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## Quantum Computing and Quantum Parallelism

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**Quantum Computing and Quantum Parallelism**Dan C. Marinescu and Gabriela M. Marinescu School of Computer Science University of Central Florida Orlando, Florida 32816, USA**Acknowledgments**• The material presented is from the book Approaching Quantum Computing by Dan C. Marinescu and Gabriela M. Marinescu ISBN 013145224X, Prentice Hall, July 2004. • Work supported by National Science Foundation grants MCB9527131, DBI0296107,ACI0296035, and EIA0296179. San Mallo, June 2004**Contents**• I. Computing and the Laws of Physics • II. The Strange World of Quantum Mechanics • III. Quantum Computing and Communication • IV Hilbert Spaces and Tensor Products • V. Qubits • VI.Quantum Gates and Quantum Circuits • VII. Quantum Parallelism • VIII. Deutsch’s Problem • IX. Bell States, Teleportation, and Dense Coding • X. Summary San Mallo, June 2004**Technological limits – density and speed**• For the past two decades we have enjoyed Gordon Moore’s law the speed doubles every 18 months. • But all good things may come to an end… • We are limited in our ability to increase • the density of solid-state circuits due to: • power dissipation and • quantum effects. • the speed of a computing device due to: • density San Mallo, June 2004**Technological limits – reliability**• Reliability will also be affected • to increase the speed we need increasingly smaller circuits (light needs 1 ns to travel 30 cm in vacuum) • smaller circuits systems consisting only of a few particles subject to Heissenberg’s uncertainty San Mallo, June 2004**Energy/operation**• If there is a minimum amount of energy dissipated to perform an elementary operation, then to increase the speed, thus the number of operations performed each second, we require at least a linear increase of the amount of energy dissipated by the device. • The computer technology vintage year 2000 requires some 3 x 10-18 Joules per elementary operation. San Mallo, June 2004**The effect of increasing the speed upon the power**consumption • Assume that: • the minimum amount of energy dissipated to perform an elementary operation is reduced 100-fold (this may not be technologically feasible) • the speed of a solid state device is increased 1,000 fold • Then we shall see a 10 (ten) fold increase in the amount of power needed by a solid state device operating at a 1,000 times higher speed. San Mallo, June 2004**Power dissipation, circuit density, and speed**• In 1992 Ralph Merkle from Xerox PARC calculated that a 1 GHz computer operating at room temperature, with 1018 gates packed in a volume of about 1 cm3 would dissipate 3 MW of power. • A small city with 1,000 homes each using 3 KW would require the same amount of power; • A 500 MW nuclear reactor could only power some 166 such circuits. San Mallo, June 2004**Heat generation…**• The heat produced by a super dense computing engine is proportional with the number of elementary computing circuits, thus, with the volume of the engine. • If the devices are densely packed in a sphere of radius rthe heat dissipated grows as the cube of the radius. San Mallo, June 2004**Heat removal**• If the devices are densely packed in a sphere of radius r, then the surface of the sphere is proportional with the square of the radius. • To prevent the destruction of the engine we have to remove the heat through a surface surrounding the device. • Our ability to remove heat increases as the square of the radius while the amount of heat increases with the cube of the radius of the computing device. San Mallo, June 2004**Contents**• I. Computing and the Laws of Physics • II. The Strange World of Quantum Mechanics • III. Quantum Computing and Communication • IV Hilbert Spaces and Tensor Products • V. Qubits • VI.Quantum Gates and Quantum Circuits • VII. Quantum Parallelism • VIII. Deutsch’s Problem • IX. Bell States, Teleportation, and Dense Coding • X. Summary San Mallo, June 2004**A happy marriage…**• Quantum computing and quantum information theorya product of a happy marriage between two of the greatest scientific achievements of the 20th century • quantum mechanics • stored program computers San Mallo, June 2004**Quantum**• Quantum Latin word meaning “some quantity”. • In physics used with the same meaning as the word discrete in mathematics, i.e., some quantity or variable that can take only sharply defined values as opposed to a continuously varying quantity. • The concepts continuum and continuous are known from geometry and calculus. San Mallo, June 2004**Quantum mechanics**• Quantum mechanics is a mathematical model of the physical world. • Quantum properties such as • uncertainty, • interference, and • entanglement do not have a correspondent in classical physics. San Mallo, June 2004**Heissenberg’s uncertainty principle**• The position and the momentum of a quantum particle cannot be determined with arbitrary precision. h=1.054 10-34 J second reduced Planck’s constant San Mallo, June 2004**Max Born’s Nobel prize lecture, Dec. 11, 1954**• “... Quantum Mechanics shows that not only the determinism of classical physics must be abandoned, but also the naive concept of reality which looked upon atomic particles as if they were very small grains of sand. At every instant a grain of sand has a definite position and velocity. This