DOCSLIB.ORG

Methods for Efficient Synthesis of Large Reversible Binary and Ternary Quantum Circuits and Applications of Linear Nearest Neighbor Model

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Methods for Efficient Synthesis of Large Reversible Binary and Ternary Quantum Circuits and Applications of Linear Nearest Neighbor Model Portland State University PDXScholar Dissertations and Theses Dissertations and Theses Spring 5-30-2013 Methods for Efficient Synthesis of Large Reversible Binary and Ternary Quantum Circuits and Applications of Linear Nearest Neighbor Model Maher Mofeid Hawash Portland State University Follow this and additional works at: https://pdxscholar.library.pdx.edu/open_access_etds Part of the Other Electrical and Computer Engineering Commons Let us know how access to this document benefits ou.y Recommended Citation Hawash, Maher Mofeid, "Methods for Efficient Synthesis of Large Reversible Binary and Ternary Quantum Circuits and Applications of Linear Nearest Neighbor Model" (2013). Dissertations and Theses. Paper 1090. https://doi.org/10.15760/etd.1090 This Dissertation is brought to you for free and open access. It has been accepted for inclusion in Dissertations and Theses by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. Methods for Efficient Synthesis of Large Reversible Binary and Ternary Quantum Circuits and Applications of Linear Nearest Neighbor Model by Maher Mofeid Hawash A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical and Computer Engineering Dissertation Committee: Marek Perkowski, Chair Douglas Hall Malgorzata Chrzanowska-Jeske Christof Teuscher John Caughman Portland State University 2013 Abstract This dissertation describes the development of automated synthesis algorithms that construct reversible quantum circuits for reversible functions with large number of variables. Specifically, the research area is focused on reversible, permutative and fully specified binary and ternary specifications and the applicability of the resulting circuit to the physical limitations of existing quantum technologies. Automated synthesis of arbitrary reversible specifications is an NP hard, multi- objective optimization problem, where 1) the amount of time and computational resources required to synthesize the specification, 2) the number of primitive quantum gates in the resulting circuit (quantum cost), and 3) the number of ancillary qubits (variables added to hold intermediate calculations) are all minimized while 4) the number of variables is maximized. Some of the existing algorithms in the literature ignored objective 2 by focusing on the synthesis of a single solution without the addition of any ancillary qubits while others attempted to explore every possible solution in the search space in an effort to discover the optimal solution (i.e., sacrificed objective 1 and 4). Other algorithms resorted to adding a huge number of ancillary qubits (counter to objective 3) in an effort minimize the number of primitive gates (objective 2). In this dissertation, I first introduce the MMDSN algorithm that is capable of synthesizing binary specifications up to 30 variables, does not add any ancillary variables, produces better quantum cost (8-50% improvement) than algorithms which limit their search to a single solution and within a minimal amount of time compared to i algorithms which perform exhaustive search (seconds vs. hours). The MMDSN algorithm introduces an innovative method of using the Hasse diagram to construct candidate solutions that are guaranteed to be valid and then selects the solution with the minimal quantum cost out of this subset. I then introduce the Covered Set Partitions (CSP) algorithm that expands the search space of valid candidate solutions and allows for exploring solutions outside the range of MMDSN. I show a method of subdividing the expansive search landscape into smaller partitions and demonstrate the benefit of focusing on partition sizes that are around half of the number of variables (15% to 25% improvements, over MMDSN, for functions less than 12 variables, and more than 1000% improvement for functions with 12 and 13 variables). For a function of n variables, the CSP algorithm, theoretically, requires n times more to synthesize; however, by focusing on the middle k (k<n) partitions, the CSP algorithms only requires k times the amount of time required by MMDSN which typically yields lower quantum cost. I also show that using a Tabu search for selecting the next set of candidate from the CSP subset results in discovering solutions with even lower quantum costs (up to 10% improvement over CSP with random selection). In Chapters 9 and 10 I question the predominant methods of measuring quantum cost and its applicability to physical implementation of quantum gates and circuits. I counter the prevailing literature by introducing a new standard for measuring the performance of quantum synthesis algorithms by enforcing the Linear Nearest Neighbor Model (LNNM) ii constraint, which is imposed by the today’s leading implementations of quantum technology. In addition to enforcing physical constraints, the new LNNM quantum cost (LNNQC) allows for a level comparison amongst all methods of synthesis; specifically, methods which add a large number of ancillary variables to ones that add no additional variables. I show that, when LNNM is enforced, the quantum cost for methods that add a large number of ancillary qubits increases significantly (up to 1200%). I also extend the Hasse based method to the ternary and I demonstrate synthesis of specifications of up to 9 ternary variables (compared to 3 ternary variables that existed in the literature). I introduce the concept of ternary precedence order and its implication on the construction of the Hasse diagram and the construction of valid candidate solutions. I also provide a case study comparing the performance of ternary logic synthesis of large