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### Quantum Computing and Quantum Parallelism

ContentsContentsContentsContentsContentsContentsContents

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

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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

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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

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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.

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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.

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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.

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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.

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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.

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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

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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

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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.

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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.

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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

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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”.

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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.

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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.

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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.

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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,.

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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.

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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.

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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.

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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

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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.

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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?

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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?

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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.

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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.

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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.

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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.

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A mathematical model to describe the state of a quantum system

are complex numbers

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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

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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.

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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.

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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?

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Multiple measurements in different bases

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Measurements in multiple bases

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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.

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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.

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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.

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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..

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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

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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.

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Frontier(s)…from Webster’s unabridged dictionary.

- The part of a settled or civilized country nearest to an unsettled or uncivilized region.
- Any new or incompletely investigated field of learning or thought.

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What is a Quantum computer?

- A device that harnesses quantum physical phenomena such as entanglement and superposition.
- The laws of quantum mechanics differ radically from the laws of classical physics.
- The unit of information, the qubit can exist as a 0, or 1, or, simultaneously, as both 0 and 1.

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Does quantum computing represent the frontiers of computing?

- Is it for real? Can we actually build quantum computers? - Very likely, but it will take some time….
- If so, what would a quantum computer allow us to do that is either unfeasible or impractical with today’s most advanced systems? – Exact simulation of physical systems, among other things.
- Once we have quantum computers do we need new algorithms? – Yes, we need quantum algorithms.
- Is it so different from our current thinking that it requires a substantial change in the way we educate our students? – Yes, it does.

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Quantum computers: now and then

- All we have at this time is a 7 qubit quantum computer able to compute the prime factors of a small integer, 15.
- To break a code with a key size of 1024 bits requires more than 3000 qubits and 108 quantum gates.

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Approximate computer simulation of physical systems

- Eniac and the Manhattan project. The first programs to run, simulation of physical processes.
- Computer simulation – new approach to scientific discovery, complementing the two well established methods of science: experiment and theory.
- Approximate simulation – based upon a model that abstracts some properties of interest of a physical system.

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Exact simulation of physical systems

- How far do we want to go at the microscopic level? Molecular, atomic, quantum? - All of the above.
- What about cosmic level? - Yes, of course.
- Is it important? - - Yes (Feynman, 1981) .
- Who will benefit? –
- Natural sciences physics, chemistry, biology, astrophysics, cosmology,….
- Applications nanotechnology, smart materials, drug design,…

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Large problem state space

- From black hole thermodynamics – a system enclosed by a surface with area A has N(A) observable states with

c = 3 x1010 cm/sec

h = 1.054 x 10-34 Joules/second

G = 6.672 x 10-8 cm3 g-1 sec-2

For an object with a radius of 1 Km N(A) = e80

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Quantum Parallelism

- In quantum systems the amount of parallelism increases exponentially with the size of the system, thus with the number of qubits. For example, a 21 qubit quantum computer is twice as powerfulas as a 20 qubit
- An exponential increase in the power of a quantum computer requires 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.

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Quantum key distribution

- To ensure confidentiality, data is often encrypted.
- We need for reliable methods for the distribution of the encryption keys.
- The problem physical difficulty to detect the presence of an intruder when communicating through a classical communication channel.

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Quantum key distribution setup

- Alice and Bob connected via two communication channels:
- a classical one, and
- a quantum one.
- Alice sends photons via the quantum channel to Bob. A photon may be prepared with
- vertical/horizontal (VH) or
- diagonal polarization (DG).
- Alice and Bob also exchange messages over the classical channel.
- Eve eavesdrops on both communication channels.

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Information encoding

- A photon may be used to transmit information on a quantum communication channel.
- The classical binary information may be encoded as follows:
- We can use a photon with vertical/horizontal (VH) polarization
- 1 a photon with vertical polarization
- 0 a photon with a horizontal polarization.
- Alternatively we may use a photon with diagonal (DG) polarization
- 1 a photon with 45 deg. polarization, and
- 0 a photon with a 135 deg. polarization.

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The measurements of photons sent over the quantum communication channel

- Bob uses a calcite crystal to separate photons with different polarization. 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.
- Whenever Eve eavesdrops on the quantum communication channel she performs a measurement thus, she alters the state of the photons.

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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.

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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.

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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’.

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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.

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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.

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- 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

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Hilbert space

- A vector space over the field of complex numbers with an inner product (a norm).
- In mathematics Hilbert spaces are infinite-dimensional.
- In quantum mechanics finite-dimensional Hilbert spaces.
- The basis vectors in a
- two-dimensional Hilbert space | 0> and |1>.
- four- dimensional Hilbert space | 00>, |01>, |10>, and |11>.
- eight-dimensional Hilbert space | 000>, |001>, |010>, |011>, |100>, |101>, |110>, |101>, and |111>.

