1 / 126

Introduction to Quantum Computing and Quantum Information Theory

Introduction to Quantum Computing and Quantum Information Theory. Dan C. Marinescu and Gabriela M. Marinescu Computer Science Department University of Central Florida Orlando, Florida 32816 , USA. Acknowledgments. The material presented is from the book Lectures on Quantum Computing

ike
Télécharger la présentation

Introduction to Quantum Computing and Quantum Information Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Introduction to Quantum Computing andQuantum Information Theory Dan C. Marinescu and Gabriela M. Marinescu Computer Science Department University of Central Florida Orlando, Florida 32816, USA

  2. Acknowledgments • The material presented is from the book Lectures on Quantum Computing by Dan C. Marinescu and Gabriela M. Marinescu Prentice Hall, 2004 • Work supported by National Science Foundation grants MCB9527131, DBI0296107,ACI0296035, and EIA0296179. ISCIS Antalya November 5, 2003

  3. Contents • I. Computing and the Laws of Physics • II. A Happy Marriage; Quantum Mechanics & Computers • III. Qubits and Quantum Gates • IV. Quantum Parallelism • V. Deutsch’s Algorithm • VI. Bell States, Teleportation, and Dense Coding • VII. Summary ISCIS Antalya November 5, 2003

  4. Technological limits • For the past two decades we have enjoyed Gordon Moore’s law. But all good things may come to an end… • We are limited in our ability to increase • the density and • the speed of a computing engine. • 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 uncertainty ISCIS Antalya November 5, 2003

  5. 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 a liner 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. • Even if this limit is reduced say 100-fold we shall see a 10 (ten) times increase in the amount of power needed by devices operating at a speed 103 times larger than the sped of today's devices. ISCIS Antalya November 5, 2003

  6. 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. ISCIS Antalya November 5, 2003

  7. Talking about the heat… • 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. The heat dissipated grows as the cube of the radius of the device. • To prevent the destruction of the engine we have to remove the heat through a surface surrounding the device. Henceforth, our ability to remove heat increases as the square of the radius while the amount of heat increases with the cube of the size of the computing engine. ISCIS Antalya November 5, 2003

  8. ISCIS Antalya November 5, 2003

  9. Contents • I. Computing and the Laws of Physics • II. A Happy Marriage; Quantum Mechanics & Computers • III. Qubits and Quantum Gates • IV. Quantum Parallelism • V. Deutsch’s Algorithm • VI. Bell States, Teleportation, and Dense Coding • VII. Summary ISCIS Antalya November 5, 2003

  10. A happy marriage… • The two greatest discoveries of the 20-th century • quantum mechanics • stored program computers produced quantum computing and quantum information theory ISCIS Antalya November 5, 2003

  11. Quantum; Quantum mechanics • Quantum is a Latin word meaning some quantity. In physics it is 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. For example, on a segment of a line there are infinitely many points, the segment consists of a continuum of points. This means that we can cut the segment in half, and then cut each half in half, and continue the process indefinitely. • Quantum mechanics is a mathematical model of the physical world ISCIS Antalya November 5, 2003

  12. Heissenberg uncertainty principle • Heisenberg uncertainty principle says we cannot determine both the position and the momentum of a quantum particle with arbitrary precision. • In his Nobel prize lecture on December 11, 1954 Max Born says about this fundamental principle of Quantum Mechanics : ``... It 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.'' ISCIS Antalya November 5, 2003

  13. A revolutionary approach to computing and communication • We need to consider a revolutionary rather than an evolutionary approach to computing. • Quantum theory does not play only a supporting role by prescribing the limitations of physical systems used for computing and communication. • Quantum properties such as • uncertainty, • interference, and • entanglement form the foundation of a new brand of theory, the quantum information theory where computational and communication processes rest upon fundamental physics. ISCIS Antalya November 5, 2003

  14. Milestones in quantum physics • 1900 - Max Plank presents the black body radiation theory; the quantum theory is born. • 1905 - Albert Einstein develops the theory of the photoelectric effect. • 1911 - Ernest Rutherford develops the planetary model of the atom. • 1913 - Niels Bohr develops the quantum model of the hydrogen atom. • 1923 - Louis de Broglie relates the momentum of a particle with the wavelength • 1925 - Werner Heisenberg formulates the matrix quantum mechanics. • 1926 - Erwin Schrodinger proposes the equation for the dynamics of the wave function. ISCIS Antalya November 5, 2003

  15. Milestones in quantum physics (cont’d) • 1926 - Erwin Schrodinger and Paul Dirac show the equivalence of Heisenberg's matrix formulation and Dirac's algebraic one with Schrodinger's wave function. • 1926 - Paul Dirac and, independently, Max Born, Werner Heisenberg, and Pasqual Jordan obtain a complete formulation of quantum dynamics. • 1926 - John von Newmann introduces Hilbert spaces to quantum mechanics. • 1927 - Werner Heisenberg formulates the uncertainty principle. ISCIS Antalya November 5, 2003

