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**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

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**ISCIS Antalya November 5, 2003**

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 MCB , DBI ,ACI , and EIA ISCIS Antalya November 5, 2003

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**ISCIS Antalya November 5, 2003**

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

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**ISCIS Antalya November 5, 2003**

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

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**ISCIS Antalya November 5, 2003**

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 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

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**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

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**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

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**ISCIS Antalya November 5, 2003**

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**ISCIS Antalya November 5, 2003**

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

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**ISCIS Antalya November 5, 2003**

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

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**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

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**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

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**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

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**Milestones in quantum physics**

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

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**Milestones in quantum physics (cont’d)**

Erwin Schrodinger and Paul Dirac show the equivalence of Heisenberg's matrix formulation and Dirac's algebraic one with Schrodinger's wave function. Paul Dirac and, independently, Max Born, Werner Heisenberg, and Pasqual Jordan obtain a complete formulation of quantum dynamics. John von Newmann introduces Hilbert spaces to quantum mechanics. Werner Heisenberg formulates the uncertainty principle. ISCIS Antalya November 5, 2003

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**Milestones in computing and information theory**

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

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**Milestones in quantum computing**

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

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**Deterministic versus probabilistic photon behavior**

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**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

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**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

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State description ISCIS Antalya November 5, 2003

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**A mathematical model to describe the state of a quantum system**

are complex numbers ISCIS Antalya November 5, 2003

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**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

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**Multiple measurements**

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**Measurements in multiple bases**

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**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

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**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

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**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

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**A photon coincidence experiment**

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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

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**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

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**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

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**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

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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

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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

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**ISCIS Antalya November 5, 2003**

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

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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

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**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

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**ISCIS Antalya November 5, 2003**

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

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**ISCIS Antalya November 5, 2003**

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

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**ISCIS Antalya November 5, 2003**

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

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Ortonormal basis ISCIS Antalya November 5, 2003

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One qubit are complex numbers ISCIS Antalya November 5, 2003

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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 ISCIS Antalya November 5, 2003

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Other states of a qubit ISCIS Antalya November 5, 2003

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**A different basis for one qubit**

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Qubit measurement ISCIS Antalya November 5, 2003

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**The Boch sphere representation of one qubit**

A qubit in a superposition state is represented as a vector connecting the center of the Bloch sphere with a point on its periphery. The two probability amplitudes can be expressed using Euler angles. ISCIS Antalya November 5, 2003

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Two qubits Represented as vectors in a 2-dimensional Hilbert space with four basis vectors When we measure a pair of qubits we decide that the system it is in one of four states with probabilities ISCIS Antalya November 5, 2003

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Two qubits ISCIS Antalya November 5, 2003

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Measuring two qubits Before a measurement the state of the system consisting of two qubits is uncertain (it is given by the previous equation and the corresponding probabilities). After the measurement the state is certain, it is 00, 01, 10, or 11 like in the case of a classical two bit system. ISCIS Antalya November 5, 2003

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**Measuring two qubits (cont’d)**

What if we observe only the first qubit, what conclusions can we draw? We expect that the system to be left in an uncertain sate, because we did not measure the second qubit that can still be in a continuum of states. The first qubit can be 0 with probability 1 with probability ISCIS Antalya November 5, 2003

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**Measuring two qubits (cont’d)**

Call the post-measurement state when we measure the first qubit and find it to be 0. Call the post-measurement state when we measure the first qubit and find it to be 1. ISCIS Antalya November 5, 2003

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**Measuring two qubits (cont’d)**

Call the post-measurement state when we measure the second qubit and find it to be 0. Call the post-measurement state when we measure the second qubit and find it to be 1. ISCIS Antalya November 5, 2003

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**Bell states - a special state of a pair of qubits**

If and When we measure the first qubit we get the post measurement state When we measure the second qubit we get the post mesutrement state ISCIS Antalya November 5, 2003

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**This is an amazing result!**

The two measurements are correlated, once we measure the first qubit we get exactly the same result as when we measure the second one. The two qubits need not be physically constrained to be at the same location and yet, because of the strong coupling between them, measurements performed on the second one allow us to determine the state of the first. ISCIS Antalya November 5, 2003

