LECTURES IN YEAR 2011

LECTURES BEFORE MIDTERM



WEEK ONE.



MANDATORY SLIDES AND VIDEOS:

  1. Introduction to class. Administrativia, texbooks, grading. SOP and ESOP logic circuits. Use of Karnaugh Maps to minimize Sum of Products (SOP) and Exclusive Sum of Products (ESOP) circuit to realize reversible and quantum logic. EXOR logic identities to be used in transformations.

  2. VIDEO LECTURE BY PERKOWSKI Perkowski_033005.wmv VIDEO: Introductory lecture by Marek Perkowski. Classical circuits. SOP, ESOP. EXOR algebra laws. Set Covering and Petrick function.

  3. Motivation to quantum computing. Overview and history. Topics to be discussed. What is quantum computation and why important. The power of quantum computers. Feynman: "Nobody understands quantum mechanics" quote. Introduction to quantum logic circuits. The Beam-Splitter. The interferometer. Double-Slit Interference. Two-slit experiment with electrons. More experimental data on quantum interference. A New theory. Probability Amplitude and Measurement. Quantum Operations. Linear Algebra. Abstractions of gates and circuits using linear algebra. Circuits with more than one qubit. Entanglement. Measuring more-than-one qubits. Classical versus reversible versus quantum circuits. Quantum Adder and its components. Converting binary logic circuits to reversible. Classical reversible gates - notations. Quantum Gates: NOT, Square-Root-of-NOT, Hadamard. Phase shift. Universal gates. CNOT, Toffoli. Controlled gates. Realization of a universal gate (Toffoli) using 2*2 quantum primitives (controlled-V). Analysis and simulation of quantum circuits. Algebraic analysis using matrix multiplication and Kronecker products.

  4. VIDEOS OF PERKOWSKI LECTURE RELATED TO THIS MATERIAL.
    1. Perkowski_040405.wmv Design of ESOP circuit. Analysis of quantum cascades. CV and CV+ circuits. Quantum arrays. Unitary matrices. Fundamentals of quantum circuits. Quantum gatses. Mirror circuits. Heisenberg and Dirac notations. Simple measurements.
    2. Perkowski_040605.wmv Quantum gates and circuits. Continuation. Problems.

  5. VIDEOS OF DAVID DEUTSCH LECTURES ON QUANTUM COMPUTING FUNDAMENTALS.
    1. Lecture 1 by David Deutsch about qubit. This lecture introduces quantum mechanics from the point of experiments, rather than in an abstract axiomatic way as in my lectures above. Watching these slides you will understand the physical background of measurements. This video demonstrates some specific philosophy worthy to learn. It is interesting to see how the creator of quantum computing thinks about quantum mechanics. The complexity of this lecture is medium.

    2. Lecture 2 by David Deutsch quantum interference. This lecture will help to understand deeper the interference and measurement. Interesting experiments are shown. The complexity of this lecture is medium.



MANDATORY BOOK READING:

Read sections 1.1 to 1.8 (inclusive) from chapter 1 of the Marinescu book. Read sections 1.13 and 14 from Marinescu.



AUXILIARY BOOK READING:

Read all chapter 4. If you have trouble with some page, skip it. Look to the similarity to classical computers and digital circuits.



AUXILIARY SLIDES AND VIDEOS:
    You really are not required to read any of the videos or slides given below. However, some of them are very interesting and are good introduction to more advanced material and webpages about current research.

  1. Michio Kaku Talks About Artificial Intelligence and Quantum Computing. Watch about "Parallel Universes and Multiple Dimensions". Easy and entertaining. Find on the webpage below. Under title "Michio Kaku Talks about Artificial Intelligence". There are few more also very good.
  2. Deutsch Lectures Excellent. Please watch all these videos. If you do not understand them now, you will understand them with the end of the class.

    The first lecture video is about qubit, quantumness and strangeness of quantum mechanics with many worlds interpretation. It has a more general measurement model than we introduced in class so far.

    The second lecture is about superposition and it shows some famous quantum mechanics experiments.

    Next 4 videos you can watch in future, but they are still of interest if you want how the Grover algorithm works for which we will be designing oracles in the first part of the class.
    1. Flying with Photons. Lecture by Helen Czerski. format .wmv This is easy and entertaining discussion of early Einstein's work in quantum mechanics and wave/particle controversy. Presentation for high school students.

    2. Videos about quantum physics on Google. Be careful to watch some other stuff which is not science and just uses name "quantum".
      Watch "Quantum Physics Double Slit Experiment. THIS IS GREATE VIDEO. High School Level.
      "Dr. Quantum sheds a little light on dimensions." ANOTHER MIND EXTENDING EXPERIENCE. High School level
      "Seth Lloyd's Quantum Computer". VISIONARY AND FUNNY
      Watch "Measurement of a classical variable on a quantum system".
      Schroedinger cats and decoherence in quantum measurement".
      "Quantum Computer running."
      "A quantum trajectory formalism for continuous variable teleportation".
      "Quantum optics of single atoms".
      "Perimeter Institute - Harnessing the Quantum World.
      "Entanglement in quantum information processing."
      " Richard Feynman lectures: Physics 02 - Quantum Behaviour."
      "Quantum non-demolition measurements".
      "Richard Dawkins Discusses Quantum Theory".
      "Is quantum Mechanics non-local?"
      "22C3: Quantum Entanglement".
      "6.4. Quantum Numbers".
      "Quantum Computer plays SudoQ".
      "What is quantum tunelling?"

    3. Introductory Lecture to Quantum Computing by Vazirani. You should have two windows on the screen. One with his video and other with his PDF slides. This lecture presents much material in a short mathematical way. Its second part is about complexity of quantum algorithms which we will not cover soon.

    4. Three Lectures by Hans Bethe at Cornell University. Excellent lectures on quantum mechanics by one of its creators. Easy. These lectures have minimum of mathematics and much of interesting history of quantum physics. They include perhaps most of what you need to know about quantum mechanics (physics) to study quantum computing. What else is really needed mathematically for us, will be introduced in the class in whole detail. Watching these slides is not mandatory but it will be very helpful to see the quantum phenomena presented in the class in a full physical context.

