DOCSLIB.ORG

Introduction to Machine Learning and Quantum Computing CHEM 584

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to Machine Learning and Quantum Computing CHEM 584 Introduction to Machine Learning and Quantum Computing CHEM 584: Lecture Notes Prof. Victor S. Batista Zoom: Meeting ID: 943 8610 8716, Passcode: victor Tuesdays and Thursdays 11:35 am – 12:50 pm Yale University - Department of Chemistry 1 Contents 1 Syllabus 6 2 Feedforward Neural Networks 7 2.1 Gradient Descent . 8 2.1.1 Stochastic Gradient Descent . 9 2.1.2 Exercise: Gradient Descent . 9 2.2 Colab, Python, Tensorflow, Keras and Pytorch . 10 2.3 Activation Functions . 11 2.3.1 Linear Activation . 11 2.3.2 Non-linear Activation . 11 2.3.2.1 Classification and Non-linear Regression Problems . 12 2.3.2.1.1 Optional Exercise: . 12 2.3.3 Vanishing Gradient Problem . 13 2.3.4 Validation, Cross Validation and Bootstrapping . 14 2.3.4.1 Exercise: Bootstrapping . 15 2.4 Tutorial Assignment on Hammett Neural Networks with Keras/TensorFlow . 15 2.5 Tutorial Regressive Models for Chemical Predictions with Scikit-Learn . 15 2.6 Prediction of Molecular Toxicity by Linear Classification with DeepChem . 15 2.6.1 Training . 16 2.6.2 Overfitting Problem . 16 2.6.3 Regularization . 17 3 Clustering and Regression Algorithms 18 3.1 Random Forest . 18 3.1.1 Entropy and Gini . 19 3.2 K-means Algorithm . 19 3.3 K-Nearest Neighbors Algorithm . 20 3.4 Unsupervised Classification Assignment: K-means and Random Forest . 20 4 Convolutional Neural Networks (CNN): Alphafold 21 5 Graph Convolutional Networks (GCN) 24 5.1 Propagation Rules . 25 5.2 Prediction of NMR Chemical Shifts by Graph Convolutional Networks . 25 5.3 Prediction of Solubilities by Graph Convolutional Networks with DeepChem . 26 5.4 Introduction to classification by Graph Convolutional Networks with DeepChem . 26 6 Recurrent Neural Networks (RNN) 27 7 Autoencoders 29 7.1 RNN, CNN and Multi-Head Attention Autoencoders . 30 7.1.1 Attention Mechanism . 31 7.2 Variational Autoencoders (VAE) and Generative Adversarial Networks (GAN) . 32 7.2.1 Maximum Mean Discrepancy Method . 33 7.3 Time Series Prediction: Dynamical Mode Decomposition ............ 34 7.4 Hybrid Quantum-Classical Neural Network ..................... 39 7.5 Variational Optimization with Hybrid Neural Networks . 40 2 7.5.1 Quantum Computation . 40 7.5.2 Hybrid Quantum-Classical Computation . 40 7.5.3 Variational Quantum Eigensolver . 40 8 Qubits and Gates 41 8.1 Single qubit gates . 41 8.2 Rotations . 42 8.3 Multiple qubits . 42 8.4 The CNOT gate . 43 8.5 Hadamard Gate . 43 9 Superconducting Circuits: IBM Quantum Computer 44 9.1 Transmon: Capacitively Shunted Junction . 46 9.2 Kerr Hamiltonian ..................................... 49 9.3 SQUID: Tunable Junction . 51 9.4 Cooper Pair Box: Charge Qubit . 52 9.4.1 CPB Eigenstates . 52 9.4.2 NMR Hamiltonian . 53 9.4.3 State Preparation and Control . 54 9.5 Split Cooper Pair Box . 55 9.6 Transmon Coupled to a Resonator . 56 9.6.1 Quantization . 57 9.6.2 Resonant and Dispersive Limits ........................ 58 9.6.2.1 Resonant Limit: . 58 9.6.2.2 Dispersive Limit: . 58 10 Dicke Model and Jaynes-Cummings Hamiltonian 60 11 Grover’s Algorithm 61 11.1 Supplement I: Average Deviation Caused by the Oracle . 66 11.2 Supplement II: Optimal Number of Queries . 68 11.3 Supplement III: Average Success Probability of Grover’s Algorithm . 