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?
Transcript
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Quantum ComputationThe Physics of Information J. Caleb Wherry Austin Peay State University Departments of Computer Science, Mathematics, & Physics
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Outline Classical Computation History Babbage, ENIAC, Vacuum Tubes, & the Transistor Moore’s Law Computation & Complexity Theory Cbits, Logic Gates, & the Circuit Model Moore’s Law Revisited Quantum Computation Mathematical Formalisms (Linear Algebra & Quantum Mechanics) Qubits, Quantum Gates, & the Quantum Circuit Model BQP & the Power of Q.C. Quantum Q.C. Implementations NMR, Iron Trap, Superconducting Qubits, & Topological Q.C. Quantum Algorithms Grover’s Search & Shor’sFactoring Algorithms Other Computational Paradigms Zeno’s Computer Relativity Computer Closed Timelike Curve Computation DNA Computing 2
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Computation & Complexity Theory 8 What is computation?
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Computation & Complexity Theory 9 Computation A process following a well-defined model that is understood and can be expressed in an algorithm, protocol, network topology, etc. Computational Complexity The measure of the resources (e.g. time, space, basic operations, energy) used by a computation. Measured as a function of the input size. Turing Machine A very simplistic computer in which computations can be executed on. Tape – Infinitely Long. Finite Alphabet. Head – Reads/Writes, Moves Tape 1 Cell L/R. Table – Finite Set of Instructions. State Register – Current Finite State of TM. Strong Church-Turing Thesis A probabilistic Turing machine (e.g. a classical computer that can make fair coin flips) can efficiently simulate any realistic model of computing.
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Mathematical Formalisms 17 Qubit – Quantum Bit Orthonormal Basis Set Superposition of 0 & 1 |0 + |1 |0 |1 |0 | |1 E.g. = | Qubits: Photons, Electrons, Ions, etc. *Spin of above particles. Bloch Sphere
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Mathematical Aside 18 Where do qubits live? lives in a Hilbert Space H . | H is a complete Vector Space with a defined inner product. What does complete mean? Formal definition: a space is complete if every Cauchy Sequence converges to a point within the set. But what does that mean? Fields: N, Q, R, C, H
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Mathematical Formalisms 19 Quantum Logic Gates = Linear Transformations Pauli Matrices Hadamard Gate Pauli-X Pauli-Y Hadamard Pauli-Z
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Quantum Algorithms 27 Grover’s Search Normal amount of time a database search takes? N items takes O(n) searches. Grover’s Search takes O( SQRT(N) ) searches for N items1
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Quantum Algorithms 28 Shor’s Factoring Fastest Classical Factoring Algorithm: General Number Field Sieve O(e^((log N)^1/3 (log log N)^2/3)) Shor’s Algorithm Factors in: O(log(N)^3) Exponential Speedup!
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Other Computational Paradigms 30 Zeno’s Computer STEP 1 STEP 2 Time (seconds) STEP 3 STEP 4 STEP 5
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31 Other Computational Paradigms Relativity Computer DONE
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Other Computational Paradigms 32 Closed Timelike Curve Computation 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.
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Other Computational Paradigms 33 DNA Computing
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References 34 [1] Arora, S., Barak, B., “Computational Complexity: A Modern Approach.” [2] Bernstein, E., Vazirani, U., “Quantum Complexity Theory.” [3] Chuang, I., “Quantum Algorithms and their Implementations: QuISU – An Introduction for Undergraduates.” [4] Lloyd, S., “Quantum Information Science.” [5] Nielson, M., Chuang, I., “Quantum Computation and Quantum Information.” [6] Images Courtesy of Wikipedia. [7] Thanks to Scott Aaronson & Michele Mosca for Slide Inspirations & Figures.
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