is not the case with an electron. If the position is determined with increasing accuracy, the possibility of ascertaining its velocity becomes less and vice versa”. San Mallo, June 2004**Quantum theory and computing and communication**• Quantum theory • Does not play only a supporting role by prescribing the limitations of physical systems used for computing and communication • It provides a revolutionary rather than an evolutionary approach to computing and communication. San Mallo, June 2004**Milestones in quantum physics**• 1900 - Max Plank black body radiation theory; the foundation of quantum theory. • 1905 - Albert Einstein the theory of the photoelectric effect. • 1911 - Ernest Rutherford the planetary model of the atom. • 1913 - Niels Bohr the quantum model of the hydrogen atom. • 1923 - Louis de Broglie relates the momentum of a particle with the wavelength. • 1925 - Werner Heisenberg the matrix quantum mechanics. San Mallo, June 2004**Milestones in quantum physics (cont’d)**• 1926 - Erwin Schrödinger Schrödinger’sequation for the dynamics of the wave function. • 1926 - Erwin Schördinger and Paul Dirac show the equivalence of Heisenberg's matrix formulation and Dirac's algebraic one with Schrödinger's wave function. • 1926 - Paul Dirac and, independently, Max Born, Werner Heisenberg, and Pascual Jordan obtain a complete formulation of quantum dynamics. • 1926 - John von Newmannintroduces Hilbert spaces to quantum mechanics. • 1927 - Werner Heisenberg the uncertainty principle. San Mallo, June 2004**Milestones in computing and information theory**• 1936 - Alan Turingthe Universal Turing Machine, UTM. • 1936 - Alonzo Church ``every function which can be regarded as computable can be computed by an universal computing machine''. • 1945 – J. Presper Eckert and John Macauly ENIAC, the world's first general purpose computer. • 1946 - John von Neumann the von Neumann architecture. • 1948 - Claude Shannon ``A Mathematical Theory of Communication’’. • 1953 - UNIVAC I the first commercial computer,. San Mallo, June 2004**Milestones in quantum computing –the pioneers**• 1961 - Rolf Landauercomputation is physical; studies heat generation. • 1973 - Charles Bennet logical reversibility of computations. • 1981 - Richard Feynman physical systems including quantum systems can be simulated exactly with quantum computers. • 1982 - Peter Beniof develops quantum mechanical models of Turing machines. San Mallo, June 2004**Milestones in quantum computing**• 1984 - Charles Bennet and Gilles Brassard quantum cryptography. • 1985 - David Deutschreinterprets the Church-Turing conjecture. • 1993 - Bennet, Brassard, Crepeau, Josza, Peres, Wooters quantum teleportation. • 1994 - Peter Shora clever algorithm for factoring large numbers. San Mallo, June 2004**Can we observe quantum effects with simple experimental**setups? • Experiments with light beams. • Beam splitters and cascaded beam-splitters. • Photon polarization and an experiment with polarization filters. • Multiple measurements indifferent bases • A photon coincidence experiment • What do we notice • Non-deterministic behavior • Strange effects that cannot be explained using classical models of physics. San Mallo, June 2004**Can we construct a mathematical model to explain the results**of the experiments? • The model of photon behavior: • non-deterministic • captures superposition effects • captures the effect of the measurement process • superposition probability rule San Mallo, June 2004**Light**• Light a form of electromagnetic radiation. • The electric and magnetic field • oscillate in a plane perpendicular to the direction of propagation and • are perpendicular to each other. • The dual, wave and corpuscular, nature of light: • Diffraction phenomena can only be explained assuming a wave-like behavior • The photoelectric effect corpuscular/granular nature of light. The light consists of quantum particles called photons. San Mallo, June 2004**Beam splitters: deterministic versus probabilistic photon**behavior • Beam splitter a half silvered mirror. Part of an incident beam of light is transmitted and part is reflected. • What happens when we send a single photon to a beam splitter? San Mallo, June 2004**A single beam splitter**• Either detector D1 or detector D2 will record the arrival of a photon • How do we explain this behavior? probabilistic/genetic model? • We repeat the experiment involving a single photon over and over again D1 and D2 record about the same number of events. • Does a photon carry a gene? • one with a “transmit” gene D2 • one with a “reflect'' gene D1? San Mallo, June 2004**Cascaded beam splitters**• The experiment we send a single photon, repeat the experiment many times, and count the number of events registered by each detector. • If the gene theory is true the photon is either reflected by the first beam splitter or transmitted by all of them. Only the first and last detectors in the chain are expected to register events (each one of them should register an equal number of events). • The experiment shows all detectors have an equal chance to register an event. San Mallo, June 2004**The polarization of light**• Is given by the electric field vector • Linearly polarized light the electric filed oscillates along any straight line in a plane perpendicular to the direction of propagation: • vertical/horizontal polarization • diagonal: 45/135 deg polarization • Circularly