70 12 Iterative Power Algorithm: Classical Amplitude Amplification 71 12.1 Convergence . 71 13 Bernstein-Vazirani Algorithm: Exponential Speedup from Superpositions 72 13.1 Hadamard Transform of Arbitrary Strings . 73 14 Phase Kickback 74 15 Hadamard Gate with Beam Splitters: Mach-Zehnder interferometer 74 16 Deutsch Algorithm 79 17 Simon’s Algorithm 80 3 18 Quantum Fourier Transform 85 18.1 Properties of the Fourier transform .......................... 86 18.2 Quantum Phase Estimation .............................. 90 18.3 Period Finding ....................................... 98 18.4 Shor’s Algorithm ..................................... 99 19 Appendix I: Golden Rule 101 19.1 Monochromatic Plane Wave .............................. 101 20 Appendix II: Coherent States 105 20.1 Overlap . 106 20.2 Closure . 106 20.3 Wavefunctions . 107 20.4 Expectation Values . 107 20.4.1 Optical Equivalence Theorem . 108 20.4.2 P-representation of the density operator . 108 20.4.2.1 Pure coherent state . 109 20.4.2.2 Pure number state . 110 20.4.3 P-representation of operators . 111 20.5 Dynamics . 111 20.6 Parity Operator . 112 21 Appendix III: Python 114 21.1 A Brief Note on Python Versions . 114 21.1.1 Basics of Python . 114 21.1.1.1 Basic data types . 114 21.1.1.1.1 Numbers . 115 21.1.1.1.2 Booleans . 115 21.1.1.1.3 Strings . 116 21.1.1.2 Containers . 117 21.1.1.2.1 Lists . 117 21.1.1.2.2 Slicing . 118 21.1.1.2.3 Loops . 118 21.1.1.2.4 List comprehensions: . 119 21.1.1.2.5 Dictionaries . 119 21.1.1.2.6 Sets . 121 21.1.1.2.7 Tuples . 122 21.1.1.3 Functions . 123 21.1.1.4 Classes . 124 21.1.1.5 Modules . 126 21.1.2 Numpy . 126 21.1.2.1 Arrays . 126 21.1.2.2 Array indexing . 128 21.1.2.3 Datatypes . 131 21.1.2.4 Array math . 131 21.1.2.5 Broadcasting . 134 21.1.3 Matplotlib . 137 21.1.3.1 Plotting . 137 4 21.1.3.2 Subplots . 139 21.2 Torch tensor . 140 5 1 Syllabus Machine learning and quantum computing have emerged as leading technologies of the twenty- first century and are expected to be increasingly applied to address a wide variety of chemical and materials science challenges. The goal of this course is to introduce fundamental concepts of machine learning and quantum computing to chemists and materials science students through an overview of algorithms, computational methods, and applications. It is intended to empower students to engage with this emerging field and foster the growing field of artificial intelligence for accelerated scientific discoveries in the molecular and physical sciences. Textbooks. Recommended textbooks for this class are: R1:"Pattern Recognition and Machine Learning" by Christopher M. Bishop (Springer, 2006). (pdf) (matlab) R2: "Deep Learning" by Ian Goodfellow, Yoshua Bengio and Aaron Courville. (pdf) (github). R3: "Deep Learning for Coders with Fastai and PyTorch: AI Applications Without a PhD" by Jeremy Howard and Sylvain Gugger. (github) R4: "Dive into Deep Learning" by Jeremy Howard and Sylvain Gugger. (pdf) R5: "Quantum Computation and Quantum Information" by Michael A Nielsen and Isaac L. Chuang (Cambridge). R6: "An Introduction to Quantum Computing" by Phillip Kaye, Raymond Laflamme and Michele Mosca (Oxford University Press). (pdf) R7: "Learn Quantum Computation Using Qiskit" and notebook. Our lecture notes will be updated according to the pace of the course and suggestions.