    5. History and future of computing. The goal of these slides is only to show you how quickly the computer technology is changing and that we have to be brave to look to new oportunities.

    6. History and future of computing. Part 2.
    7. Videos and slides from the 2003 Quantum Summer School. The following lectures took place at the 2003 PIMS-MITACS Summer School on Quantum Information Science, held June 23 - 27, 2003, at the University of Calgary. Very good lecture, but you need a player
    8. Quantum Mechanics and Quantum Physics. This is a short of easy explanations, on high school level, of many topics of quantum physics.
    9. Quantum and Electronics Videos. You have to login.
    10. University of Wisconsin Workshop on Quantum Computation.
      Richard Cleve, University of Waterloo. WATCH THIS VIDEO
      David DiVincenzo, IBM Watson Research Center.
      Seth Lloyd, MIT
      Very instructional, the first lecture is easy. Other are above class level if you want to understand them fully.


WEEK TWO.


MANDATORY SLIDES:
  1. Reversible gates and circuits. Introduction. Reversible gates for low power. Moore's Law. Predictions. Research issues in Reversible and Quantum Computing. Landauer Principle. From Irreversible to reversible gates and circuits. Reversible gates as constraints. Analysis of Fredkin and Toffoli universal reversible gates. Bennet's research and the concept of reversible computer. Mirror circuits. Direction of research in quantum and reversible circuits. SWAP gates. Linear and Affine Reversible circuits. Structures of reversible cascades. Example of linear reversible circuit: Converter of binary to Gray code. Ancilla bits and garbage outputs. Quantum Array notation versus reversible circuit notation. Use of ancilla bits to create subfunctions of 3*3 and 4*4 Fredkin gates. Various types of quantum cascades. Generalized gates. Analysis of Miller gate. Designing reversible adder in various notations. Searching and Backtracking methods to design quantum cascades. Reversible Circuits and Permutations. Even and Odd permutations. Reversible De Morgan Laws. Search algorithms for quantum cascades. IDA search. Designing oracles for Grover and other quantum algorithms.

  2. VIDEO LECTURE BY PERKOWSKI, CONTINUATION OF PREVIOUS LECTURE AND THIS LECTURE.
      Perkowski_041105.wmv Quantum arrays andd gates. Reversible circuits.

    1. Perkowski_041305.wmv Jeff presentation about quantum dot technology. Perkowski - continuation about reversible and quantum circuits and their applications. Explanation of homeworks and projects.

  3. Billiard Ball Model and Conservative gates. Reversible and Quantum Logic fundamentals from the Logic Synthesis point of view. Review: Atomic-scale computation. Information loss is an energy loss. Information is physical. Billard Ball model. New notations for reversible functions. Reversible logic constraints. To develop intuitive feeling for reversible circuits design. Conservative and balanced functions. Gates in Billiard Ball model - examples. Interaction gate. Switch gate. Feynman gate in BBM. Fredkin gate - new notations. Fredkin gate from switch gates. Illustrations of conservative property. Minimal Full Adder from Interaction Gates. Adder from Switch gates. Adder from Fredkin gates. Fredkin gate from Interaction Gates. Concluding on BBM.

  4. VIDEO LECTURE BY PERKOWSKI ON THIS TOPICS Perkowski_041805.wmv Reversible logic synthesis methods. Billiard Ball model.

  5. VISUALIZATIONS OF BILLIARD BALL MODEL. bounce_001f.avi
  6. bounce_010f.avi
  7. bounce_011f.avi
  8. bounce_100f.avi
  9. bounce_101f.avi
  10. bounce_110f.avi
  11. bounce_111f.avi


NON-MANDATORY BUT HIGHLY RECOMMENDED VIDEO AND SLIDES:
  1. Introduction to Quantum Computing video and slides by Rod Van Meter. Download and print slides, when you listen to video, add more text to your printouts. This lecture talks about
  2. chapter1.mp4 Lecture 1 about quantum computing by Prof. Ivan Deutsch

  3. Lecture1 in PDF. Slides to Lecture 1 of prof. Ivan Deutsch.



MANDATORY BOOK READING:

Read entire chapter 4 from Marinescu. You should be able to build similar and more complex circuits. You should be able to solve all problems at the end of chapter 4. You should be able to design an oracle for graph coloring problem or other similar Constraint Satisfaction Problem such as SEND+MORE=MONEY Problem.

AUXILIARY BOOK READING:

Read the entire chapter 3. If you do not understand some physics, skip it and concentrate on circuit design ideas.

WEEK THREE.


MANDATORY SLIDES:
  1. KFDDs or Kronecker Functional Decision Diagrams and their use in reversible circuits. Review of Binary Decision Diagrams (BDD) and Kronecker Functional Decision Diagrams (KFDD). Reduction types. Examples of Ordered KFDDs (OKFDDs). Diagrams with complemented edges. Operations on diagrams. Use of simulated annealing to find best diagrams.

  2. VIDEO LECTURE BY PERKOWSKI ON BDD, KFDD, SYMMETRIC FUNCTIONS AND LATTICES. Perkowski_042005.wmv This lecture extends on slides. There is more material on synthesis and more examples. The methods to design quantum arrays directly from decision diagrams and lattices.

  3. Reversible Circuits Synthesis with Garbage. Review of reversible gates. Kerntopf gates. Reed-Muller Notation for reversible logic. Using mirrors. Synthesis of inverse circuits. Synthesis from Decision Diagrams. Diagrams with linear and reversible preprocessors. Uses of reversiblity in synthesis. Use of ancilla bits. Composition and decomposition methods. Synthesis of an adder - various approaches. Role of search. Ashenhurst-Curtis and levelized decompositions. Open research topics.

  4. VIDEO PERKOWSKI LECTURE ABOUT THIS MATERIAL: Perkowski_042705.wmv Kerntopf gates, Decision Diagrams, decompositions. Design for test of quantum circuits. Synthesis from KFDDs. Backtracking and advanced methods of designing quantum arrays.

  5. Basic MMD (Miller, Maslov and Dueck) synthesis algorithm for binary reversible cascades. Toffoli gate family. Basic algorithm. Bidirectional algorithm. Improvements. Templates. Statistics. Open Problems and current works.

