Quantum Computation: The Mathematics of Information


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  • Where does computation “start”? Many people have differing view points on this matter.
  • I will start with the first mechanical type calculating machine.Blaise Pascal (1623) – ”Pascaline” – Did simple adding/subtracting/multiplyingGerman Guy (1673) – “Step Reckoner” – Also did simple adding/subtracting/multiplying-Charles Babbage (1823) – “Difference Engine” - ?? - “Analytical Engine” - ???-Other inventions after these but I won’t mention them.
  • -ENIAC (finished in 1946) “Electronic Numerical Integrator And Computer” - Constructed by Department of Defense - First used for Hydrogen Bomb calculations - Most famous for missile guidance calculations-Vacuum Tubes - How computation was done on ENIAC - Very cumbersome & clunky - Lots of energy needed to use, not efficient- What changed computation around this ??
  • -Biggest revolution of the modern computational age-Enables computers to be etched on chips, much much smaller than vacuum tubes.-These were the first transistors implemented in 1954 by T.I.-What “law” was noticed shortly after the invention of the transistor?
  • -Moore’s Law!-Every 18 months the number of transistors able to be etched on a chip doubles.-Not a “law”, just an observation-Moore’s Law standstill will be presented in later slides, will come back to it!
  • -TM has unlimited tape so unbounded amount of storage & time.
  • -What most people think of computational complexity theory…-Explain some of the classes.-L -uses a log amount of memory with respect to input size-P -uses polynomial amount of time to solve problem-NP -non-deterministic polynomial time -decision problem -answer can be checked in polynomial time
  • -How computational complexity actual looks.-Set inclusion diagram
  • -Since classical bits can be either 1OR 0, each string of n bits represents n amounts of information with 2^n possibilities.-Why do we use a 2-state system? Why not 3? 4? 1000? -distinguish-ability! -error correction
  • -Describe FANOUT & FANIN (Electrical Engineering Terms)-Discard bits to make reversible (doesn’t really apply to classical computation)
  • Gates “glued” together.
  • -Now that we know how classical computation works, what is its future?-Moore’s Law Shows a trend that has to stop, why?-Physical Constraints, not engineering constraints. -Quantum effects -Special Relativity-Next slide – Quantum Computation!
  • -Orthonormal Basis Set sometimes called Computational Basis Set-Only showing pure states -mixed states cannot be decomposed into PSI and they live INSIDE the bloch sphere.-measurements kill superposition, collapse wave function/state vector
  • -Integers,Rationals, Reals, Complex, Quaternions-Sum = PI^2/6
  • -There are universal sets of gates, you usually pick 2 gates from above and then one 3-level entangling gate. -This will give an approximate circuit to within delta error.-Only a single qubit picture, can’t see this pic for composite systems, too complex.
  • -Describe this experiment as a wave & particle of light.-Describe interference & Double Slit Experiment
  • -Double slit experiment (explain)-What will the probabilities be of measuring a particle (or wave?)
  • - Talk about quantum mechanics & interpretations of quantum mechanicsLocality is out the windowEPR Paradox -Einstein, Podolsky, Rosen
  • -Used in quantum cryptography
  • -nqubits = 2^n amounts of information.-Why???
  • -Remember BQP though.-Some error associated with it.
  • -Describe factoring-GNFS is sub-exponential-Uses superposition, entanglement, & quantum Fourier transform.
  • -Based on Zeno’s Paradoxes -Zeno’s Dichotomy Paradox -”To get to a point, you have to get half way there, to get half way you have to get a fourth, etc. Thus, you never reach where you want to go.”-What is wrong with this paradigm?-Planck Scale, infinite amount of energy to do computation
  • -Special Relativity Laws-Travel away from light at “near” light speed -Observer only ages 10 years but computation goes for 100 years or so
  • “Time travel computer”Scott Aaronson & John WatrousWhat are the implications of this paradigm?
  • -Similar analog to classical parallel computation-Distribute datasets over DNA strain and compute.-Why is this good?
  • Quantum Computation: The Mathematics of Information

    1. 1. Quantum ComputationThe Mathematics of Information<br />J. Caleb Wherry<br />Austin Peay State University<br />Departments of Computer Science, Mathematics, & Physics<br />
    2. 2. Outline<br />Classical Computation<br />History<br />Babbage, ENIAC, Vacuum Tubes, & the Transistor<br />Moore’s Law<br />Computation & Complexity Theory<br />Cbits, Logic Gates, & the Circuit Model<br />Moore’s Law Revisited<br />Quantum Computation<br />Mathematical Formalisms (Linear Algebra & Quantum Mechanics)<br />Qubits, Quantum Gates, & the Quantum Circuit Model<br />BQP & the Power of Q.C.<br />Quantum Q.C. Implementations<br />NMR, Iron Trap, Superconducting Qubits, & Topological Q.C.<br />Quantum Algorithms<br />Grover’s Search & Shor’s Factoring Algorithms<br />Other Computational Paradigms<br />Zeno’s Computer<br />Relativity Computer<br />Closed Timelike Curve Computation<br />DNA Computing<br />2<br />
    3. 3. 3<br />Classical Computation<br />
    4. 4. History<br />4<br />Difference Engine - 1823<br />Pascaline - 1623<br />Step Reckoner - 1673<br />
    5. 5. History<br />5<br />ENIAC - 1946<br />Vacuum Tubes<br />
    6. 6. History<br />6<br />Texas Instruments 1954 Transistor<br />
    7. 7. History<br />7<br />Moore’s Law<br />
    8. 8. Computation & Complexity Theory<br />8<br />What is computation?<br />
    9. 9. Computation & Complexity Theory<br />9<br />Computation<br /> A process following a well-defined model that is understood and can be expressed in an algorithm, protocol, network topology, etc.<br />Computational Complexity<br /> The measure of the resources (e.g. time, space, basic operations, energy) used by a computation. Measured as a function of the input size.<br />Turing Machine<br /> A very simplistic computer in which computations can be executed on. <br />Tape – Infinitely Long. Finite Alphabet.<br />Head – Reads/Writes, Moves Tape 1 Cell L/R.<br />Table – Finite Set of Instructions.<br />State Register – Current Finite State of TM.<br />Strong Church-Turing Thesis <br />A probabilistic Turing machine (e.g. a classical computer that can make fair coin flips) can efficiently simulate any realistic model of computing.<br />
    10. 10. Computation & Complexity Theory<br />10<br />
    11. 11. Computation & Complexity Theory<br />11<br />
    12. 12. Cbits, Logic Gates, & the Circuit Model<br />12<br />Classical Bits<br /><ul><li> 2-state system (Boolean Algebra)
    13. 13. Possible states: 0 or 1 (Off or On)
    14. 14. 0 -> No voltage
    15. 15. 1 -> 0.5 voltage</li></ul>If we have n classical bits, how much information do we have?<br />
    16. 16. Cbits, Logic Gates, & the Circuit Model<br />13<br />Basic Classical Logic Gates<br />Logic Gates<br /><ul><li>{One,Two}-ary Operations on our Boolean Algebra
    17. 17. Universal set of gates: (AND, NOT, & FANOUT)
    18. 18. What does universal mean?