Load more

Recommended publications
  • Direct Dispersive Monitoring of Charge Parity in Offset-Charge
    PHYSICAL REVIEW APPLIED 12, 014052 (2019) Direct Dispersive Monitoring of Charge Parity in Offset-Charge-Sensitive Transmons K. Serniak,* S. Diamond, M. Hays, V. Fatemi, S. Shankar, L. Frunzio, R.J. Schoelkopf, and M.H. Devoret† Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA (Received 29 March 2019; revised manuscript received 20 June 2019; published 26 July 2019) A striking characteristic of superconducting circuits is that their eigenspectra and intermode coupling strengths are well predicted by simple Hamiltonians representing combinations of quantum-circuit ele- ments. Of particular interest is the Cooper-pair-box Hamiltonian used to describe the eigenspectra of transmon qubits, which can depend strongly on the offset-charge difference across the Josephson element. Notably, this offset-charge dependence can also be observed in the dispersive coupling between an ancil- lary readout mode and a transmon fabricated in the offset-charge-sensitive (OCS) regime. We utilize this effect to achieve direct high-fidelity dispersive readout of the joint plasmon and charge-parity state of an OCS transmon, which enables efficient detection of charge fluctuations and nonequilibrium-quasiparticle dynamics. Specifically, we show that additional high-frequency filtering can extend the charge-parity life- time of our device by 2 orders of magnitude, resulting in a significantly improved energy relaxation time T1 ∼ 200 μs. DOI: 10.1103/PhysRevApplied.12.014052 I. INTRODUCTION charge states, like a usual transmon but with measurable offset-charge dispersion of the transition frequencies The basic building blocks of quantum circuits—e.g., between eigenstates, like a Cooper-pair box. This defines capacitors, inductors, and nonlinear elements such as what we refer to as the offset-charge-sensitive (OCS) Josephson junctions [1] and electromechanical trans- transmon regime.
    [Show full text]
  • A Scanning Transmon Qubit for Strong Coupling Circuit Quantum Electrodynamics
    ARTICLE Received 8 Mar 2013 | Accepted 10 May 2013 | Published 7 Jun 2013 DOI: 10.1038/ncomms2991 A scanning transmon qubit for strong coupling circuit quantum electrodynamics W. E. Shanks1, D. L. Underwood1 & A. A. Houck1 Like a quantum computer designed for a particular class of problems, a quantum simulator enables quantitative modelling of quantum systems that is computationally intractable with a classical computer. Superconducting circuits have recently been investigated as an alternative system in which microwave photons confined to a lattice of coupled resonators act as the particles under study, with qubits coupled to the resonators producing effective photon–photon interactions. Such a system promises insight into the non-equilibrium physics of interacting bosons, but new tools are needed to understand this complex behaviour. Here we demonstrate the operation of a scanning transmon qubit and propose its use as a local probe of photon number within a superconducting resonator lattice. We map the coupling strength of the qubit to a resonator on a separate chip and show that the system reaches the strong coupling regime over a wide scanning area. 1 Department of Electrical Engineering, Princeton University, Olden Street, Princeton 08550, New Jersey, USA. Correspondence and requests for materials should be addressed to W.E.S. (email: [email protected]). NATURE COMMUNICATIONS | 4:1991 | DOI: 10.1038/ncomms2991 | www.nature.com/naturecommunications 1 & 2013 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms2991 ver the past decade, the study of quantum physics using In this work, we describe a scanning superconducting superconducting circuits has seen rapid advances in qubit and demonstrate its coupling to a superconducting CPWR Osample design and measurement techniques1–3.