MANDATORY TEXTBOOK READING:

You should be now experts on all material from chapters 3 and 4. Read them again, if necessary. You should be also able to build a practical oracle for Grover algorithm as a permutative circuit.


AUXILIARY SLIDES:
  1. Introduction to Quantum Information Decision Diagrams. QUIDDs. Research projects at Michigan. Quantum Circuits simulation using QuIDDs. Operations on Multiple Qubits. Patterns of blocks. QUIDD structure. QUIDD data representation. Examples. QUIDDs versus ADDs. QUIDD operations. Tensor product. Other operations. Simulation results on Grover algorithm. Current work.
    Most of this lecture is auxiliary. We will discuss only few concepts, such as the quantum diagram itself.

  2. VIDEO - AUXILIARY. Perkowski_042505.wmv MMD algorithm. Quantum Array transformations and design of large gates. The last material is in slides at the end of the class, under title: Transformation rules by Iwama, Kambayashi and Yamashita.

  3. Perkowski_032805.wmv Lectures by Dr. Maslov and Prof. Miller about their research from year 2005. Other lectures from the quantum circuits symposium organized at PSU.



MIDTERM 1.



This midterm will be announced soon. It will take place not in class time. May be on Friday.

This midterm is open book and you have plenty of time. It will take place when the class will be able to do the following:
  1. Understand the qubit, superposition and Bloch Sphere. Be able to understand how to use Bloch sphere to show operation of gates, measurements and multi-valued states.
  2. Know the following gates: Pauli rotations, Hadamard, Square Root of Not, Square Root of Not adjoint. CNOT = Feynman, SWAP, Toffoli, Fredkin. Big controlled gates such as Toffoli with 4 controls. Make yourself a creepsheet.
  3. Basic quantum measurements.
  4. Entanglement and no-cloning.
  5. building oracles for Constraint Satisfaction, mapping and Satisfiability problems.
  6. Billiard Ball model.
  7. The concepts of linear, affine, conservative and balanced functions.
  8. Representations of binary, ternary, quaternary and 6-valued logic. Representation of fuzzy logic. Fredkin gate for mV and fuzzy logic.
  9. Analysis of arbitary quantum circuit: Kronecker product, matrix product. For both binary and ternary circuits.
  10. Calculate any quantum state inside the multi-qubit (multi-qudit) quantum circuit.
  11. You do not have to remember details of logic synthesis methods for quantum circuits, but you should be able to design a (non-minimal) permutative quantum circuit from initial specification using any method or a combination of methods.

WEEK FOUR.

MANDATORY TEXTBOOK READING:

Read chapter 2 from Marinescu. Review chapter 3, if necessary. Review chapter (appendix) from Marinescu about linear algebra, if necessary.

MANDATORY VIDEOS.
  1. Lecture 4 by David Deutsch about the Schroedinger Picture of Quantum Mechanics.


    AUXILIARY VIDEOS.
    1. Lectures by Ivan Deutsch about Quantum Computing.
      Introduction to quantum computing.
      Formal Structure and Quantum Mechanics.
      Entanglement.
      Qubits and Quantum Circuits.
      Quantum Algorithms: Shor. Grover. Simulation.
      Noise. Decoherence. Errors. Error correction. Basic codes. Fault-tolerance.
      Physical Implementations.
MANDATORY SLIDES AND VIDEOS:
  1. EXPERIMENTS IN QUANTUM MECHANICS. Slides in PPT. Classical Versus quantum experiments. Experiment with Bullets. Experiment with Waves. Two Slit experiment. Two Slit experiment with observation. Stern-Gerlach experiment. Conclusion from Experiments. Technological limits. Energy/Operation. Power dissipation, circuit density, and speed. Reducing heat is important. Quantum and computers: a happy marriage. Heisenberg Uncertainty principle. Milestones in quantum physics. Milestones in Computing and Information Theory. Milestones in Quantum Computing. Deterministic versus probabilistic photon behavior. The puzzling nature of light. One more formalism for quantum states. Multiple experiments related to superposition and uncertainty. Measurement in multiple bases. Measurement of superposition states. The superposition probability rule. The experiment with 2 beam splitters illustrating it. A photon coincidence experiment.

    Also, this material is partially in David Deutch second video.

    1. video doubleslit.mpeg
    2. bounces.mov Bouncing and energy visualization.
    3. gauss.mpeg
    4. harmonicosc.mpeg
    5. reflectionless.mpeg
    6. resonance.mpeg
    7. squarewell.mpeg
  2. VIDEO LECTURE BY PERKOWSKI.
    1. Perkowski_050905.wmv This lecture has also project explanation and homework ideas.

  3. STERN-GERLACH EXPERIMENT. Slides in PPT. Quantum Computing Mathematics and Postulates. Requirements on Mathematics apparatus. What is Quantum Mechaanics ? Linear algebra. What is useful in QM? Dirac notation. Inner products. Hilbert space. Unitary operator. Tensor Product. Postulates in QM. What is Qubit mathematically? Bloch sphere again. Qubit in Stern-Gerlach experiment.

  4. MORE ON QUANTUM CIRCUITS AND COMPLEXITY. Slides in PPT. Church-Turing thesis. Church-Turing-Deutsch Principle. Models of quantum computation. Quantum Circuit model. Pauli gates. Review of other gates. Review of mirror and garbage removal. Introduction to Deutsch problem. Putting information in the phase. Quantum Algorithm for Deutsch problem. Universality in the quantum circuit model. Summary of the quantum circuit model. Quantum complexity classes. Alternate models of quantum computation.



AUXILIARY VIDEO. WATCH THEM FOR FUN ONLY.
  1. Nova, introductory videos about modern physics, including quantum. Watch the " Quantum Cafe".
  2. Physics for High School Teachers.


MANDATORY VIDEO. WATCH FOR SECOND CLASS MEETING.

  1. Lecture 3 by David Deutsch about measurement.
  2. Introduction to Quantum Algorithms vide and slides by Rod Van Meter. Download and print slides, when you listen to video, add more text to your printouts. Quantum Parallelism. Search. Deutsch. Deutsch-Jozsa. Grover. Shor. Grover vs Shor. Quantum Fourier Transform. Order Finding. Factoring. Quantum Algorithms. Quantum Error Correction. Role of Phase.