    19. 19. Are they reversible?
    20. 20. What does reversible mean?</li></li></ul><li>Cbits, Logic Gates, & the Circuit Model<br />14<br />
    21. 21. Moore’s Law Revisited<br />15<br />Moore’s Law<br />
    22. 22. 16<br />Quantum Computation<br />
    23. 23. Mathematical Formalisms<br />17<br />Qubit – Quantum Bit<br />Orthonormal Basis Set<br />Superposition of 0 & 1<br />|0 + |1<br />|0<br />|1<br />|0<br /><br />|<br />|1<br />E.g.<br /><br />=<br />|<br />Qubits: Photons, Electrons, Ions, etc.<br />*Spin of above particles.<br />Bloch Sphere<br />
    24. 24. Neat Mathematics<br />18<br />Where do qubits live?<br />|<br /><br />lives in a Hilbert Space H .<br />H is a complete Vector Space with a defined inner product. <br />What does complete mean?<br />Formal definition: a space is complete if every Cauchy Sequence converges to a point within the set.<br />But what does that mean?<br />Fields: N, Q, R, C, H<br />
    25. 25. Mathematical Formalisms<br />19<br />Quantum Logic Gates = Linear Transformations<br />Pauli Matrices<br />Hadamard Gate<br />Pauli-X <br />Pauli-Y <br />Hadamard<br />Pauli-Z <br />
    26. 26. Mathematical Formalisms<br />20<br />Quantum Weirdness<br />Superposition<br />Entanglement<br />Teleportation<br />
    27. 27. Mathematical Formalisms<br />21<br />Quantum Weirdness I<br />Superposition & Interference<br />
    28. 28. Mathematical Formalisms<br />22<br />Quantum Weirdness I<br />Superposition & Interference<br />
    29. 29. Mathematical Formalisms<br />23<br />Quantum Weirdness II<br />Entanglement – EPR Paradox<br />“Spookiness at a distance” - Einstein <br />
    30. 30. Mathematical Formalisms<br />24<br />Quantum Weirdness III<br />Teleportation<br />
    31. 31. BQP & the Power of Q.C.<br />17<br />If we have nqubits, how much information do we have?<br />
    32. 32. Quantum Implementations<br />26<br />NMR<br />Ion Trap<br />Superconducting Qubits<br />Topological Q.C.<br />
    33. 33. Quantum Algorithms<br />27<br />Grover’s Search<br />Normal amount of time a database search takes?<br />N items takes O(n) searches.<br />Grover’s Search takes O( SQRT(N) ) searches for N items.<br />
    34. 34. Quantum Algorithms<br />28<br />Shor’s Factoring<br />Fastest Classical Factoring Algorithm:<br />General Number Field Sieve<br />O(e^((log N)^1/3 (log log N)^2/3))<br />Shor’s Algorithm Factors in:<br /> O(log(N)^3)<br />Exponential Speedup!<br />
    35. 35. 29<br />Other Computational Paradigms<br />
    36. 36. Other Computational Paradigms<br />30<br />Zeno’s Computer<br />STEP 1<br />STEP 2<br />Time (seconds)<br />STEP 3<br />STEP 4<br />STEP 5<br />
    37. 37. 31<br />Other Computational Paradigms<br />Relativity Computer<br />DONE<br />
    38. 38. Other Computational Paradigms<br />32<br />Closed Timelike Curve Computation<br />S. Aaronson and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent, Proceedings of the Royal Society A 465:631-647, 2009. arXiv:0808.2669. <br />
    39. 39. Other Computational Paradigms<br />33<br />DNA Computing<br />
    40. 40. References<br />34<br />[1] Arora, S., Barak, B., “Computational Complexity: A Modern Approach.”<br />[2] Bernstein, E., Vazirani, U., “Quantum Complexity Theory.”<br />[3] Chuang, I., “Quantum Algorithms and their Implementations: QuISU – An Introduction for Undergraduates.”<br />[4] Lloyd, S., “Quantum Information Science.”<br />[5] Nielson, M., Chuang, I., “Quantum Computation and Quantum Information.”<br />[6] Images Courtesy of Wikipedia.<br />[7] Thanks to Scott Aaronson & Michele Mosca for Slide Inspirations & Figures. <br />
    41. 41. Questions & Comments<br />35<br />Questions?<br />Comments?<br />
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