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Machine Learning for Designing Fast Quantum Gates University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2016-01-26 Machine Learning for Designing Fast Quantum Gates Zahedinejad, Ehsan Zahedinejad, E. (2016). Machine Learning for Designing Fast Quantum Gates (Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/26805 http://hdl.handle.net/11023/2780 doctoral thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY Machine Learning for Designing Fast Quantum Gates by Ehsan Zahedinejad A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN PHYSICS AND ASTRONOMY CALGARY, ALBERTA January, 2016 c Ehsan Zahedinejad 2016 Abstract Fault-tolerant quantum computing requires encoding the quantum information into logical qubits and performing the quantum information processing in a code-space. Quantum error correction codes, then, can be employed to diagnose and remove the possible errors in the quantum information, thereby avoiding the loss of information. Although a series of single- and two-qubit gates can be employed to construct a quan- tum error correcting circuit, however this decomposition approach is not practically desirable because it leads to circuits with long operation times. An alternative ap- proach to designing a fast quantum circuit is to design quantum gates that act on a multi-qubit gate. Here I devise quantum control schemes to design high-fidelity single-shot multi-qubit (up to three) quantum gates. Quantum control task is to steer quantum dynamics towards closely realizing specific quantum operation by varying the external control parameters (external field) such that the resultant evolution closely approximates the desired evolution. A set of instructions that determines the control parameters, and hence the e↵ectiveness of the control scheme, is called a policy. Machine learning algorithms can be employed to find successful policies for designing quantum gates. In particular, we employ supervised machine learning techniques to generate these successful policies. Finding successful policies is a feasibility problem for which optimization algo- rithms can be employed. Greedy algorithms are at the heart of machine learning tech- niques. They converge faster onto a successful policy and require less-computational resource than non-greedy algorithms. However, there is no guarantee that greedy algorithms succeed to a feasible solution when there exist constraints on i) gate op- eration time ii) computational resources, and iii) experimental resources. Our results show the failure of standard greedy machine learning algorithms and ii the superiority of non-greedy machine learning algorithms over greedy ones for de- signing quantum logic gates, when there exist constraints on the quantum system. We have also observed the failure of existing greedy and non-greedy techniques for designing high-fidelity three-qubit gates. Hence, we devised our machine learning technique called Subspace-Selective Self-adaptive Di↵erential Evolution (SuSSADE). Each three-qubit gate designed by SuSSADE operates as fast as an entangling two- qubit gate under the same experimental constraints. Preface We have two published papers [1, 2], the content of which are used in the appropriate sections of my thesis. The main portion of the papers are used in Chapters 5 and 6eitherverbatimorwithsomerequiredmodifications.Inordertokeeptheflowof di↵erent subjects smooth in my thesis, I have used the background and introduction parts of the papers in the early chapters of my thesis. Here I list the two papers which are published based on this work: 1. Ehsan Zahedinejad, Sophie Schirmer, and Barry C. Sanders. Evo- lutionary algorithms for hard quantum control. Physical Review A, 90:032310, Sep 2014. arXiv.org:1403.0943 2. Ehsan Zahedinejad, Joydip Ghosh, and Barry C. Sanders. High-fidelity single-shot To↵oli gate via quantum control. Physical Review Letters, 114:200502, May 2015. arXiv.org:1501.04676 In order to make the material taken from our publications and used in my thesis more informative, I have modified and introduced some changes into the context of previously published work. The changes that I made to those materials taken from our publications and used in the body of my thesis are listed as below: Chapter 2: Section 2.4.3 contains some sentences verbatim (but not • explicitly marked) from the introduction sections of [1, 2]. Chapter 3: Sections 3.3 and 3.11 transcribed from [1]. • Chapter 5: Mostly contains material from [1]. The following changes • are made to the content to make the chapter more informative: – Whenever found appropriate, a sentence is added to refer to the introduction and background chapters. All figures iv in this chapter are cited to the corresponding figures in the paper. – In Section 5.1 the first paragraph is completely modified. The gate operation time T is changed to ⇥ to