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The tensor product of two vectors in a two-dimensional Hilbert space

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Two vectors in a four-dimensional Hilbert space

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The outer product of two vectors in a four-dimensional Hilbert space

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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.

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- 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

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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)

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State description

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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
- in state with probability

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Qubit measurement

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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

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Two qubits

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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.

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Measuring two qubits (cont’d)

- What if we observe only the first qubit, what conclusions can we draw?
- We expect 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

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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.

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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.

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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

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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.

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Entanglement (Vërschrankung)

- Discovered by Schrödinger.
- 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.
- Einstein called entanglement “Spooky action at a distance".

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Physical embodiment of a qubit

- The photon with tow independent polarizations, horizonatal and vertical.
- The electron with two independent spin values, +1/2 and -1/2.
- Quantum dots
- Small devices that contain a tiny droplet of free electrons.
- They are fabricated in semiconductor materials and have typical dimensions between nanometres to a few microns.
- The size and shape of these structures and therefore the number of electrons they contain, can be precisely controlled; a quantum dot can have anything from a single electron to a collection of several thousands.
- The binary information is encoded as the presence/absence of electrons.

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Physical embodiment of a qubit (cont’d)

- A two-level atom in an optical cavity.
- Two internal states of an ion in a trap.
- Others

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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.
- The spin quantum number s may assume integer and half-integer values.
- The spin is quantized 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.

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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

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The Stern-Gerlach experiment with hydrogen atoms

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The spin of the electron

- The electron has spin s = 1 /2
- The spin projection can assume the values
- + ½ spin up, and
- -1/2 spin down.

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Communication with entangles particles

- Even when separated two entangled particles continue to interact with one another.
- Basic idea. Consider three particles
- Two particles (particle 2 and particle 3) in an anti-correlated state (spin up and spin down).
- 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.

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- 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

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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 irretrievable loss of information.

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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

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One qubit gates

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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.

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Identity transformation, Pauli matrices, Hadamard

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Two-qubit gates

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Tensor products and outer products

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The two input qubits of a two qubit gates

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A two qubit gate:CNOT

- Two inputs
- Control
- Target
- The control qubit transferred to the output as is.
- The target qubit
- Unaltered if the control qubit is 0
- Flipped (0 1 and 1 0) if the control qubit is 1.

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The transfer matrix of the CNOT gate

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The transformation performed by CNOT gate

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The output of CNOT gate

- CNOT preserves the control qubitthe first and the second component of the input vector are replicated in the output vector.
- CNOT flips the target qubitthe third and fourth component of the input vector become the fourth and respectively the third component of the output vector.
- CNOT gate is reversible. The control qubit is replicated at the output and knowing it we can reconstruct the target input qubit.

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Three qubit gates

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The 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

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The transformation performed by the Fredkin gate

- |000> |000>
- |001> |001>
- |010> |010>
- |011> |101> flip the two target qubits when the control qubit is 1
- |100> |100>
- |101> |011> flip the two target qubits when the control qubit is 1
- |110> |111>
- |111> |110>

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The transfer matrix of the Fredkin gate

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The transfer matrix of the Fredkin gate

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The 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.

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The transformation performed by the Toffoli gate

- |000> |000>
- |001> |001>
- |010> |010>
- |011> |001>
- |100> |100>
- |101> |101>
- |110> |111> when both control qubits are 1
- |111> |110> then the target qubit is flipped

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The transfer matrix of the Toffoli gate

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The transfer matrix of a Toffoli gate

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

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Controlled H gate

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Generic one qubit controlled gate

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- 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

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Problem formulation

- Consider a circuit to compute f(x).
- x is a binary n-tuple there are 2n possible values of x.
- a gate array is used to compute f(x) in one time step
- We wish to compute the values of f(x) for the 2n values of x using
- a classical circuit
- a quantum circuit
- Classical system
- Sequential computation using a single gate array we need 2n time steps
- Parallel computation using 2n gate arrays we need a single time step

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Problem formulation (cont’d)

- Consider first the case n=1, x takes only two values
- x=0
- x=1
- We’ll show that the output of a quantum circuit is a superposition of f(0) and f(1)
- Both values, f(0) and f(1) are available
- But…..once we measure the output of the quantum circuit we can obtain only one of them

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A quantum circuit to compute f(x)

- 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

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A quantum circuit to compute f(x) (cont’d)

- If the second input is zero then the transformation done by the circuit is

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A quantum circuit (cont’d)

- We 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).

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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.
- We start with n qubits, each in state |0> and we apply a Walsh-Hadamard transformation.

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- 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

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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

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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.

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- 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

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Quantum circuit to create Bell states

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- 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

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Final remarks

- 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.
- 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.

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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.

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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.

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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.

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Summary

- Quantum computing and quantum information theory is truly an exciting field.
- It is too important to be left to the physicists alone….

San Mallo, June 2004

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