Quantum Computing Marek Perkowski Part of Computational Intelligence Course 2007

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<ul><li> Slide 1 </li> <li> Quantum Computing Marek Perkowski Part of Computational Intelligence Course 2007 </li> <li> Slide 2 </li> <li> Introduction to Quantum Logic and Reversible/Quantum Circuits Marek Perkowski </li> <li> Slide 3 </li> <li> Historical Background and Links Quantum Computation &amp; Quantum Information Computer Science Information Theory Cryptography Quantum Mechanics Study of information processing tasks that can be accomplished using quantum mechanical systems Digital Design </li> <li> Slide 4 </li> <li> What is quantum computation? Computation with coherent atomic-scale dynamics. The behavior of a quantum computer is governed by the laws of quantum mechanics. </li> <li> Slide 5 </li> <li> Why bother with quantum computation? Moores Law: We hit the quantum level 2010~2020. Quantum computation is more powerful than classical computation. More can be computed in less timethe complexity classes are different! </li> <li> Slide 6 </li> <li> The power of quantum computation In quantum systems possibilities count, even if they never happen! Each of exponentially many possibilities can be used to perform a part of a computation at the same time. </li> <li> Slide 7 </li> <li> Nobody understands quantum mechanics No, youre not going to be able to understand it.... You see, my physics students dont understand it either. That is because I dont understand it. Nobody does.... The theory of quantum electrodynamics describes Nature as absurd from the point of view of common sense. And it agrees fully with an experiment. So I hope that you can accept Nature as She is -- absurd. Richard Feynman </li> <li> Slide 8 </li> <li> Absurd but taken seriously (not just quantum mechanics but also quantum computation) Under active investigation by many of the top physics labs around the world (including CalTech, MIT, AT&amp;T, Stanford, Los Alamos, UCLA, Oxford, lUniversit de Montral, University of Innsbruck, IBM Research...) In the mass media (including The New York Times, The Economist, American Scientist, Scientific American,...) Here. </li> <li> Slide 9 </li> <li> Quantum Logic Circuits </li> <li> Slide 10 </li> <li> A beam splitter Half of the photons leaving the light source arrive at detector A; the other half arrive at detector B. </li> <li> Slide 11 </li> <li> A beam-splitter The simplest explanation is that the beam-splitter acts as a classical coin-flip, randomly sending each photon one way or the other. </li> <li> Slide 12 </li> <li> An interferometer Equal path lengths, rigid mirrors. Only one photon in the apparatus at a time. All photons leaving the source arrive at B. WHY? </li> <li> Slide 13 </li> <li> Possibilities count There is a quantity that well call the amplitude for each possible path that a photon can take. The amplitudes can interfere constructively and destructively, even though each photon takes only one path. The amplitudes at detector A interfere destructively; those at detector B interfere constructively. </li> <li> Slide 14 </li> <li> Calculating interference Arrows for each possibility. Arrows rotate; speed depends on frequency. Arrows flip 180 o at mirrors, rotate 90 o counter-clockwise when reflected from beam splitters. Add arrows and square the length of the result to determine the probability for any possibility. </li> <li> Slide 15 </li> <li> Double slit interference </li> <li> Slide 16 </li> <li> Quantum Interference : Amplitudes are added and not intensities ! </li> <li> Slide 17 </li> <li> Interference in the interferometer Arrows flip 180 o at mirrors, rotate 90 o counter-clockwise when reflected from beam splitters </li> <li> Slide 18 </li> <li> Quantum Interference The simplest explanation must be wrong, since it would predict a 50-50 distribution. How to create a mathematical model that would explain the previous slide and also help to predict new phenomena? Two beam-splitters </li> <li> Slide 19 </li> <li> More experimental data </li> <li> Slide 20 </li> <li> A new theory The particle can exist in a linear combination or superposition of the two paths </li> <li> Slide 21 </li> <li> Probability Amplitude and Measurement If the photon is measured when it is in the state then we get with probability and |1&gt; with probability of |a 1 | 2 </li> <li> Slide 22 </li> <li> Quantum Operations The operations are induced by the apparatus linearly, that is, if and then </li> <li> Slide 23 </li> <li> Quantum Operations Any linear operation that takes states satisfying and maps them to states satisfying must be UNITARY </li> <li> Slide 24 </li> <li> Linear Algebra is unitary if and only if </li> <li> Slide 25 </li> <li> Linear Algebra corresponds to </li> <li> Slide 26 </li> <li> Linear Algebra corresponds to </li> <li> Slide 27 </li> <li> Linear Algebra corresponds to </li> <li> Slide 28 </li> <li> Abstraction The two position states of a photon in a Mach-Zehnder apparatus is just one example of a quantum bit or qubit Except when addressing a particular physical implementation, we will simply talk about basis states and and unitary operations like and </li> <li> Slide 29 </li> <li> wherecorresponds to and corresponds to </li> <li> Slide 30 </li> <li> An arrangement like is represented with a network like </li> <li> Slide 31 </li> <li> More than one qubit If we concatenate two qubits we have a 2-qubit system with 4 basis states and we can also describe the state as or by the vector 1 </li> <li> Slide 32 </li> <li> More than one qubit In general we can have arbitrary superpositions where there is no factorization into the tensor product of two independent qubits. These states are called entangled. </li> <li> Slide 33 </li> <li> Entanglement Qubits in a multi-qubit system are not independentthey can become entangled. To represent the state of n qubits we use 2 n complex number amplitudes. </li> <li> Slide 34 </li> <li> Measuring multi-qubit systems If we measure both bits of we getwith probability </li> <li> Slide 35 </li> <li> Measurement | | 2, for amplitudes of all states matching an output bit-pattern, gives the probability that it will be read. Example: 0.316|00 + 0.447|01 + 0.548|10 + 0.632|11 The probability to read the rightmost bit as 0 is |0.316| 2 + |0.548| 2 = 0.4 Measurement during a computation changes the state of the system but can be used in some cases to increase efficiency (measure and halt or continue). </li> <li> Slide 36 </li> <li> Sources Mosca, Hayes, Ekert, Lee Spector in collaboration with Herbert J. Bernstein, Howard Barnum, Nikhil Swamy {lspector, hbernstein, hbarnum, nikhil_swamy}@hampshire.edu} School of Cognitive Science, School of Natural Science Institute for Science and Interdisciplinary Studies (ISIS) Hampshire College Origin of slides: John Hayes, Peter Shor, Martin Lukac, Mikhail Pivtoraiko, Alan Mishchenko, Pawel Kerntopf, Mosca, Ekert </li> </ul>

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