    [Show full text]
  • A Radix-4 Chrestenson Gate for Optical Quantum Computation
    A Radix-4 Chrestenson Gate for Optical Quantum Computation Kaitlin N. Smith, Tim P. LaFave Jr., Duncan L. MacFarlane, and Mitchell A. Thornton Quantum Informatics Research Group Southern Methodist University Dallas, TX, USA fknsmith, tlafave, dmacfarlane, mitchg @smu.edu Abstract—A recently developed four-port directional coupler is described in Section III followed by the realization of the used in optical signal processing applications is shown to be coupler with optical elements including its fabrication and equivalent to a radix-4 Chrestenson operator, or gate, in quantum characterization in Section IV. Demonstration of the four-port information processing (QIP) applications. The radix-4 qudit is implemented as a location-encoded photon incident on one of coupler as a radix-4 Chrestenson gate is presented in Section V the four ports of the coupler. The quantum informatics transfer and a summary with conclusions is found in Section VI. matrix is derived for the device based upon the conservation of energy equations when the coupler is employed in a classical II. QUANTUM THEORY BACKGROUND sense in an optical communications environment. The resulting transfer matrix is the radix-4 Chrestenson transform. This result A. The Qubit vs. Qudit indicates that a new practical device is available for use in the The quantum bit, or qubit, is the standard unit of information implementation of radix-4 QIP applications or in the construction for radix-2, or base-2, quantum computing. The qubit models of a radix-4 quantum computer. Index Terms—quantum information processing; quantum pho- information as a linear combination of two orthonormal basis tonics; qudit; states such as the states j0i and j1i.
    [Show full text]
  • Electrical Characterisation of Ion Implantation Induced Defects in Silicon Based Devices for Quantum Applications
    Electrical Characterisation of Ion Implantation Induced Defects in Silicon Based Devices for Quantum Applications Aochen Duan Supervised by Professor Jeffrey C. McCallum and Doctor Brett C. Johnson School of Physics The University of Melbourne Australia 1 Abstract Quantum devices that leverage the manufacturing techniques of silicon-based classical computers make them strong candidates for future quantum computers. However, the demands on device quality are much more stringent given that quantum states can de- cohere via interactions with their environment. In this thesis, a detailed investigation of ion implantation induced defects generated during device fabrication in a regime relevant to quantum device fabrication is presented. We identify different types of defects in Si using various advanced electrical characterisation techniques. The first experimental technique, electrical conductance, was used for the investigation of the interface state density of both n- and p-type MOS capacitors after ion implantation of various species followed by a rapid thermal anneal. As precise atomic placement is critical for building Si based quantum computers, implantation through the oxide in fully fabricated devices is necessary for some applications. However, implanting through the oxide might affect the quality of the Si/SiO2 interface which is in close proximity to the region in which manipulation of the qubits take place. Implanting ions in MOS capacitors through the oxide is a model for the damage that might be observed in other fabricated devices. It will be shown that the interface state density only changes significantly after a fluence of 1013 ions/cm2 except for Bi in p-type silicon, where significant increase in interface state density was observed after a fluence of 1011 Bi/cm2.
    [Show full text]
  • Technology-Dependent Quantum Logic Synthesis and Compilation
    Southern Methodist University SMU Scholar Electrical Engineering Theses and Dissertations Electrical Engineering Fall 12-21-2019 Technology-dependent Quantum Logic Synthesis and Compilation Kaitlin Smith Southern Methodist University, [email protected] Follow this and additional works at: https://scholar.smu.edu/engineering_electrical_etds Part of the Other Electrical and Computer Engineering Commons Recommended Citation Smith, Kaitlin, "Technology-dependent Quantum Logic Synthesis and Compilation" (2019). Electrical Engineering Theses and Dissertations. 30. https://scholar.smu.edu/engineering_electrical_etds/30 This Dissertation is brought to you for free and open access by the Electrical Engineering at SMU Scholar. It has been accepted for inclusion in Electrical Engineering Theses and Dissertations by an authorized administrator of SMU Scholar. For more information, please visit http://digitalrepository.smu.edu. TECHNOLOGY-DEPENDENT QUANTUM LOGIC SYNTHESIS AND COMPILATION Approved by: Dr. Mitchell Thornton - Committee Chairman Dr. Jennifer Dworak Dr. Gary Evans Dr. Duncan MacFarlane Dr. Theodore Manikas Dr. Ronald Rohrer TECHNOLOGY-DEPENDENT QUANTUM LOGIC SYNTHESIS AND COMPILATION A Dissertation Presented to the Graduate Faculty of the Lyle School of Engineering Southern Methodist University in Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy with a Major in Electrical Engineering by Kaitlin N. Smith (B.S., EE, Southern Methodist University, 2014) (B.S., Mathematics, Southern Methodist University, 2014) (M.S., EE, Southern Methodist University, 2015) December 21, 2019 ACKNOWLEDGMENTS I am grateful for the many people in my life who made the completion of this dissertation possible. First, I would like to thank Dr. Mitch Thornton for introducing me to the field of quantum computation and for directing me during my graduate studies.