be consis- tent with the rest of the thesis. The rest of this section is unchanged. – Subsection 5.3.1 is modified to clarify our choice of ex- ternal pulse. A sentence is added to refer to Section 3.5 for more clarity. – In Subsection 5.3.2 we have removed the details of evo- lutionary algorithms. A sentence is added to refer to Secs. 3.11 and 3.10 in which we discussed the optimiza- tion algorithms in detail. – Subsection 5.3.3 is modified slightly by adding a sentence which refers to the evolution equation 3.4. Chapter 6 contains some of the material from [2]. All the result for the To↵oli gate are reproduced from [2] with proper citations to the original publication. Other figures are based on our new results which will be published in [3]. The following are the list of the changes made to the content to make the chapter more informative: Section 6.1: The first paragraph includes the first paragraph of [2] with • some modifications to include two other three-qubit gates (i. e. CNOT- NOT and Fredkin gates). Section 6.1: The fourth paragraph copied from [2]. • Subsection 6.4.6 transcribed from [2] and modified extensively to be • more clear. We added a sentence to refer to original DE algorithm in Section 3.11. Section 6.5 is copied verbatim from [2]. • Subsection 6.6.1: The second paragraph is transcribed from [2]. • Subection 6.8.3 discussion on distortion of control pulses is transcribed • from [2]. Acknowledgements IwouldliketoexpressmyspecialappreciationtomyadvisorBarrySanders,forhis endless support, enthusiasm, and insight during my PhD studies. I would like to thank him for allowing me to follow my interest in research and to become an independent researcher. Your advice on both research and my career have been priceless. I also like to thank my co-supervisor Dennis Salahub for his subtle guidance on my research on molecular dynamics Simulation. I am deeply indebted to Joydip Ghosh whose insight and valuable ideas helped me to complete this work. I am also grateful for the expertise of my recent scientific collaborators, Sophie Schirmer, Nathan Babcock. I specially thank Doug Phillips for helping me to become familiar with the world of supercomputers during the first year of my PhD program. For insightful discussions and words of encouragement, I wish to thank Jonny Johannes, Pantita Palittapongarnpim, and Hamidreza Kaviani. I am grateful to Lucia Wang, Nancy Jing Lu, Tracy Korsgaard, and Gerri Zannet for their administration assistance during my PhD studies. IacknowledgeWestgridComputeCanadaforprovidingcomputationalresources to enable this work. I acknowledge the Murray Fraser Memorial Graduate program and Eyes High International Doctoral Scholarship and support from Natural Sciences and Engineering Research Council of Canada. I would not be here without the moral and physical support of my great parents Ghorban Zahedinejad and Nosrat Zahedi. Words cannot express my feelings, nor my thanks for all your sacrifice in my life. I would like to thank my brothers and sister Ali, Mohammad and Elahe. Just saying thank you will never repay your kindness. Last but not the least I would like to thank my beautiful wife, Samira Hafezi for her support and care during the stressful time of my PhD studies. Your support and vii encouragement was in the end what made this dissertation possible. I cannot imagine what would I do without you in my life! Table of Contents Abstract .................................... ii Preface ..................................... iv Acknowledgements .............................. vii TableofContents................................ ix List of Tables . xii ListofFigures.................................. xiii List of Symbols . xvii 1Introduction................................ 1 1.1 Motivation . 1 1.2 Research problem . 3 1.3 Research objective . 4 1.4 Physical models . 4 1.5 Approach ................................. 5 1.6 Summary of the research achievements . 9 1.7 Overview of chapters . 10 2QuantumInformationProcessing:APrelude..............13 2.1 Quantum information theory . 13 2.2 Classical and quantum bits . 14 2.3 Classical logic gates . 16 2.4 Quantum logic gates . 17 2.4.1 Single-qubit gates: Pauli matrices and the Hadamard . 18 2.4.2 Controlled-NOT(CNOT)gate. 19 2.4.3 Three-qubit gates: To↵oli and Fredkin . 20 2.5 Universalquantumgates . 22 2.6 Quantum error correction . 22 2.7 Fault-tolerant quantum computing and threshold theorem . 24 2.8 Quantum noise . 24 2.8.1 Decoherence-induced noise . 25 2.8.2 Quantum noise and quantum operation . 26 2.8.3 Quantumoperationsandenvironment . 27 2.8.4 Operator-sum representation . 27 3QuantumControl:Background.....................29 3.1 Optimal control theory: Application in quantum systems . 29 3.2 Control fields: Time- vs. frequency-domains . 31 3.3 Quantum control . 32 3.4 Machine learning: A quantum control tool .
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