functions using both a CUDA graphic processor with 1024 cores and an Intel i7 processor with 8 cores. In the process of exploring large ternary functions I introduce, to the literature, eight families of ternary benchmark functions along with a Multiple Valued file specification (the Extended Quantum Specification XQS). I also introduce a new composite quantum gate, the multiple valued Swivel gate, which swaps the information of qubits around a centrally located pivot point. In summary, my research objectives are as follows: • Explore and create automated synthesis algorithms for reversible circuits both in binary and ternary logic for large number of variables. iii • Study the impact of enforcing Linear Nearest Neighbor Model (LNNM) constraint for every interaction between qubits for reversible binary specifications. • Advocate for a revised metric for measuring the cost of a quantum circuit in concordance with LNNM, where, on one hand, such a metric would provide a way for balanced comparison between the various flavors of algorithms, and on the other hand, represents a realistic cost of a quantum circuit with respect to an ion trap implementation. • Establish an open source repository for sharing the results, software code and publications with the scientific community. With the dwindling expectations for a new lifeline on silicon-based technologies, quantum computations have the potential of becoming the future workhorse of computations. Similar to the automated CAD tools of classical logic, my work lays the foundation for creating automated tools for constructing quantum circuits from reversible specifications. iv Dedication I dedicate this dissertation to My loving parents Mofeid and Raefah Hawash; My wife Lisa for her unwavering support; My wonderful children Jared, Sarra and Amani; My brother Shaher; My Sisters Shaheera, Maheera, Mohaya & Majdoleen; And my family and friends around the globe. v Acknowledgments I offer my sincere gratitude to many individuals for their support, advice and motivation to help fulfill my aspiration to attain a PhD in Electrical and Computer Engineering. I am deeply indebted to Dr. Marek A. Perkowski for his transcendent patronage and mentorship throughout the years of work on my PhD and for his interminable motivation and inspirations to help me blaze forward. I am greatly impressed by the tremendous effort he expends for the sake of his students specially the time and energy he spends in nurturing middle and high school students in quantum computing and robotics. I have personally benefited throughout the years from the lectures and discussions in his weekly Friday meeting and from working directly with him in developing ideas, reviewing results and drafting papers for publishing. I am committed to, in addition to absorbing his well of knowledge, by following in Dr. Perkowski’s footsteps in mentoring young students to attain their own scientific achievement. I would also like to extend my deepest appreciations to the committee members who have given me valuable feedback for improving the quality of results from my simulation of quantum logic synthesis and for introducing me to new concepts. I am grateful for Dr. Douglas Hall for the multiple roles that he took as my graduate advisor and a supportive member of my committee. Dr. Malgorzata Chrzanowska-Jeske has been extremely helpful in reviewing my dissertation and giving me valuable feedback in regards to organization and content. I am greatly appreciative of her insightful questions and vi constructive advice during my comprehensive exam and dissertation proposal. I cannot thank Dr. Christof Teuscher enough for expanding my field of vision by introducing me to the stimulating and promising nanotechnology, which I hope to dedicate more time to learn and explore in the future. Since my first
Load more

Recommended publications
  • Development of Tool for Mapping Conventional Circuit to Reversible
    Development of Tool for Mapping Conventional Circuit to Reversible Logic A Dissertation Submitted in Partial fulfillment for the award of the Degree of MASTER OF TECHNOLOGY in Department of Computer Science Engineering (With specialization in COMPUTER SCIENCE & ENGINEERING) Supervisor Submitted By Dr. S.C. Jain Piyush Gautam (Professor) Enrolment No.: 10E2UCCSM4XT607 DEPARTMENT OF COMPUTER SCIENCEENGINEERING UNIVERSITY COLLEGE OF ENGINEERING RAJASTHAN TECHNICAL UNIVERSITY KOTA (RAJASTHAN) March 2014 CANDIDATE’S DECLARATION I hereby declare that the work, which is being presented in the Dissertation, entitled “Development of Tool for Mapping Conventional Circuit to Reversible Logic” in partial fulfillment for the award of Degree of “Master of Technology” in Dept. of Computer Science Engineering with Specialization in Computer Science, and submitted to the Department of Computer Science Engineering, University College of Engineering, Kota, Rajasthan Technical University is a record of my own investigations carried under the Guidance of Dr. S.C. Jain, Department of Computer Science Engineering, University College of Engineering, Kota. I have not submitted the matter presented in this Dissertation anywhere for the award of any other Degree. Piyush Gautam Computer Science & Engineering Enrolment No.: 10E2UCCSM4XT607 University College of Engineering, Kota (Rajasthan) Under Guidance of Dr. S. C. Jain Professor, Department of Computer Science & Engineering University College of Engineering, Kota (Rajasthan) ii CERTIFICATE This is to certify that this Dissertation entitled “Development of Tool for Mapping Conventional Circuit to Reversible Logic” has been successfully carried out by Piyush Gautam (Enrolment No.:10E2UCCSM4XT607), under my supervision and guidance, in partial fulfillment of the requirement for the award of Master of Technology Degree in Computer Science & Engineering from University College of Engineering, Rajasthan Technical University, Kota for the year 2010-2012.