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Entanglement Entanglement is an elegant, almost exact translation of the German term Verschrankung used by Schrodinger who was the first to recognize this quantum effect. An entangled pair is a single quantum system in a superposition of equally possible states. The entangled state contains no information about the individual particles, only that they are in opposite states. The important property of an entangled pair is that the measurement of one particle influences the state of the other particle. Einstein called that “Spooky action at a distance". ISCIS Antalya November 5, 2003

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The spin In quantum mechanics the intrinsic angular moment, the spin, is quantized and the values it may take are multiples of the rationalized Planck constant. The spin of an atom or of a particle is characterized by the spin quantum number s , which may assume integer and half-integer values. For a given value of s the projection of the spin on any axis may assume 2s + 1 values ranging from - s to $ + s by unit steps, in other words the spin is quantized. ISCIS Antalya November 5, 2003

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More about the spin There are two classes of quantum particles fermions - spin one-half particles such as the electrons. The spin quantum numbers of fermions can be s=+1/2 and s=-1/2 bosons - spin one particles. The spin quantum numbers of bosons can be s=+1, s=0, and s=-1 ISCIS Antalya November 5, 2003

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**The Stern-Gerlach experiment with hydrogen atoms**

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**Physical embodiment of a qubit**

The electron with tow independent spin values, +1/2 and -1/2 The photon, with tow independent polarizations, horizonatla and vertical ISCIS Antalya November 5, 2003

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**The spin of the electron**

The electron has spin s = 1 /2 and the spin projection can assume the values $ + ½ referred to as spin up, and -1/2 referred to as spin down. ISCIS Antalya November 5, 2003

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Light and photons Light is a form of electromagnetic radiation; the wavelength of the radiation in the visible spectrum varies from red to violet. Light can be filtered by selectively absorbing some color ranges and passing through others. A polarization filter is a partially transparent material that transmits light of a particular polarization. ISCIS Antalya November 5, 2003

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Photons Photons differ from the spin 1/2 electrons in two ways: (1) they are massless and (2) have spin $1$. A photon is characterized by its vector momentum (the vector momentum determines the frequency) and polarization. In the classical theory light is described as having an electric field which oscillates either vertically, the light is x-polarized, or horizontally, the light is y-polarized in a plane perpendicular to the direction of propagation, the z-axis. The two basis vectors are |h> and |v> ISCIS Antalya November 5, 2003

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**Vertically and horizontally polarized photons**

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Polarization filters A polarization filter is a partially transparent material that transmits light of a particular polarization. If we set the axis of a polarization filter to let pass y-polarized light, then all photons in the state |v> will be absorbed in the filter and only the photons in state |h> will pass through. If the axis of the polarization filter is set to let pass x-polarized light, then all photons in state |h> will be absorbed and only photons in state | v > will pass through. When the polarization filter is set at angle with respect to the coordinate system of the incoming beam of light the emerging photons are in a superposition state ISCIS Antalya November 5, 2003

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**The effect of a polarization filter**

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**Communication with entangles particles**

Even when separated two entangled particles continue to interact with one another. Particle 2 and particle 3 in an anti-correlated state (spin up and spin down). Then if we measure particle 1 and particle 2 and set them in an anti-correlated state, then particle 1 ends up in the same state particle 3 was initially. ISCIS Antalya November 5, 2003

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Classical gates Implement Boolean functions. Are not reversible (invertible). We cannot recover the input knowing the output. This means that there is an iretriviable loss of information ISCIS Antalya November 5, 2003

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One qubit gates Transform an input qubit into an output qubit Characterized by a 2 x 2 matrix with complex coefficients ISCIS Antalya November 5, 2003

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One qubit gates I identity gate; leaves a qubit unchanged. X or NOT gate transposes the components of an input qubit. Y gate. Z gate flips the sign of a qubit. H the Hadamard gate. ISCIS Antalya November 5, 2003

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One qubit gates ISCIS Antalya November 5, 2003

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**Identity transformation, Pauli matrices, Hadamard**

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**Tensor products and ``outer’’ products**

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CNOT a two qubit gate Two inputs Control Target The control qubit is transferred to the output as is. The target qubit Unaltered if the control qubit is 0 Flipped if the control qubit is 1. ISCIS Antalya November 5, 2003

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**The two input qubits of a two qubit gates**

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Two qubit gates ISCIS Antalya November 5, 2003

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Two qubit gates ISCIS Antalya November 5, 2003

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**Final comments on the CNOT gate**