polarized light the electric field vector moves along a circle in a plane perpendicular to the direction of propagation: • Right-hand polarization counterclockwise rotation • Left-hand polarization clockwise rotation • Elliptically polarized light the electric field vector moves along an ellipse in a plane perpendicular to the direction of propagation. San Mallo, June 2004**An experiment with polarization analyzers/filters**• A polarization analyzer or polarized filter A partially transparent material that transmits light of a particular polarization. • We perform an experiment involving: • A source S of linearly polarized light of intensity I. • A screen E where we measure the intensity of incoming beam of light. • There types of polarization filters: • A vertical polarization • B horizontal polarization • C a 45 degree polarization • Each photon has a random orientation of the polarization vector. San Mallo, June 2004**The puzzling observations**• Without any filter the measured intensity is I. • When we introduce a vertically polarized filter between the source and the screen the measured intensity is I/2. • When we introduce a horizontally polarized filter between the vertically polarized filter and the screen the measured intensity is 0. • When we introduce a 45 deg polarized filter between the vertically polarized filter and the horizontally polarized filter the measured intensity isI/8. San Mallo, June 2004**A mathematical model to describe the state of a quantum**system are complex numbers San Mallo, June 2004**Superposition and uncertainty**• In this model a state is a superposition of two basis states, “0” and “1” or and (Dirac’s notation) • This state is unknown before we make a measurement. • After we perform a measurement the system is no longer in an uncertain state but it is in one of the two basis states: the probability of observing the outcome 1 the probability of observing the outcome 0 San Mallo, June 2004**The measurement of superposition states**• The polarization of a photon is described by a unit vector on a two-dimensional vector space with basis • | 0 > and • | 1>. • Measuring the polarization is equivalent to projecting the random vector onto one of the two basis vectors. • Thus after a measurement each photon is forced to choose between one of the two basis states. San Mallo, June 2004**Does the model explain the results?**• When filter A with vertical polarization is inserted between the source S and the screen E all photons are forced to choose between vertical and horizontal polarization. About half of them reach E because they choose vertical polarization the measuredintensity is about I/2. • When filter B with horizontal polarization is inserted between A and E then none of the incoming photons (all have horizontal polarization) reach E the measured intensity is 0. San Mallo, June 2004**A puzzling question**• Why when filter C with a 45 deg. polarization is inserted between A and B, the measured intensity is intensity is about I / 8? San Mallo, June 2004**Multiple measurements in different bases**San Mallo, June 2004**Measurements in multiple bases**San Mallo, June 2004**The answer to the puzzling question in the polarization**filters experiment • When we insert C, the 45 deg filter we force a measurement in a new base (45/135 degree). • About half of the I/2 photons with vertical polarization • (emerging from filter A) pass through filter B and exit with a 45 degree polarization. • Then these I/4 photons are measured again in new basis (Vertical/Horizontal) and about half of them choose a horizontal polarization. They pass through filter B. • Thus, the intensity of the measured light is now I/8. San Mallo, June 2004**The superposition probability rule**• If an event may occur in two or more indistinguishable ways • For classical systems Bayes rules: • In quantum mechanics the probability amplitude of the event is the sum of the probability amplitudes of each case considered separately (sometimes known as Feynman’s rule). • An experiment illustrating the superposition probability rule. San Mallo, June 2004**The experiment**• We observe experimentally that • a photon emitted by S1 is always detected by D1 and never by D2 and • one emitted by S2 is always detected by D2 and never by D1. • A photon emitted by one of the sources S1 or S2 may take one of four different paths shown on the next slide, depending whether • it is transmitted, or • reflected by each of the two beam splitters. San Mallo, June 2004**A photon coincidence experiment**• One source emits two photons simultaneously into two separate beams. • Each beam is directed by a reflecting mirror to one of two beam splitters. • There are two detectors. We never observe a coincidence.. San Mallo, June 2004**Contents**• I. Computing and the Laws of Physics • II. The Strange World of Quantum Mechanics • III. Quantum Computing and Communication • IV Hilbert Spaces and Tensor Products • V. Qubits • VI.Quantum Gates and Quantum Circuits • VII. Quantum Parallelism • VIII. Deutsch’s Problem • IX. Bell States, Teleportation, and Dense Coding • X. Summary San Mallo, June 2004**The new frontier in computing and communication**• Applications of quantum computing and quantum information theory: • Exact simulation of systems with a very large state space. • Quantum algorithms based upon quantum parallelism. • Quantum key distribution. San Mallo, June 2004