    [Show full text]
  • Higher Levels of the Transmon Qubit
    Higher Levels of the Transmon Qubit MASSACHUSETTS INSTITUTE OF TECHNirLOGY by AUG 15 2014 Samuel James Bader LIBRARIES Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2014 @ Samuel James Bader, MMXIV. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created. Signature redacted Author........ .. ----.-....-....-....-....-.....-....-......... Department of Physics Signature redacted May 9, 201 I Certified by ... Terr P rlla(nd Professor of Electrical Engineering Signature redacted Thesis Supervisor Certified by ..... ..................... Simon Gustavsson Research Scientist Signature redacted Thesis Co-Supervisor Accepted by..... Professor Nergis Mavalvala Senior Thesis Coordinator, Department of Physics Higher Levels of the Transmon Qubit by Samuel James Bader Submitted to the Department of Physics on May 9, 2014, in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics Abstract This thesis discusses recent experimental work in measuring the properties of higher levels in transmon qubit systems. The first part includes a thorough overview of transmon devices, explaining the principles of the device design, the transmon Hamiltonian, and general Cir- cuit Quantum Electrodynamics concepts and methodology. The second part discusses the experimental setup and methods employed in measuring the higher levels of these systems, and the details of the simulation used to explain and predict the properties of these levels. Thesis Supervisor: Terry P. Orlando Title: Professor of Electrical Engineering Thesis Supervisor: Simon Gustavsson Title: Research Scientist 3 4 Acknowledgments I would like to express my deepest gratitude to Dr.
    [Show full text]
  • In Situ Quantum Control Over Superconducting Qubits
    ! In situ quantum control over superconducting qubits Anatoly Kulikov M.Sc. A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland in 2020 School of Mathematics and Physics ARC Centre of Excellence for Engineered Quantum Systems (EQuS) ABSTRACT In the last decade, quantum information processing has transformed from a field of mostly academic research to an applied engineering subfield with many commercial companies an- nouncing strategies to achieve quantum advantage and construct a useful universal quantum computer. Continuing efforts to improve qubit lifetime, control techniques, materials and fab- rication methods together with exploring ways to scale up the architecture have culminated in the recent achievement of quantum supremacy using a programmable superconducting proces- sor { a major milestone in quantum computing en route to useful devices. Marking the point when for the first time a quantum processor can outperform the best classical supercomputer, it heralds a new era in computer science, technology and information processing. One of the key developments enabling this transition to happen is the ability to exert more precise control over quantum bits and the ability to detect and mitigate control errors and imperfections. In this thesis, ways to efficiently control superconducting qubits are explored from the experimental viewpoint. We introduce a state-of-the-art experimental machinery enabling one to perform one- and two-qubit gates focusing on the technical aspect and outlining some guidelines for its efficient operation. We describe the software stack from the time alignment of control pulses and triggers to the data processing organisation. We then bring in the standard qubit manipulation and readout methods and proceed to describe some of the more advanced optimal control and calibration techniques.