    [Show full text]
  • Chapter 2 Quantum Gates
    Chapter 2 Quantum Gates “When we get to the very, very small world—say circuits of seven atoms—we have a lot of new things that would happen that represent completely new opportunities for design. Atoms on a small scale behave like nothing on a large scale, for they satisfy the laws of quantum mechanics. So, as we go down and fiddle around with the atoms down there, we are working with different laws, and we can expect to do different things. We can manufacture in different ways. We can use, not just circuits, but some system involving the quantized energy levels, or the interactions of quantized spins.” – Richard P. Feynman1 Currently, the circuit model of a computer is the most useful abstraction of the computing process and is widely used in the computer industry in the design and construction of practical computing hardware. In the circuit model, computer scien- tists regard any computation as being equivalent to the action of a circuit built out of a handful of different types of Boolean logic gates acting on some binary (i.e., bit string) input. Each logic gate transforms its input bits into one or more output bits in some deterministic fashion according to the definition of the gate. By compos- ing the gates in a graph such that the outputs from earlier gates feed into the inputs of later gates, computer scientists can prove that any feasible computation can be performed. In this chapter we will look at the types of logic gates used within circuits and how the notions of logic gates need to be modified in the quantum context.
    [Show full text]
  • Quantum Computers: a Brief Overview
    Quantum Computers: A Brief Overview Merciadri Luca [email protected] Abstract. A promising technology is the “quantum computers,” and this paper gives a general overview about this subject. Keywords: quantum computers. 1 Introduction c being a number. [10] If c = φ ψ the complex con- jugate is h | i A promising technology is the “quantum computers.” c∗ = φ ψ ∗ = ψ φ . (2) More and more scientists are interested in it because h | i h | i The state of a physical system is identified with of the performances’ enhancement it could bring to a ray in a complex separable1 Hilbert space, denoted the today’s computing world. by , or equivalently, by a point in the projective Quantum computers require quantum logic, which HilbertH space of the system. [14] is fundamentally different to classical Boolean logic. Each vector in the ray is called a “ket.” Evidently, This difference leads to a greater efficiency of quan- in the ket ψ , ψ can be replaced by any symbols, let- tum computation over its classical counterpart. [13] ters, numbers,i or even words. The ket can be viewed However, many articles about quantum comput- as a column-vector and (given a basis for the Hilbert ers are still a bit difficult to understand for the “abso- space) written out in components, lute beginner.” Thus, the main aim of this paper is to give a simple overview, without coming into technical ^ ψ = (c , c , ), (3) details, of quantum computing, basing principally on | i 0 1 · · · [3,9,10,13]. when the considered Hilbert space is finite- dimensional.
