the doors of physics for me, my former supervisor Daniel Nagaj for introducing me to quantum computing and Nathan Wiebe for his ongoing mentorship. I am also thankful to my friends and colleagues for all the conversations that enriched me academically. I would not be able to carry out the research in this thesis without my collaborators Ryan Babbush, Yudong Cao, Matthias Degroote, Craig Gidney, Peter D Johnson, Alan´ Aspuru-Guzik, Ian D Kivlichan, Guang Hao Low, Tim Menke, Jonathan Olson, Borja Peropadre, Sadegh Raeisi, Jonathan Romero, Nicolas PD Sawaya, Sukin Sim, Yuval Sanders, Artur Scherer and Libor Veis. I thank Mike and Ophelia Lazaridis Fellowship, University of Waterloo, Michele Mosca and Macquarie University for financial support. viii Lastly, I want to thank my parents for raising me, to Danielka for her friendship since the day she was born and to Yuval for his love and caffeine supply. ix Table of Contents List of Figures xiv List of Abbreviations xv 1 Introduction 1 1.1 Quantum Computing Hardware in 2019 . .2 1.2 Quantum Computing Software in 2019 . .4 1.3 Overview . .5 1.4 Notation . .6 2 Background 8 2.1 Preliminaries . .9 2.1.1 Quantum Gates and the Circuit Model . .9 2.1.2 Complexity . 11 2.1.3 Error Correction and the Threshold Theorem . 12 2.2 A Brief Overview of Quantum Algorithms . 14 2.2.1 Coherent Classical Computation . 15 2.2.2 Phase Estimation and Eigenstate Preparation . 16 2.2.3 Grover Search and Amplitude Amplification . 19 2.2.4 Oblivious Amplitude Amplification . 22 2.3 Hamiltonian Evolution Simulation . 22 x 2.3.1 The Hamiltonian Oracles . 24 2.3.2 Lower Bounds on Hamiltonian Simulation . 26 2.3.3 Simulation Based on Trotterization . 28 2.3.4 Sparse Matrix Decomposition . 29 2.3.5 The Quantum Walk Approach . 31 2.3.6 Hamiltonian Simulation with Linear Combination of Unitaries . 34 2.3.7 Quantum Signal Processing and Qubitization . 36 2.3.8 Generalization of the Hamiltonian Simulation Problem 38 3 Quantum Sort and Shuffle 40 3.1 How to Shuffle a Deck of Cards . 41 3.2 Quantum Approach to a Shuffle . 43 3.3 Preparing a Uniform Superposition With the Quantum FY Shuffle . 44 3.3.1 Initialization . 47 3.3.2 FY Blocks . 48 3.3.3 Disentangling index from input ............ 51 3.3.4 Complexity Analysis of the FY shuffle . 52 3.4 Symmetrization Through Quantum Sorting . 53 3.4.1 Quantum Sorting Network . 55 3.4.2 Quantum Comparator . 59 3.4.3 Analysis of ‘Delete Collisions’ Step . 65 3.4.4 Complexity Analysis of the Shuffle via Sorting . 67 3.5 Applications . 68 3.6 Conclusion . 70 xi 4 Simulation of Time-Dependent Hamiltonians 71 4.1 Unitary Evolution Under a Time-dependent Hamiltonian . 72 4.2 Framework . 73 4.2.1 Oracles . 73 4.2.2 Enabling Oblivious Amplitude Amplification . 75 4.3 Algorithm Overview . 77 4.3.1 Evolution Discretization . 77 4.3.2 Linear Combination of Unitaries . 81 4.4 Preparation of Auxiliary Registers . 84 4.4.1 Clock Preparation Using Compressed Rotation En- coding . 86 4.4.2 Clock Preparation Using a Quantum Sort . 88 4.4.3 Completing the State Preparation . 90 4.5 Complexity Requirements . 95 4.5.1 Complexity for Scenario 1 . 96 4.5.2 Complexity for Scenario 2 . 98 4.6 Results . 99 4.7 Applications . 100 4.8 Conclusion . 102 5 Training and Tomography with Quantum Boltzmann Machines 103 5.1 Quantum Machine Learning . 104 5.2 Boltzmann Machines . 105 5.3 Training Quantum Boltzmann Machines . 110 5.3.1 Quantum Training Set . 112 5.3.2 Golden-Thompson Training . 114 5.3.3 Commutator Training . 116 xii 5.3.4 Relative Entropy Training .