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
-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
Quantum ComputationThe Mathematics of Information<br />J. Caleb Wherry<br />Austin Peay State University<br />Departments of Computer Science, Mathematics, & Physics<br />
Computation & Complexity Theory<br />8<br />What is computation?<br />
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 />
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 />
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 />
Other Computational Paradigms<br />33<br />DNA Computing<br />
References<br />34<br /> Arora, S., Barak, B., “Computational Complexity: A Modern Approach.”<br /> Bernstein, E., Vazirani, U., “Quantum Complexity Theory.”<br /> Chuang, I., “Quantum Algorithms and their Implementations: QuISU – An Introduction for Undergraduates.”<br /> Lloyd, S., “Quantum Information Science.”<br /> Nielson, M., Chuang, I., “Quantum Computation and Quantum Information.”<br /> Images Courtesy of Wikipedia.<br /> Thanks to Scott Aaronson & Michele Mosca for Slide Inspirations & Figures. <br />