  16. Milestones in computing and information theory • 1936 - Alan Turing dreams up the Universal Turing Machine, UTM. • 1936 - Alonzo Church publishes a paper asserting that ``every function which can be regarded as computable can be computed by an universal computing machine''. • 1945 - ENIAC, the world's first general purpose computer, the brainchild of J. Presper Eckert and John Macauly becomes operational. • 1946 - A report co-authored by John von Neumann outlines the von Neumann architecture. • 1948 - Claude Shannon publishes ``A Mathematical Theory of Communication’’. • 1953 - The first commercial computer, UNIVAC I. ISCIS Antalya November 5, 2003

  17. Milestones in quantum computing • 1961 - Rolf Landauer decrees that computation is physical and studies heat generation. • 1973 - Charles Bennet studies the logical reversibility of computations. • 1981 - Richard Feynman suggests that physical systems including quantum systems can be simulated exactly with quantum computers. • 1982 - Peter Beniof develops quantum mechanical models of Turing machines. • 1984 - Charles Bennet and Gilles Brassard introduce quantum cryptography. • 1985 - David Deutsch reinterprets the Church-Turing conjecture. • 1993 - Bennet, Brassard, Crepeau, Josza, Peres, Wooters discover quantum teleportation. • 1994 - Peter Shor develops a clever algorithm for factoring large numbers. ISCIS Antalya November 5, 2003

  18. Deterministic versus probabilistic photon behavior ISCIS Antalya November 5, 2003

  19. The puzzling nature of light • If we start decreasing the intensity of the incident light we observe the granular nature of light. Imagine that we send a single photon. Then either detector D1 or detector D2 will record the arrival of a photon. • If we repeat the experiment involving a single photon over and over again we observe that each one of the two detectors records a number of events. • Could there be hidden information, which controls the behavior of a photon? Does a photon carry a gene and one with a ``transmit'' gene continues and reaches detector D2 and another with a ``reflect'' gene ends up at D1? ISCIS Antalya November 5, 2003

  20. The puzzling nature of light (cont’d) • Consider now a cascade of beam splitters. As before, we send a single photon and repeat the experiment many times and count the number of events registered by each detector. • According to our theory we expect the first beam splitter to decide the fate of an incoming photon; the photon is either reflected by the first beam splitter or transmitted by all of them. Thus, only the first and last detectors in the chain are expected to register an equal number of events. • Amazingly enough, the experiment shows that all the detectors have a chance to register an event. ISCIS Antalya November 5, 2003

  21. State description ISCIS Antalya November 5, 2003

  22. A mathematical model to describe the state of a quantum system are complex numbers ISCIS Antalya November 5, 2003

  23. Superposition and uncertainty • In this model a state is a superposition of two basis states, “0” and “1” • 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. • is the probability of observing the outcome “1” • is the probability of observing the outcome “0” ISCIS Antalya November 5, 2003

  24. Multiple measurements ISCIS Antalya November 5, 2003

  25. Measurements in multiple bases ISCIS Antalya November 5, 2003

  26. Measurements of superposition states • The polarization of a photon is described by a unit vector on a two-dimensional space with basis | 0 > and | 1>. • Measuring the polarization is equivalent to projecting the random vector onto one of the two basis vectors. • Source S sends randomly polarized light to the screen; the measured intensity is I. • The filter A with vertical polarization is inserted between the source and the screen an the intensity of the light measured at E is about I/2. • Filter B with horizontal polarization is inserted between A and E. The intensity of the light measured at E is now 0. • Filter C with a 45 deg. polarization is inserted between A and B. The intensity of the light measured at E is about 1 / 8. ISCIS Antalya November 5, 2003

  27. ISCIS Antalya November 5, 2003

  28. The superposition probability rule • If an event may occur in two or more indistinguishable ways then the probability amplitude of the event is the sum of the probability amplitudes of each case considered separately (sometimes known as Feynmann rule). ISCIS Antalya November 5, 2003

  29. ISCIS Antalya November 5, 2003

  30. The experiment illustrating the superposition probability rule • In certain conditions, 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. ISCIS Antalya November 5, 2003

  31. ISCIS Antalya November 5, 2003

  32. A photon coincidence experiment ISCIS Antalya November 5, 2003

  33. A glimpse into the world of quantum computing and quantum information theory • Quantum key distribution • Exact simulation of systems with a very large state space • Quantum parallelism ISCIS Antalya November 5, 2003

  34. Quantum key distribution • To ensure confidentiality, data is often encrypted. The most reliable encryption techniques are based upon one time pads whereby the encryption key is used for one session only and then discarded. • Thus, there exists the need for reliable and effective methods for the distribution of the encryption keys. The problem rests on the physical difficulty to detect the presence of an intruder when communicating through a classical communication channel. • To date, secure and reliable methods for cryptographic key distribution have largely eluded the cryptographic community in spite of considerable research effort and ingeniousness. ISCIS Antalya November 5, 2003