    [Show full text]
  • Arxiv:Quant-Ph/0512071 V1 9 Dec 2005 Contents I.Ipoeet Ntekmprotocol KLM the on Improvements III
    Linear optical quantum computing Pieter Kok,1,2, ∗ W.J. Munro,2 Kae Nemoto,3 T.C. Ralph,4 Jonathan P. Dowling,5,6 and G.J. Milburn4 1Department of Materials, Oxford University, Oxford OX1 3PH, UK 2Hewlett-Packard Laboratories, Filton Road Stoke Gifford, Bristol BS34 8QZ, UK 3National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan 4Centre for Quantum Computer Technology, University of Queensland, St. Lucia, Queensland 4072, Australia 5Hearne Institute for Theoretical Physics, Dept. of Physics and Astronomy, LSU, Baton Rouge, Louisiana 6Institute for Quantum Studies, Department of Physics, Texas A&M University (Dated: December 9, 2005) Linear optics with photo-detection is a prominent candidate for practical quantum computing. The protocol by Knill, Laflamme and Milburn [Nature 409, 46 (2001)] explicitly demonstrates that efficient scalable quantum computing with single photons, linear optical elements, and projec- tive measurements is possible. Subsequently, several improvements on this protocol have started to bridge the gap between theoretical scalability and practical implementation. We review the original proposal and its improvements, and we give a few examples of experimental two-qubit gates. We discuss the use of realistic components, the errors they induce in the computation, and how they can be corrected. PACS numbers: 03.67.Hk, 03.65.Ta, 03.65.Ud Contents References 36 I. Quantum computing with light 1 A. Linear quantum optics 2 B. N-port interferometers and optical circuits 3 C. Qubits in linear optics 4 I. QUANTUM COMPUTING WITH LIGHT D. Early optical quantum computers and nonlinearities 5 Quantum computing has attracted much attention over II.
    [Show full text]
  • Arxiv:1903.12615V1
    Encoding an oscillator into many oscillators Kyungjoo Noh,1, 2, ∗ S. M. Girvin,1, 2 and Liang Jiang1, 2, y 1Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA 2Yale Quantum Institute, Yale University, New Haven, Connecticut 06520, USA Gaussian errors such as excitation losses, thermal noise and additive Gaussian noise errors are key challenges in realizing large-scale fault-tolerant continuous-variable (CV) quantum information processing and therefore bosonic quantum error correction (QEC) is essential. In many bosonic QEC schemes proposed so far, a finite dimensional discrete-variable (DV) quantum system is encoded into noisy CV systems. In this case, the bosonic nature of the physical CV systems is lost at the error-corrected logical level. On the other hand, there are several proposals for encoding an infinite dimensional CV system into noisy CV systems. However, these CV-into-CV encoding schemes are in the class of Gaussian quantum error correction and therefore cannot correct practically relevant Gaussian errors due to established no-go theorems which state that Gaussian errors cannot be corrected by Gaussian QEC schemes. Here, we work around these no-go results and show that it is possible to correct Gaussian errors using GKP states as non-Gaussian resources. In particular, we propose a family of non-Gaussian quantum error-correcting codes, GKP-repetition codes, and demonstrate that they can correct additive Gaussian noise errors. In addition, we generalize our GKP-repetition codes to an even broader class of non-Gaussian QEC codes, namely, GKP-stabilizer codes and show that there exists a highly hardware-efficient GKP-stabilizer code, the two-mode GKP-squeezed-repetition code, that can quadratically suppress additive Gaussian noise errors in both the position and momentum quadratures.
    [Show full text]
  • Superconducting Phase Qubits
    Noname manuscript No. (will be inserted by the editor) Superconducting Phase Qubits John M. Martinis Received: date / Accepted: date Abstract Experimental progress is reviewed for superconducting phase qubit research at the University of California, Santa Barbara. The phase qubit has a potential ad- vantage of scalability, based on the low impedance of the device and the ability to microfabricate complex \quantum integrated circuits". Single and coupled qubit ex- periments, including qubits coupled to resonators, are reviewed along with a discus- sion of the strategy leading to these experiments. All currently known sources of qubit decoherence are summarized, including energy decay (T1), dephasing (T2), and mea- surement errors. A detailed description is given for our fabrication process and control electronics, which is directly scalable. With the demonstration of the basic operations needed for quantum computation, more complex algorithms are now within reach. Keywords quantum computation ¢ qubits ¢ superconductivity ¢ decoherence 1 Introduction Superconducting qubits are a unique and interesting approach to quantum computation because they naturally allow strong coupling. Compared to other qubit implementa- tions, they are physically large, from » 1 ¹m to » 100 ¹m in size, with interconnection topology and strength set by simple circuit wiring. Superconducting qubits have the advantage of scalability, as complex circuits can be constructed using well established integrated-circuit microfabrication technology. A key component of superconducting qubits is the Josephson junction, which can be thought of as an inductor with strong non-linearity and negligible energy loss. Combined with a capacitance, coming from the tunnel junction itself or an external element, a inductor-capacitor resonator is formed that exhibits non-linearity even at the single photon level.