WEEK FIVE.

ADDITIONAL SLIDES FOR EXAM.

  1. Quantum Circuits. Intro to Deutsch. Slides in PPT. From Nielsen. What is Quantum Computation. Church-Turing Thesis. Church-Turing-Deutsch Principle. Models of Quantum Computation. Quantum Circuit Model. Simple Quantum Logic Gates and Identities. How to calculate Classical Logic functions on Quantum Computers. Remvoving Garbage. Example: Deutsch Problem and Algorithm. Putting Information in Phase. Universality in quantum circuit model. Summary of the quantum circuit model. Quantum Complexity classes. Alternate models of quantum computation.

  2. Superdense coding. Teleportation. Slides in PPT. How much information in n qubits? Holevo Theorem. Superdense coding. How it works? Use of Bell basis. Complete Circuit. Incomplete Measurements. Teleportation. Measuring the first qubit of a two-qubit system. Teleportation scenario. Teleportation circuit. What Alice does. Bob's adjustement procedure. Circuit for teleportation. No-cloning theorem.

  3. Deutsch and Complexity Properties. Slides in PPT Comparison of classical and quantum circuits. Factoring problem. Simulating classical circuits with quantum circuits. Simulating probabilistic algorithms. Simulating quantum by classical. Complexity classes. Query scenario. Deutsch algorithm revisited.

  4. Intro to quantum postulates. Slides in PPT. Requirement of mathematical apparatus. Quantum Mechanics. Linear Algebra. Bra and Ket. Hilbert space fundamentals. Inner Products. Duals as row vectors. Unitary operators. Tensor products. What is qubit mathematically. Basis and linear operators. Bloch sphere and unit circle as useful visualizations. Postulates of quantum mechanics - first view. Postulate 3 in rough form. Irrelevance of global phase. Revised postulate 1. Multiple-qubit systems. Postulate 4. Entanglement. Some conventions implicit in postulate 4. Entangled as opposed to separable states. Composite quantum systems. On More example of Toffoli gate realization.

  5. 2005-q-0024a-review-algebra.ppt This is old lecture with more slides and better explanations. Use this to learn for exam.

  6. 2005-q-0024b-Postulates-of-quantum-mechanics.ppt This is old lecture with more slides and better explanations. Use this to learn for exam.

  7. Density operators intro. Slides in PPT. Density and ensamble. Why we learn this? Review of outer products. Projection operators. Probability of being in a state and trace. Mathematics of trace. Density matrix. Examples. Measurement using the density matrix. Dynamics and the density matrix. Examples. How the density matrix changes during the measurement. Density matrix is positive. Examples of properties and problems to solve.

  8. Density matrices, traces, partial traces. Operators and Measurements. Slides in PPT. This is advanced material on traces. Part of this was not covered in the lecture. What was not covered in the lecture in class will be not on the exam. Density matrices of pure states. Trace of a matrix. Notations and formulas. Mixture of pure states. Density matrices of mixed states. Operationally indistinguishable states. Applying Unitary Operator to a Density Matrix of a Pure State. Applying Unitary Operator to a Density Matrix of Mixed Pure State. Operators on Density Matrices of Mixed States. How do quantum operations work using density matrices. More examples of density matrices. Properties of density matrices. Use of density matrix and trace to calculate the probability of obtaining a given state in measurement (for pure states). The same for mixed states. Use of spectral decomposition. Taxonomy of various normal matrices used in quantum computing - their properties. Unitary and Hermitian matrices. Positive semidefinite matrices. Bloch Sphere review. Polar coordinates. General Quantum Operations. Decoherence operator. Trine state measurement operator. Partial trace as a general quantum operation. Distinquishing mixed states. Simultaneously diagonalizable matrices. Basic properties of trace. Partial trace. Derivation of formulas. Partial trace can be calculated in arbitrary basis. Methods to calculate the partial trace. Examples of partial trace. Calculating matrices of partial traces. Unitary transformations do not change the local density matrix. Distant transformations do not change the local density matrix. Principle of implicit measurement. POVM. The measurement postulate in terms of observables. Example of using observables. What can be measured in quantum mechanics. Von Neumann measurement in computational basis. What is expected value of an observable? Partial measurements. Most general measurements. More examples of von Neumann Measurements. How to implement Von Neumann measurements? Bell base measurements. Examples. Destructive and non-destructive measurements. Simulations among operations. Simulations of POVM. Separable states. Continuous time evolution.

  9. Spectral Decomposition. Slides in PPT. Michele Mosca on Dirac and Spectral. Review and extension of Dirac notation. Pauli matrices in new notation. Exponense and spectral decomposition. Spectral decomposition theorem. Spectral decomposition of NOT gate. Columns and rows of matrices. Verifying eigenvalues and eigenvectors. Why is spectral decomposition useful. Matrix notation for spectral decomposition. Von Neumann measurement in computational basis. Polar decomposition. Gram-Schmidt Orthogonalization.

  10. More details about the Bloch Sphere. In PDF format.

MANDATORY SLIDES AND VIDEOS.

  1. ONE IN FOUR QUANTUM CIRCUIT. INTRODUCTION TO GROVER. Slides in PPT. Detailed illustration of the simplest possible application of Grover algorithm. This is a good introduction to using quantum simulator such as Quidpro. You can also perform all these simulations in MATLAB.

  2. REVIEW OF LINEAR ALGEBRA FOR QUANTUM COMPUTING. Slides in PPT. Linear algebra in quantum mechanics. Lecture objectives. Complex numbers. Complex Exponentiation. Qubit. Properties of qubit. Abstract Vector Spaces. Vectors. Hilbert Spaces. Dirac's Ket notation. Spanning Set and basic vectors. Bases and linear independence. Linear operators. Pauli Matrices. Examples of operators. Inner Products of vectors. Formalisms for inner products. Norms. Outer products. Eigenvectors and eigenvalues. Diagonal representation of matrices. Adjoint operators. Normal and Hermitian operators. Unitary operators. Hermitian operators. Tensor Products of vector spaces. Tensor Products of operators. Properties of tensor products. Functions of operators. Trace and commutator. Polar Decomposition. More on Inner Products. Review.