  35. Quantum key distribution setup • Alice and Bob are connected via two communication channels, a quantum and a classical one. Eve eavesdrops on both. • The photons prepared by Alice may have vertical/horizontal (VH) or diagonal polarization (DG). • The photons with vertical/horizontal (VH) polarization may be used to transmit binary information as follows: a photon with vertical polarization may transmit a 1 while one with a horizontal polarization may transmit a 0. • Similarly, those with diagonal (DG) polarization may transmit binary information, 1 encoded as a photon with 45 deg. polarization, and 0 encoded as a photon with a 135 deg. polarization. • Bob uses a calcite crystal to separate photons with different polarization. Shown is the case when the crystal is set up to separate vertically polarized photons from the horizontally polarized ones. To perform a measurement in the DG basis the crystal is oriented accordingly. ISCIS Antalya November 5, 2003

  36. ISCIS Antalya November 5, 2003

  37. The quantum key distribution algorithm of Bennett and Brassard (BB84) • Alice selects n, the approximate length of the encryption key. Alice generates two random strings a and b, each of length (4+ )n. By choosing sufficiently large Alice and Bob can ensure that the number of bits kept is close to 2n with a very high probability. • A subset of length n of the bits in string a will be used as the encryption key and the bits in string $b$ will be used by Alice to select the basis (VH) or (DG) for each photon sent to Bob. ISCIS Antalya November 5, 2003

  38. BB84 (cont’d) • Alice encodes the binary information in string a based upon the corresponding values of the bits in string b. • For example, if the i-th bit of string b is • 1 then Alice selects Vertical-Horizontal (VH) polarization. If VH is selected, then • a 1 in the i-th position of string a is sent as a photon with vertical polarization (V), and • a $0$ as a photon with horizontal (H) polarization; • 0 then Alice selects Diagonal (DG) polarization. If DG is selected, then • a 1 in the i-th position of string a is sent as a photon with a 45 deg. polarization, and • a $0$ as a photon with 135 deg. polarization. • Both Alice and Bob use the same encoding convention for each of the bases. ISCIS Antalya November 5, 2003

  39. BB84 (cont’d) • In turn, Bob picks up randomly (4+ )n bits to form a string b’. He uses one of the two basis for the measurement of each incoming photon in string a based upon the corresponding value of the bit in string b’. • For example, a 1 in the i-th position of b’ implies that the i-th photon is measured in the DG basis, while a 0 requires that the photon is measured in the VH basis. • As a result of this measurement Bob constructs the string a’. ISCIS Antalya November 5, 2003

  40. BB84 (cont’d) • Bob uses the classical communication channel to request the string b and Alice responds on the same channel with b. Then Bob sends Alice string b’ on the classical channel. • Alice and Bob keep only those bits in the set {a, a’} for which the corresponding bits in the set {b, b’} are equal. Let us assume that Alice and Bob keep only 2n bits. ISCIS Antalya November 5, 2003

  41. BB84 (cont’d) • Alice and Bob perform several tests to determine the level of noise and eavesdropping on the channel. The set of 2n bits is split into two sub-sets of n bits each. • One sub-set will be the check bits used to estimate the level of noise and eavesdropping, and • The other consists of the {\it data} bits used for the quantum key. • Alice selects n check bits at random and sends the positions and values of the selected bits over the classical channel to Bob. Then Alice and Bob compare the values of the check bits. If more than say t bits disagree then they abort and re-try the protocol. ISCIS Antalya November 5, 2003

  42. State space dimension of classical and quantum systems • Individual state spaces of n particles combine quantum mechanically through the tensor product. If X and Y are vectors, then • their tensor product X Y is also a vector, but its dimension is: dim(X) x dim(Y) • while the vector product X x Y has dimension dim(X)+dim(Y). • For example, if dim(X)= dim(Y)=10, then the tensor product of the two vectors has dimension 100 while the vector product has dimension 20. ISCIS Antalya November 5, 2003

  43. Quantum computers • In quantum systems the amount of parallelism increases exponentially with the size of the system, thus with the number of qubits. This means that the price to pay for an exponential increase in the power of a quantum computer is a linear increase in the amount of matter and space needed to build the larger quantum computing engine. • A quantum computer will enable us to solve problems with a very large state space. ISCIS Antalya November 5, 2003

  44. Contents • I. Computing and the Laws of Physics • II. A Happy Marriage; Quantum Mechanics & Computers • III. Qubits and Quantum Gates • IV. Quantum Parallelism • V. Deutsch’s Algorithm • VI. Bell States, Teleportation, and Dense Coding • VII. Summary ISCIS Antalya November 5, 2003

  45. One qubit • Mathematical abstraction • Vector in a two dimensional complex vector space (Hilbert space) • Dirac’s notation ket  column vector bra  row vector bra  dual vector (transpose and complex conjugate) ISCIS Antalya November 5, 2003

  46. Ortonormal basis ISCIS Antalya November 5, 2003

  47. One qubit are complex numbers ISCIS Antalya November 5, 2003

  48. A bit versus a qubit • A bit • Can be in two distinct states, 0 and 1 • A measurement does not affect the state • A qubit • can be in state or in state or in any other state that is a linear combination of the basis state • When we measure the qubit we find it • in state with probability • in state with probability ISCIS Antalya November 5, 2003

  49. ISCIS Antalya November 5, 2003

  50. Other states of a qubit ISCIS Antalya November 5, 2003

More Related