CNOT preserves the control qubit (the first and the second component of the input vector are replicated in the output vector) and flips the target qubit (the third and fourth component of the input vector become the fourth and respectively the third component of the output vector). The CNOT gate is reversible. The control qubit is replicated at the output and knowing it we can reconstruct the target input qubit. ISCIS Antalya November 5, 2003

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Fredkin gate Three input and three output qubits One control Two target When the control qubit is 0 the target qubits are replicated to the output 1 the target qubits are swapped ISCIS Antalya November 5, 2003

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Toffoli gate Three input and three output qubits Two control One target When both control qubit are 1 the target qubit is flipped otherwise the target qubit is not changed. ISCIS Antalya November 5, 2003

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**Toffoli gate is universal. It may emulate an AND and a NOT gate**

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Controlled H gate ISCIS Antalya November 5, 2003

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**Generic one qubit controlled gate**

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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

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A quantum circuit Given a function f(x) we can construct a reversible quantum circuit consisting of Fredking gates only, capable of transforming two qubits as follows The function f(x) is hardwired in the circuit ISCIS Antalya November 5, 2003

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**A quantum circuit (cont’d)**

If the second input is zero then the transformation done by the circuit is ISCIS Antalya November 5, 2003

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**A quantum circuit (cont’d)**

Now apply the first qubit through a Hadamad gate. The resulting sate of the circuit is The output state contains information about f(0) and f(1). ISCIS Antalya November 5, 2003

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Quantum parallelism The output of the quantum circuit contains information about both f(0) and f(1). This property of quantum circuits is called quantum parallelism. Quantum parallelism allows us to construct the entire truth table of a quantum gate array having 2n entries at once. In a classical system we can compute the truth table in one time step with 2n gate arrays running in parallel, or we need 2n time steps with a single gate array. We start with n qubits, each in state |0> and we apply a Welsh-Hadamard transformation. ISCIS Antalya November 5, 2003

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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

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Deutsch’s problem Consider a black box characterized by a transfer function that maps a single input bit x into an output, f(x). It takes the same amount of time, T, to carry out each of the four possible mappings performed by the transfer function f(x) of the black box: f(0) = 0 f(0) = 1 f(1) = 0 f(1) = 1 The problem posed is to distinguish if ISCIS Antalya November 5, 2003

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**A quantum circuit to solve Deutsch’s problem**

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Evrika!! By measuring the first output qubit qubit we are able to determine performing a single evaluation. ISCIS Antalya November 5, 2003

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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

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**Quantum circuit to create Bell states**

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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

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Final remarks A tremendous progress has been made in the area of quantum computing and quantum information theory during the past decade. Thousands of research papers, a few solid reference books, and many popular-science books have been published in recent years in this area. The growing interest in quantum computing and quantum information theory is motivated by the incredible impact this discipline could have on how we store, process, and transmit data and knowledge in this information age. ISCIS Antalya November 5, 2003

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**Final remarks (cont’d)**

Computer and communication systems using quantum effects have remarkable properties. Quantum computers enable efficient simulation of the most complex physical systems we can envision. Quantum algorithms allow efficient factoring of large integers with applications to cryptography. Quantum search algorithms speedup considerably the process of identifying patterns in apparently random data. We can guarantee the security of our quantum communication systems because eavesdropping on a quantum communication channel can always be detected. ISCIS Antalya November 5, 2003

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**Final remarks (cont’d)**

It is true that we are years, possibly decades away from actually building a quantum computer requiring little if any power at all, filling up the space of a grain of sand, and computing at speeds that are unattainable today even by covering tens of acres of floor space with clusters made from tens of thousands of the fastest processors built with current state of the art solid state technology. ISCIS Antalya November 5, 2003

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**Final remarks (cont’d)**

All we have at the time of this writing is a seven qubit quantum computer able to compute the prime factors of a small integer, 15. Building a quantum computer faces tremendous technological and theoretical challenges. At the same time, we witness a faster rate of progress in quantum information theory where applications of quantum cryptography seem ready for commercialization. Recently, a successful quantum key distribution experiment over a distance of some 100 km has been announced. ISCIS Antalya November 5, 2003

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**ISCIS Antalya November 5, 2003**

Summary Quantum computing and quantum information theory is truly an exciting field. It is too important to be left to the physicists alone…. ISCIS Antalya November 5, 2003

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