  3. SPECTRAL DECOMPOSITION. Slides in PPT. Dirac notation. Pauli Matrices. Spectral Decomposition. Verifying eigenvectors and eigenvalues. Why is spectral decomposition useful. Use of matrix notation. Von Neuman Measurement in the computational basis.

  4. INTRODUCTION TO NOISE AND DENSITY MATRICES. Slides in PPT. Quantum Noise. Density matrices. Outer product notation. The trace operation. Ensamble point of view. Example of calculating the density matrix. A measurement using the density matrix. Why work with density matrices? Dynamics and the density matrices. How the density matrix changes during a measurement? Characterizing the density matrix. Summary of the ensamble point of view.

  5. VIDEO LECTURE BY PERKOWSKI. Perkowski_052305.wmv Ideas about Deutsch algorithm. Links to Walsh. Variants of Deutsch. Testing. Deutsch-Jozsa. Density Matrices. Traces. Measurement notation. NP problems and introduction to Grover.

WEEK SIX.

MANDATORY SLIDES AND VIDEOS.

  1. MEASUREMENT. OBSERVABLES. COMMUNICATION. Slides in PPT. The measurement postulate formulated in terms of observables. An example of observables in action. What can be measured in quantum mechanics?

  2. DENSITY MATRICES AND MEASUREMENTS. Slides in PPT. Von Neuman Measurement in the computational basis. Partial measurements. Von Neuman measurements. How to implement them? Another approach. Example: Bell basis change. Bell measurement. Most general measurement. Trace of a matrix. Density matrices. Mixture of pure states. Density matrix of a mixed state. Spectral Decomposition. Partial trace. Distant Transformations don't change the local density matrix. Principle of implicit measurement. Partial trace using matrices.

  3. POSTULATES OF QUANTUM MECHANICS. Slides in PPT. Another Example of Toffoli realization. Linear operators. Eigenvalues and Eigenvectors. Exam Problems. Unitary transformations. Postulates of quantum mechanics. History. Postulate 1: state space. Systems and subsystems. Closed versus open systems. Concrete versus abstract systems. States and state spaces. Qubit as the simplest state space. Distinquishability of states. Revised postulate 1. Distinquishability of states - more precisely. State vectors and Hilbert space. Postulate 2: Evolution. Example of Evolution: Hadamard gate. Time Evolution. Wavefunctions. Schroedinger wave equation. Features of wave equation. Heisenberg and Schroedinger view of Postulate 2. The Schroedinger Equation. The Hamiltonian Matrix in Schroedinger equation. The Evolution matrix in Schroedinger equation. Postulate 3: Quantum Measurements. Computational Basis - a reminder. Example of measurement in different bases. Simplified Bloch Sphere to illustrate bases and measurements. Probability and measurement. Observables. Density operators. Use of Duals and Inner Products in Measurements. Duals as row vectors. General Measurement. Review of Bloch Sphere. Postulate 3: rough form. The measurement problem. More examples how measurement operators act on state space of a quantum system. Mixed states. Measurement of a state vector using projective measurements. Ensambles of quantum states. The density and the trace. Measurement of a density state. Reminder: Ensamble point of view. Measurement of a density state. Postulate 3: Quantum Measurement. Precise formulation. What happens to a system after the measurement? Distinguishability. Projective Measurements: Average values and Standard Deviations. Irrelevance of "global Phase". Phase. Postulate 4: Composite Systems. Compound System. Example of composition. Entanglement. Alice and Bob versus entanglement. Superdense Coding. Multiple-Qubit Systems. Postulate 4 one more time. Example. Size of Compound State Spaces. Summary. Key points to remember. General Measurement in compound spaces. Uncertainty Principle. Positive Operator - Valued Measurements (POVM).

  4. LECTURE BY PERKOWSKI Perkowski_051605.wmv Unitary Matrices - review of algebra. Universal gates. Modular arithmetics. Measurement and partial measurement. Introduction to quantum algorithms. Deutsch Algorithm in detail using many methods to show various analysis and synthesis methods for algorithms and also how to describe them using various notations.



AUXILIARY SLIDES AND VIDEOS

  1. Ivan Deutsch. chapter2.mp4 Lecture about formal models, measurement and postulates.
  2. Ivan Deutsch. QI_Lecture2.pdf Collection of slides to this lecture. Download slides or open them in a separate winodow.

    Ivan Deutsch. chapter3.mp4 Lecture about entanglement.
    Ivan Deutsch. QI_Lecture3.pdf Slides to this lecture.

WEEK SEVEN.



This will be taught by Martin Lukac. All slides are available, auxiliary slides are available and several videos are also avialable for your use. Ask Martin for help with Matlab, if you need any.

INTRODUCTION TO QUANTUM ALGORITHMS.



MANDATORY VIDEO TO READ BEFORE THE LECTURE:
  1. Lecture 4 by David Deutsch about the Deutsch Algorithm and quantum algorithms. This is the simplest algorithm which shows why and how a quantum computer is essentially more powerful than a standard computer.


MANDATORY SLIDES AND VIDEOS.

  1. QUANTUM ALGORITHMS INTRODUCTION AND DEUTSCH ALGORITHM AS AN EXAMPLE. Slides in PPT. Outline and review. Some new quantum gates. Generalized 3-qubit Toffoli by Deutsch. Barenco's 2-qubit generalized CNOT. Barenco's et all results in summary. Modular Arithmetics. Modulo versus XOR. Use of CNOT to copy classical information. More formalisms for reversible functions. Review of single qubit gates and their compositions. Review: Examples of measurement and partial measurement. Review: Global phase. Basic ideas about quantum algorithms. Speedups. Example of FFT. Computational complexity of FFT. Example: Factoring. Factoring with quantum systems. Implications of factoring and other quantum algorithms. Shor's algorithm. Shor type algorithms. Search Problems. Oracles. Quantum Searching. Quantum Counting. How do quantum algorithms work? Quantum Algorithms research. Summary on quantum algorithms. Deutsch Problem. Classical Deutsch. First explanation. Deutsch circuit. Second explanation of Quantum Deutsch. Generalizations of these ideas. Third explanation. Phase "Kick_Back" trick. Third explanation continued. Deutsch Algorithm philosophy. Deutsch with single qubit measurement. Deutsch in perspective. Extended Deutsch problems. Deutsch-Jozsa problem. Classical and Quantum DJ algorithm. Analysis.



