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Part of Computational Intelligence Course 2007 Quantum Computing Marek Perkowski Introduction to Quantum Logic and Reversible/Quantum Circuits Marek Perkowski Historical Background and Links Quantum Computation & Quantum Information Study of information processing tasks that can be accomplished using quantum mechanical systems Cryptography Quantum Mechanics Computer Science Information Theory Digital Design What is quantum computation? • • Computation with coherent atomic-scale dynamics. The behavior of a quantum computer is governed by the laws of quantum mechanics. Why bother with quantum computation? • Moore’s Law: We hit the quantum level 2010~2020. • Quantum computation is more powerful than classical computation. • More can be computed in less time—the complexity classes are different! 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. Nobody understands quantum mechanics “No, you’re not going to be able to understand it. . . . You see, my physics students don’t understand it either. That is because I don’t 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 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&T, Stanford, Los Alamos, UCLA, Oxford, l’Université de Montréal, University of Innsbruck, IBM Research . . .) • In the mass media (including The New York Times, The Economist, American Scientist, Scientific American, . . .) • Here. Quantum Logic Circuits A beam splitter Half of the photons leaving the light source arrive at detector A; the other half arrive at detector B. A beam-splitter 0 0 1 1 50% 50% The simplest explanation is that the beam-splitter acts as a classical coin-flip, randomly sending each photon one way or the other. 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? Possibilities count • There is a quantity that we’ll 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. Calculating interference • Arrows for each possibility. • Arrows rotate; speed depends on frequency. • Arrows flip 180o at mirrors, rotate 90o counter-clockwise when reflected from beam splitters. • Add arrows and square the length of the result to determine the probability for any possibility. Double slit interference Quantum Interference : Amplitudes are added and not intensities ! Interference in the interferometer Arrows flip 180o at mirrors, rotate 90o counter-clockwise when reflected from beam splitters Quantum Interference Two beam-splitters 0 0 1 1 100% 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? More experimental data 0 1 0 sin 2 2 1 cos 2 2 A new theory The particle can exist in a linear combination or superposition of the two paths 0 1 i 1 0 1 2 2 0 sin 2 2 1 cos 2 2 i ei 0 1 2 2 ei 1 i(ei 1) 0 1 2 2 Probability Amplitude and Measurement If the photon is measured when it is in the state 2 0 0 1 1 then we get 0 with probability 0 and |1> with probability of |a1|2 0 1 0 1 0 2 1 2 2 0 1 2 1 Quantum Operations The operations are induced by the apparatus linearly, i 1 that is, if 0 0 1 2 and then 2 1 i 1 0 1 2 2 i 0 0 1 1 0 0 2 i 0 1 2 1 1 1 2 1 0 0 2 1 i 0 1 2 2 1 i 1 1 2 2 Quantum Operations Any linear operation that takes states 2 2 satisfying 0 0 1 1 0 1 1 and maps them to states '0 0 '1 1 satisfying must be UNITARY ' 0 2 ' 1 2 1 Linear Algebra u00 u01 U u10 u11 is unitary if and only if u u u 00 01 t 00 UU u10 u11 u01* * u10 1 0 I * u11 0 1 * Linear Algebra 0 corresponds to 1 corresponds to 0 0 1 1 corresponds to 1 0 0 1 1 0 0 0 1 0 1 1 Linear Algebra corresponds to corresponds to i 2 1 2 1 2 i 2 1 0 i 0 e Linear Algebra 0 corresponds to i 2 1 2 1 2 i 2 1 0 i 0 e i 2 1 2 1 2 i 2 1 0 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 0 and 1 and unitary operations like H and where H corresponds to and corresponds to 1 2 1 2 1 2 1 2 1 0 i 0 e An arrangement like 0 is represented with a network like 0 H H More than one qubit If we concatenate two qubits 0 0 1 1 0 0 1 1 we have a 2-qubit system with 4 basis states 0 0 00 0 1 01 1 0 10 1 1 11 and we can also describe the state as 00 00 01 01 10 10 11 11 or by the vector 0 0 01 0 0 1 0 1 1 1 1 More than one qubit In general we can have arbitrary superpositions 00 0 0 01 0 1 10 1 0 11 1 1 2 2 2 2 00 01 10 11 1 where there is no factorization into the tensor product of two independent qubits. These states are called entangled. Entanglement • Qubits in a multi-qubit system are not independent—they can become “entangled.” • To represent the state of n qubits we use 2n complex number amplitudes. Measuring multi-qubit systems If we measure both bits of 00 0 0 01 0 1 10 1 0 11 1 1 we get x y with probability xy 2 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). Sources Origin of slides: John Hayes, Peter Shor, Martin Lukac, Mikhail Pivtoraiko, Alan Mishchenko, Pawel Kerntopf, Ekert Mosca, 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