    [Show full text]
  • Designing Gates and Architectures for Superconducting Quantum Systems
    Portland State University PDXScholar Dissertations and Theses Dissertations and Theses 6-9-2020 Designing Gates and Architectures for Superconducting Quantum Systems Sahar Daraeizadeh Portland State University Follow this and additional works at: https://pdxscholar.library.pdx.edu/open_access_etds Part of the Electrical and Computer Engineering Commons Let us know how access to this document benefits ou.y Recommended Citation Daraeizadeh, Sahar, "Designing Gates and Architectures for Superconducting Quantum Systems" (2020). Dissertations and Theses. Paper 5451. https://doi.org/10.15760/etd.7324 This Dissertation is brought to you for free and open access. It has been accepted for inclusion in Dissertations and Theses by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. Designing Gates and Architectures for Superconducting Quantum Systems by Sahar Daraeizadeh A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical and Computer Engineering Dissertation Committee: Xiaoyu Song, Chair Marek Perkowski, Co-chair Douglas Hall Steven Bleiler Anne Y. Matsuura Portland State University 2020 Abstract Large-scale quantum computers can solve certain problems that are not tractable by currently available classical computational resources. The building blocks of quantum computers are qubits. Among many different physical realizations for qubits, superconducting qubits are one of the promising candidates to re- alize gate model quantum computers. In this dissertation, we present new multi-qubit gates for nearest-neighbor superconducting quantum systems. In the absence of a physical hardware, we simulate the dynamics of the quan- tum system and use the simulated environment as a framework for test, design, and optimization of quantum gates and architectures.
    [Show full text]
  • Toffoli Gate
    Building Blocks for Quantum Computing PART II Patrick Dreher CSC801 – Seminar on Quantum Computing Spring 2018 Building Blocks for Quantum Computing 1-February-2018 1 Patrick Dreher Building Quantum Circuit Gates Using Qubits and Quantum Mechanics Building Blocks for Quantum Computing 1-February-2018 2 Patrick Dreher Classical Gates versus Quantum Gates • Quantum physics puts restrictions on the types of gates that can be incorporated into a quantum computer • The requirements that – A quantum gate must incorporate the linear superposition of pure states that includes a phase – all closed quantum state transformations must be reversible restrict the type of logic gates available for constructing a quantum computer • The classical NOT gate is reversible but the AND, OR and NAND gates are not Building Blocks for Quantum Computing 1-February-2018 3 Patrick Dreher Classical Logic Gates • There are several well known logic gates Building Blocks for Quantum Computing 1-February-2018 4 Patrick Dreher Quantum Property of Reversibility and Constraints of Gate Operations • Reversibility can be quantified mathematically through the matrix representation of the logic gate • The IDENTITY operation and NOT gates are “reversible” the classical NOT gate are reversible, the AND, OR and NAND gates are not • The outcome of the gate can be undone by applying other gates, or effectively additional matrix operations • The matrix has the property of preserving the length of vectors, implying that the matrices are unitary, thereby satisfying the Axiom 4 requirement
    [Show full text]
  • Quantum Neural Networks
    QUANTUM NEURAL NETWORKS A Dissertation by Nam H. Nguyen Master of Science, Wichita State University, 2016 Bachelor of Arts, Eastern Oregon University, 2014 Submitted to the Department of Mathematics, Statistics, and Physics and the faculty of the Graduate School of Wichita State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy May 2020 © Copyright 2020 by Nam H. Nguyen All Rights Reserved QUANTUM NEURAL NETWORKS The following faculty members have examined the final copy of this dissertation for form and content, and recommend that it be accepted in partial fulfillment of the requirement for the degree of Doctor of Philosophy with a major in Applied Mathematics. Elizabeth Behrman, Committee Chair James Steck, Committee Member Buma Fridman, Committee Member Ziqi Sun, Committee Member Terrance Figy, Committee Member Accepted for the College of Liberal Arts and Sciences Andrew Hippisley, Dean Accepted for the Graduate School Coleen Pugh, Dean iii DEDICATION To my late grandmother Phan Thi Suu , and my loving mother Nguyen Thi Kim Hoa iv ACKNOWLEDGEMENTS First and foremost, I would like to express my profound gratitude to my advisor, Dr. Elizabeth Behrman, for her advice, support and guidance toward my Ph.D. degree. She taught me not only the way to do scientific research, but also the way to become a profes- sional scientist and mathematician. Her endless encouragements and patient throughout my graduate studied have positively impacted my life. She would always allowed me to work on different research areas and ideas even if it's outside the scope of my dissertation. This allows me to learn and investigate problems in many different areas of mathematics, which have benefited me greatly.