GROVER ALGORITHM AND QUANTUM SEARCH.



MANDATORY VIDEO TO WATCH BEFORE THE LECTURE:
  1. Lecture 4 by David Deutsch about the Grover Search quantum algorithm. We use this algorithm in our project and simulation. This is perhaps the most practical quantum algorithm so far since it give a quadratic speed-up on very many problems of practical importance.
  2. VIDEO LECTURE BY PERKOWSKI. GROVER AND POSTULATES OF QUANTUM MECHANICS> Perkowski_052505.wmv



MANDATORY TEXTBOOK READING:

Read about Deutsch and Grover algorithms from chapter 5.

MANDATORY SLIDES AND VIDEOS.
  1. GROVER ALGORITHM INTRODUCTION. Slides in PPT. Classes of quantum algorithms. The concept of unsorted data base as a metaphor in quantum algorithms. Traveling Salesman problem. The quantum oracle. Possible exam problems related to Grover and oracles.

  2. GROVER ALGORITHM DETAILS. Slides in PPT. Components of Grover loop. Formulas for quantum oracle. The role of quantum oracle. Hadamard gate. Zero State Phase Shift. Circuit for Grover's algorithm. Generality. Grover iterate. Visualization of Grover transformation on amplitudes. How many Grover iterates do we need? Generalizations of Grover's Algorithm. Optimality of search algorithm.


AUXILIARY VIDEOS ABOUT QUANTUM ALGORITHMS:

  1. Video and slides by Rod Von Meter about quantum algorithms, including Grover. Select "Start video" on top left. Use both window with video and other window with slides.

  2. Ivan Deutsch. chapter5.mp4
  3. Ivan Deutsch. QI_Lecture5.pdf

WEEK EIGHT.



QUANTUM FOURIER TRANSFORM AND SHOR ALGORITHM.



MANDATORY SLIDES:

  1. QUANTUM FOURIER TRANSFORM AND INTRODUCTION TO SHOR ALGORITHM. Slides in PPT. Overvied of Shor algorithm. Discrete Fourier Transform. Quantum Fourier Transform. Review.

  2. SHOR ALGORITHM. Slides in PPT. Controlled-U gate - review. Phase estimation algorithm. RSA encyption. A little number theory reminder. Choosing a U. Choosing the initial state. Reductions. Summary of Shor algorithm.

  3. QUANTUM AUTOMATA AND COMPLEXITY CLASSES. Slides in PPT. Quantum automata and complexity. Models of computation. Finite State Systems. DFAs and matrices. Non-deterministic automata. Probabilistic automata. Language accepted. Quantum Automata. Acceptance Probabilities. Language accepted. Interference. Turing Machines. Configurations and Computations. Non-deterministic Turing Machines. Probabilistic Machines. BPPs. Quantum Turing Machines. BQP. Complexity classes.
  4. Slides about quantum Braitenberg Vehicles, Quantum Robots, Quantum Emotions and Quantum Learning.

AUXILIARY SLIDES:

  1. CONTROLLED SHIFT ALGORITHMS. Slides in PPT. Classical Versus quantum computers. Randomized classical versus quantum computers. Quantum Algorithms as Interferometry. Other ways to introduce a relative phase. Deutsch's problem. Variants of controlling _ function or shift?

  2. PHASE ESTIMATION. PERIOD FINDING. Slides in PPT. Phase estimation vs QFFT. Eigenvalue Estimation. Period Finding ala Shor. Order finding.



AUXILIARY VIDEO:

  1. Many good videos about quantum computing.


WEEK NINE.

NMR AND REALIZATION OF QUANTUM COMPUTERS IN REAL HARDWARE



MANDATORY VIDEO AND SLIDES:
  1. Video and slides of Rod Van Meter about quantum technologies.
  2. Video and slides of Rod Van Meter about quantum computer architectures.
  3. Ivan Deutsch. chapter8.mp4 Physical Implementations of quantum computers.
  4. Ivan Deutsch. QI_Lecture8.pdf Slides to this lecture.

  5. INTRODUCTION TO QUANTUM COMPUTATION TECHNOLOGIES. NMR AS AN EXAMPLE. Slides in PPT. Requirements for quantum computation. Implications of building quantum computers. Why building them is so difficult. Main contender technologies. Quantum Technology requirements. Di Vincenzo Criteria. Decoherence. Quantum Computer using a single molecule. Spins and coherence. Energy Levels splitting by magnetic fields.

  6. MAIN IDEAS OF NMR QUANTUM COMPUTER. Slides in PPT. Nuclear Magnetic Resonance. The Average Hamiltonian of an NMR sample. NMR spectrum of two spin-1/2 system. The approximate quantum state of the sample. Preparation of an effective pure state. Main ideas of NMR quantum computer. Advantages of NMR. Liquid State NMR Ensamble Computers. History. Inter-Atomic Bonds. How the NMR QC looks like? Initialization of Spins of Protons. Controlling spins. Electromagnetic fields. RF pulses. CNOT gate and machine language. RF coils and Static Field Coils. NMR in the works. Disadvantages of NMR. Example: 7-qubit quantum computer by IBM. Shor's factoring. The molecule. Pulse sequence. Results: Spectra. Circuit Simplifications. Some molecules used for NMR Quantum Computing. Physical Limitations to NMR qc.

  7. MATHEMATICS OF NMR QC. Slides in PPT. NMR quantum computers. NMR QC Physical apparatus. Hamiltonian. Computation. Experiments. Drawbacks.

  8. MORE ON NMR QUANTUM COMPUTERS. Slides in PPT. Decoherence. Quantum Errors. Classic Error Correction. Quantum Error Correction. Requirements for Quantum Computer. NMR-based QC. Structure of the system. Implementing operations. Full System architecture. Programming/Compiling. Error Correction. Architecture. Conclusions.