    [Show full text]
  • G53NSC and G54NSC Non Standard Computation Research Presentations
    G53NSC and G54NSC Non Standard Computation Research Presentations March the 23rd and 30th, 2010 Tuesday the 23rd of March, 2010 11:00 - James Barratt • Quantum error correction 11:30 - Adam Christopher Dunkley and Domanic Nathan Curtis Smith- • Jones One-Way quantum computation and the Measurement calculus 12:00 - Jack Ewing and Dean Bowler • Physical realisations of quantum computers Tuesday the 30th of March, 2010 11:00 - Jiri Kremser and Ondrej Bozek Quantum cellular automaton • 11:30 - Andrew Paul Sharkey and Richard Stokes Entropy and Infor- • mation 12:00 - Daniel Nicholas Kiss Quantum cryptography • 1 QUANTUM ERROR CORRECTION JAMES BARRATT Abstract. Quantum error correction is currently considered to be an extremely impor- tant area of quantum computing as any physically realisable quantum computer will need to contend with the issues of decoherence and other quantum noise. A number of tech- niques have been developed that provide some protection against these problems, which will be discussed. 1. Introduction It has been realised that the quantum mechanical behaviour of matter at the atomic and subatomic scale may be used to speed up certain computations. This is mainly due to the fact that according to the laws of quantum mechanics particles can exist in a superposition of classical states. A single bit of information can be modelled in a number of ways by particles at this scale. This leads to the notion of a qubit (quantum bit), which is the quantum analogue of a classical bit, that can exist in the states 0, 1 or a superposition of the two. A number of quantum algorithms have been invented that provide considerable improvement on their best known classical counterparts, providing the impetus to build a quantum computer.
    [Show full text]
  • EE214/PHYS220 Quantum Computing Lecture 1: Introduction Textbook: N
    EE214/PHYS220 Quantum Computing Lecture 1: Introduction Textbook: N. D. Mermin, Quantum computer science (Cambridge Univ. Press, 2007) (errata at http://www.lassp.cornell.edu/mermin/errata-1-12-12.pdf); http://www.lassp.cornell.edu/mermin/qcomp/CS483.html (lecture notes) Other resources: http://www.theory.caltech.edu/~preskill/ph219/ (lecture notes, Caltech course) http://inst.eecs.berkeley.edu/~cs191 (lecture notes, UC Berkeley course) M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000) G. Benenti, G. Casati, and G. Strini, Principles of Quantum Computation and Information, Vol. I: Basic Concepts (World Scientific, 2005) Prospects for Quantum Computing (QC): not really clear Pessimistic view: never, possibly limited to very small QCs as servers for quantum cryptography networks Optimistic view: in 20-50 years large-scale QCs, capable of factoring large integers Very optimistic (overoptimistic): will partially or completely replace general-purpose computers What QC can do efficiently 1) Factoring large integers (exponential speedup) Best classical: exp[ log / ], 1 3 more∼ accurately exp log log log , log base 2 1⁄3 64 2⁄3 Quantum: log (Shor’s∼ algorithm)9 2) Search in unsorted database3 (quadratic speedup) ∼ Classical: (simply check all) Quantum: (Grover’s algorithm) ∼ 3) Simulation of quantum∼ systems (for study of materials, etc.) 4) Possibly something else important (still area of active research) Current status: numbers 15 and 21 “factored” (also 143 with adiabatic QC) 14 well-entangled qubits (trapped ions, 2011), <25 qubits quantum algorithms with 9 superconducting qubits (2015), 1,000 D-Wave “qubits” Truly interdisciplinary effort: physics, engineering, computer science, mathematics Classical vs.
    [Show full text]