    [Show full text]
  • Chiral Spin Flipping Gate Implemented in IBM Quantum Experience
    Appl. Math. Inf. Sci. 11, No. 5, 1519-1526 (2017) 1519 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.18576/amis/110531 Chiral Spin Flipping Gate Implemented in IBM Quantum Experience 1,2 1 1 1 1,3, Shi-Yuan Ma , Sheng-Wen Li , Han Cai , Da-Wei Wang and Hichem Eleuch ∗ 1 Institute for Quantum Science and Engineering, Texas A&M University, College Station, TX 77843, USA 2 University of Science and Technology of China, Hefei, Anhui 230026, China 3 Department of Applied Sciences and Mathematics, College of Arts and Sciences, Abu Dhabi University, Abu Dhabi, UAE Received: 2 Jul. 2017, Revised: 25 Aug. 2017, Accepted: 28 Aug. 2017 Published online: 1 Sep. 2017 Abstract: A chiral spin flipping gate is simulated by specially designed quantum circuits. We consider using ancillary qubits to accomplish the algorithm as well as quantum circuits based on different basic operations. Three methods are applied to build these algorithms, which result in three different quantum circuits that all realize the evolution process of the system. We also implement our quantum algorithms on the IBM Quantum Experience. Keywords: Quantum computation 1 Introduction design, simulation, testing, and actual execution of an algorithm on a physical device. We know that the We are now in the frontier of a new area of quantum quantum computers need not only new hardware simulation [1,2], since Feynman [3] proposed the idea of components but also software languages and algorithms. quantum computer and envisioned the possibility of With the help of IBM QE, scientists can conveniently efficiently simulating quantum systems, significant implement their own algorithms, which will greatly progress has been made.
    [Show full text]
  • Realization of Efficient Quantum Gates with a Superconducting Qubit-Qutrit Circuit
    www.nature.com/scientificreports OPEN Realization of efcient quantum gates with a superconducting qubit- qutrit circuit Received: 15 January 2019 T. Bækkegaard1, L. B. Kristensen1, N. J. S. Loft1, C. K. Andersen 2, D. Petrosyan 3 & Accepted: 8 August 2019 N. T. Zinner1,4 Published: xx xx xxxx Building a quantum computer is a daunting challenge since it requires good control but also good isolation from the environment to minimize decoherence. It is therefore important to realize quantum gates efciently, using as few operations as possible, to reduce the amount of required control and operation time and thus improve the quantum state coherence. Here we propose a superconducting circuit for implementing a tunable system consisting of a qutrit coupled to two qubits. This system can efciently accomplish various quantum information tasks, including generation of entanglement of the two qubits and conditional three-qubit quantum gates, such as the Tofoli and Fredkin gates. Furthermore, the system realizes a conditional geometric gate which may be used for holonomic (non- adiabatic) quantum computing. The efciency, robustness and universality of the presented circuit makes it a promising candidate to serve as a building block for larger networks capable of performing involved quantum computational tasks. Richard Feynman famously suggested to simulate quantum physics with quantum computers1. Fourteen years later, Seth Lloyd proved that an array of spins with tunable interactions indeed represents a universal quantum simulator2. Dynamically controlled spin chains can realize analog quantum simulations and digital quantum computations. Several physical systems are being explored for implementing tunable spin chains in the quantum regime, including trapped ions and atoms3–5, quantum dots6 and superconducting circuits7,8.
    [Show full text]
  • Quantum Fredkin Gate
    Quantum Fredkin Gate Researchers from Griffith University and the University of Queensland have overcome one of the key challenges to quantum computing by simplifying a complex quantum logic operation. They demonstrated this by experimentally realising a challenging circuit—the quantum Fredkin gate—for the first time. [32] A team of researchers from Google, the University of the Basque Country, the University of California and IKERBASQUE, Basque Foundation for Science has devised a means for combining the two leading ideas for creating a quantum computer in one machine, offering a possible means for learning more about how to create a true quantum computer sometime in the future. [31] When future users of quantum computers need to analyze their data or run quantum algorithms, they will often have to send encrypted information to the computer. [30] Quantum systems were believed to provide perfectly secure data transmission because until now, attempts to copy the transmitted information resulted in an altered or deteriorated version of the original information, thereby defeating the purpose of the initial hack. [29] Researchers have developed a new type of light-enhancing optical cavity that is only 200 nanometers tall and 100 nanometers across. Their new nanoscale system represents a step toward brighter single-photon sources, which could help propel quantum-based encryption and a truly secure and future-proofed network. [28] Researchers at Tohoku University have, for the first time, successfully demonstrated the basic operation of spintronics-based artificial intelligence. [27] The neural structure we use to store and process information in verbal working memory is more complex than previously understood, finds a new study by researchers at New York University.
    [Show full text]