  9. ION TRAP QUANTUM COMPUTERS. Main ideas. Two-level atom as a qubit. Ion Trap Quantum Computer. Linear Ion Trap. Modes of ion oscillation. Energy levels of single ion. Controlled-phase-flip. Swap gate. Readout. Silicion based Quantum Computer. Optical Quantum Computer. What about scaling. Technical or fundamental problems? Quantum Error Correction is necessary. Summary and literature fo further reading.


WEEK TEN.



FINAL EXAM.
Final exam will be on Friday June 15. It is open book examination. It will cover the following material: Only material covered in lectures will be required.
  1. Deutsch algorithm.
  2. Deutsch-Jozsa algorithm.
  3. Grover algorithm.
  4. Eigenvalues and Eigenvectors.
  5. Quantum Hadamard Transform and Quantum Fast Fourier Transform.
  6. Density Matrix.
  7. Dirac versus Heisenberg Notation.
  8. Measurements.
  9. Superdense Coding.
  10. Teleportation.
  11. Quantum Error Correction.


QUANTUM ERROR CORRECTION.



MANDATORY SLIDES:

  1. INTRODUCTION TO QUANTUM ERROR CORRECTION. Slides in PPT. Quantum Errors. Classical Error Codes. Parity checking. Classical Error Correcting codes. Reversible networks for encoding and decoding a single bit "b". Example of operation of a classical error correcting code. Example of 3-qubit error correction. The Encoder. The Decoder. All cases taken together. Decoder without measurement. Reversible 5-qubit network for error correction. Other variant.

  2. MORE ON ERROR CORRECTION. Slides in PPT. Errors in communication. Classical Channel models. Quantum Channel models. Computing power versus error control. Basic Concepts in Error Control. Applications of Error Control. History of classical error Correcting codes. Classical Error Correction - encoding. Vector space and subspaces. Decoding in classical error correction. Role of Parity Check Matrix P. Classical Linear Error Control Codes. Linear operators. Block linear codes. Features of binary (n,k,d) LBC. Error Detection and Correction capability. Detection Capability of Linear Block Codes. Detection and Correction of (n,k) Linear Block Codes. Linear (n,k) Cyclic Codes over GF(2). Encoding a Cyclic Code. Cyclic Shifts in Cyclic codes. Cyclic property. Cyclic group in code subspace. Quantum Error correction. Why? What can go wrong. Decoherence. Role of environment. How to deal with decoherence. Decoherence times in practice. Gate completion time. Maximum number of operations before decoherence for various quantum systems. Dealing with decoherence and other sources of errors. History of Quantum Error Correction Codes. Quantum Errors. Quantum Error Correcting Codes. Commuting and anti-Commuting Quantum Operators. Pauli operations. Properties of Pauli Operators. 1-qubit Pauli group. Example: error operator in G_5. Quantum Network for correcting errors. Variants. Quantum Error Correction by Peter Shor.

  3. VIDEOS - AUXILIARY: Lecture by Prof. Daniel Lidar about decoherence in Quantum Computers. Format .wmv

  4. SHOR'S 9-QUBIT ERROR CORRECTION CODE. Slides in PPT. Review. A Simple quantum (3,1) Repetition code. Single Qubit errors. A review of simple classical error correcting codes. Why using classical error correction for correcting qubits is not trivial? Examples of quantum error correcting codes. Simple (3,1) repetition code circuit. Error Correction for 1 Bit flip. Encoder for (3,1) repetition code. How decoder works? The important idea of Syndrome. Analysis. Correcting single phase flip in (3,1) circuits. Correcting single phase flip. Initial problems avoided. Step by step analysis of decoding and correction. Shor's 9 qubit error correcting code. Basic Idea of Shor code. Architecture of Shor code. Principle of encoding in Shor code. Encoding in Shor Code. Detailed analysis. Decoder. Detailed analysis of an error. Detailed illustration of decoding. The (9,1) circuit - put it all togther. Another explanation.

  5. 5 QUBIT QUANTUM ERROR CORRECTION. Slides in PPT. How to use syndrome bits to calculate the minimum length. Encoding. Rules of flipping phase and bits. Signal after Hadamards. Step-by-Step analysis of the encoding circuit. Step-by-Step analysis of the decoding circuit. Error in phase and bit flip on 3rd qubit. Error analysis in decoder. Syndrome tables. Execution of correction based on syndromes. The 5-qubit error correcting circuit. Concatenated code. Example.

  6. QUANTUM ERROR CORRECTING METHODOLOGY. Slides in PPT. Quantum Errors. Barriers to Quantum Error Correction. Measurement destroys superpositions? Measure the error - not the data!. Redundancy, not repetition. Correcting just phase errors. Nine-qubit code. Correcting continuous rotation. Correcting all single-qubit errors. The Pauli group. Small error on every qubit.

AUXILIARY SLIDES:

  1. REVIEW OF CLASSICAL ERROR CORRECTING CODES. Slides in PPT. General model. Applications. Hierarchy of codes. Block codes. Example of (6,3,3)_2 systematic code. Error Detection with parity bit. Error Correcting One Bit Messages. Error Correcting multi-bit messages. Hamming codes. Encoding and decoding. Lower Bound on parity bits. Lower bounds. Linear Codes. Generator and parity check matrices. Advantages of linear codes. Example and "Standard Form". Relationship of G and H.The d of linear codes. Dual Codes. How to find the error locations.

  2. Auxiliary lecture about classical error correction codes. Slides in PPT. Reed-Solomon Codes. RS in real world. Applications. RS and "burst" errors. Galois Field. Discrete Fourier Transform. DFT example. Decoding. Cyclic codes. Generator and Parity Check Matrices. Generator and Parity Check Polynomials. Viewing g as a matrix. g generates cyclic codes. Viewing h as a matrix. Hamming codes revisited. Factors of X^n - 1. Another way to write g. Back to Reed-Solomon. Examples. A useful theorem. Fixing errors. Efficient decoding.

  3. Auxiliary Use Of Codes in Quantum Memory. Slides in PPT. Memory hierarchies for quantum data. Intro. Controlled Entanglement. Uncontrolled entanglement. Quantum Error Correction. Other encodings. Error Calculations. Concatenation. Examples. Teleportation. Memory hierarchies. Encoding levels in memory hierarchy.

  4. Dave Bacon on Quantum Error Correction. Slides in PPT. Future quantum systems. Fault tolerant quantum architecture. Concatenation. Threshold theorem. Concatenation and locality. Kitaev's toric codes. Local codes. Physics and Toric Codes. The Physics Guarantee. The Quantum Hard Drive?

  5. QUANTUM ERROR CORRECTION BY MICHELE MOSCA. Slides in PPT. Bit flip errors. Analysis of 6-qubit system which uses syndrome bits to correct errors. Quantum error Correction. Main Error Correction Theorem.

  6. STABILIZER CODES. Slides in PPT. Error Syndromes revisited. Stabilizer measurement? The Shor code as a Stabilizer Code. Stabilizer for nine-qudit code. Stabilizer codes. Stabilizer codes by Gottesman. Properties of a stabilizer. Stabilizer elements detect errors. Distance of a stabilizer code. Stabilizer codes correct errors. Application: 5-qubit code. Classical Linear Codes. Classical Hamming Codes. Linear Codes and Stabilizers. CSS codes. Which CSS codes are possible? Properties of CSS codes. Classical (7,4,3) Hamming code C_1. Stabilizer for Steane code. Steane Code based on Hamming code. Connection to classical codes: CSS Codes. Example of CSS code: the [7,1,3] code. Vector Space under Sympletic Inner Products. Weight and distance. Theorem for existence of Additive QECC. Stabilizer for 5-qubit [5,1,3] Code. Sympletic form for Stabilizer. Commuting group. Linear Block Codes. Standard Sympletic form. Encoder for stabilizer codes. (5,1,3) code. Stabilizer codes.

  7. HOMOLOGICAL QECC. Slides in PPT. Lecture 22 of Michele Mosca. Correcting Phase errors. Quantum error correction. c Correcting both Phase errors and bit flip errors. Correcting any error. Degenerate codes. Quantum Hamming bound.

  8. LECTURE OF A STUDENT ABOUT CLASSICAL AND QUANTUM ERROR CORRECTION. Perkowski_051105.wmv This lecture has also material about quantum testing and introduction to Deutsch algorithms. Use of phase information. One in four circuit.

  9. GALOIS FIELD AND HOMOLOGICAL CODES. Slides in PPT. Codes over GF(4). Relation to Pauli Matrices. From Sympletic Form to GF(4). Self-Orthogonal and self-dual. Toric codes of Kitaev. Toric code example. Generalized Toric codes. Homological codes. Summary and literature.

WEEK ELEVEN.



Week of finals. There will be student presentations and my lecture. on Wednesday and final exam on Friday.

THE MATERIAL BELOW IS NOT MANDATORY.



AUXILIARY VIDEOS.
  1. Video and slides of Rod Van Meter about Quantum Networking. The slides are on his webpage. You should download them and read.

  2. Ivan Deutsch. chapter4.mp4 Qubits and their applications in teleportation and similar topics.

  3. Ivan Deutsch. QI_Lecture4.pdf Slides to this lecture.

  4. Ivan Deutsch. chapter6.mp4 Decoherence. Errors. Error Correction.

  5. Ivan Deutsch. QI_Lecture6.pdf Slides to this lecture.



ADVANCED SLIDES AND VIDEOS ON DESIGNING QUANTUM CIRCUITS.



  1. Transformations of reversible circuits by Kinoshita et al from DAC 2002. How to realize many-input AND gates in quantum circuits. Main problems in Cascade Design. Previous research on cascades. Transformation rules by Iwama, Kambayashi and Yamashita. The concepts of control and target bits. Canonicity of reversible circuits. Verification versus optimization. Moving gates and transformations to the canonical Form. Transformation rules and how to apply them. Examples of using complex sets of rules for big circuits.
    The realization of big gates from the first part was covered in other lecture and is mandatory. However the set of all transforming rules is not required. It is only good to know how the properties can be proved based on them and new algorithms created.

  2. Fuzzy Reversible Circuits Building reversible fuzzy Toffoli gate.

    The fuzzy reversible and fuzzy quantum gates such as 4*4 Fredkin are discussed in the next lecture, where the gates can be built for both mv and fuzzy logic in the same way.

  3. LECTURE ABOUT SYNTHESIS OF QUANTUM FUZZY CIRCUITS.
    Perkowski_050205.wmv This lecture covers also nets, symmetric function realization and QMDDs.

  4. Symmetric reversible nets. Regular realization of symmetric binary and ternary reversible logic functions. Multi-valued Fredkin gates. Example. Using these gates to create MIN/MAX gates. Regular structures of binary and ternary min/max gates. Theorem and new structures. Use of MIN/MAX gates. Open problems and new research,

  5. Regular Lattices. Levelized structures for binary and ternary logic. Lattice structures in 2 and 3 dimensions. Three types of general expansions. New research ideas for binary and mv, reversible and quantum logic circuits.

  6. Optical Reversible. Optical Conservative reversible and nearly reversible gates. Integrated optics. Troubles with optical logic gates. Requirements for gates. The device. The dual-beam nonlinear interface. RNI - reversible nonlinear interface. Half-wave plates. Complete circuit. Interaction gate. Priese switch gate. Fredkin gate. Example. Realization of RNI Half-adder.

  7. Mitra and wave cascades. Problems for midterm in the past class. Derivatives of Perkowski's gate. Structure of wave cascades. Material for midterm 1 in the past class.

  8. Perkowski_051805.wmv

      STUDENT PROJECTS PRESENTATIONS.
      1. Faisal Khan - CSD Decomposition.
      2. Dean Pierce and Jake Biamonte - Use of QUIDPRO for test generation for reversible circuits.
      3. Sazzad Hossain - NPNP classification for reversible circuits.



ADDITIONAL ADVANCED LECTURES SLIDES AND VIDEOS

  1. 1-hamiltonian.mp4 Adiabatic Computing.

  2. Adiabatic Computing. 2-hamiltonian (1).mp4

  3. Braunstein.ppt Lecture by Prof. Braunstein about Entanglement as a fundamental resource.

  4. RealBig.WMV Using Neutral Atoms as Qubits. Lecture by a Nobel Prize Laureate.

    FOR FUN - FUTURE QUANTUM ROBOTS.

    1. Robot from Japan. .mpeg