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Quantum Mechanic A Review
Quantum Computation
Quantum Algorithms and its
Implementations
Quantum Cryptography
Quantum Information
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Quantum Mechanic Review
Quantum Computation
Quantum Algorithms and
Implementations
Quantum Cryptography
Quantum Information
MEYSAM MADANI- MEYSAM MADANI - AN INTRODUCTION TO
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QUANTUM MECHANICS
 Quantum mechanics is the body of scientific principles that explains the behaviour
of matter and its interactions with energy on the scale of atoms and subatomic
particles.
 Classical physics explains matter and energy at the macroscopic level of the scale
familiar to human experience
 On the other hand, at the end of the 19th century scientists discovered
phenomena in both the large (macro) and the small (micro) worlds that classical
physics could not explain.
 Coming to terms with these limitations led to the development of quantum
mechanics, a major revolution in physics.
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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QUANTUM MECHANIC
 Ludwig Eduard Boltzmann suggested in 1877 that the energy
levels of a physical system, such as a molecule, could be
discrete.
Ludwig Eduard Boltzmann
 In 1900, the German physicist Max Planck reluctantly
introduced the idea that energy is quantized in order to
derive a formula for the observed frequency dependence of
the energy emitted by a black body, called Planck's Law,
1. I(ν,T) is the energy per unit time ,
Max Planck
2.
h is the Planck constant;
3.
c is the speed of light in a vacuum;
4.
k is the Boltzmann constant;
5. ν is the frequency of the electromagnetic radiation
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QUANTUM MECHANIC
 Ștefan Procopiu in 1911—1913, and subsequently Niels Bohr in
1913, calculate the magnetic moment of the electron, which was
later called the "magneton";
 Subsequently made possible for both the magnetic moments of the
proton and the neutron .
Niels Bohr
 In 1905, Einstein explained the photoelectric effect by postulating
that light, or more generally all electromagnetic radiation, can be
divided into a finite number of "energy quanta" that are localized
points in space.
 These energy quanta later came to be called "photons“
 It effectively solved the problem of black body radiation attaining
infinite energy,
Albert Einstein
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COMPUTING CRYPTOGRAPHY AND INFORMATION
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QUANTUM MECHANIC
 In 1924, Louis de Broglie put forward his theory of matter
waves by stating that particles can exhibit wave
characteristics and vice versa.
 In 1925, when the German physicists Werner Heisenberg
and Adam Jonathon Davis developed matrix mechanics
Werner Heisenberg
 Erwin Schrödinger invented wave mechanics and the nonrelativistic Schrödinger equation as an approximation to the
generalized case of de Broglie's theory.
 Schrödinger subsequently showed that the two approaches
were equivalent.
 Heisenberg formulated his uncertainty principle in 1927,
Erwin Schrödinger
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QUANTUM MECHANIC
 Paul Dirac began the process of unifying quantum mechanics with
special relativity by proposing the Dirac equation for the electron.
 The Dirac equation achieves the relativistic description of the
wavefunction of an electron that Schrödinger failed to obtain.
Paul Dirac
 It predicts electron spin and led Dirac to predict the existence of
the positron.
 He also pioneered the use of operator theory, including the
influential bra-ket notation.
 John von Neumann formulated the mathematical basis for quantum
mechanics as the theory of linear operators on Hilbert spaces
John von Neumann
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
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PHOTONS
 Albert suggested that quantization was not just a mathematical
trick: the energy in a beam of light occurs in individual packets,
which are now called photons.
 The energy of a single photon is given by its frequency multiplied by
Planck's constant:
 For centuries, scientists had debated between two possible
theories of light: was it a wave or did it instead comprise a stream
of tiny particles?
 The photoelectric effect puzzling wave theory.
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
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THE PHOTOELECTRIC EFFECT
 In 1887 Heinrich Hertz observed that light can eject electrons from
metal.
 In 1902 Philipp Lenard discovered that the maximum possible energy of
an ejected electron is related to the frequency of the light, not to its
intensity.
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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QUANTUM MECHANICS
 Measurement
 Let 𝜑(𝑥, 𝑡) be the wave function where |𝜑 𝑥, 𝑡 | 2 be the
probability with particle is in position 𝑥 at time 𝑡.
 Where is the particle at time 𝑡 − 𝑒?
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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HEISENBERG - QUANTUM MECHANICS
 𝝈𝒙 𝝈𝒑 ≥ 𝒉
𝟐
Position or
velocity: ?
=2
Position: 5
=?
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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YOUNG DOUBLE SLIT INTERFERENCE
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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YOUNG DOUBLE SLIT INTERFERENCE
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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YOUNG DOUBLE SLIT INTERFERENCE
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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PHOTON
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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PHOTON
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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PHOTON
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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PHOTON
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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PHOTON
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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PHOTON
Output
B.NOT.B=Id
B
Input
B
NOT
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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Quantum Mechanic Review
Quantum Computation
Quantum Algorithms and
Implementations
Quantum Cryptography
Quantum Information
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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MODELS OF COMPUTATION
 Richard Feynman was the first to suggest, in a talk in 1981, that quantum-mechanical
systems might be more powerful than classical computers.
 Feynman asked what kind of computer could simulate physics and then argued that only
a quantum computer could simulate quantum physics efficiently.
 He focused on quantum physics rather than classical physics because, as he colorfully
put it, nature isn't classical, dammit, and if you want to make a simulation of nature,
you'd better make it quantum mechanical
 Around the same time, in “Quantum mechanical models of Turing machines that
dissipate no energy“
 Paul Benio demonstrated that quantum-mechanical systems could model Turing
machines.
 In other words, he proved that quantum computation is at least as powerful as classical
computation.
 But is quantum computation more powerful than classical computation?
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MODELS OF COMPUTATION
 David Deutsch explored this question and more in his 1985 paper
“Quantum theory, the Church Turing principle and the universal quantum
computer“
 First, he introduced quantum counterparts to both the Turing machine
and the universal Turing machine.
 He then demonstrated that the universal quantum computer can do
things that the universal Turing machine cannot, including
1. generate genuinely random numbers
2. perform some parallel calculations in a single register
3. Perfectly simulate physical systems with finite dimensional state
spaces.
 In 1989, in “Quantum computational networks", Deutsch described a
second model for quantum computation: quantum circuits.
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MODELS OF COMPUTATION
 Deutsch demonstrated that quantum gates can be combined to achieve
quantum computation in the same way that Boolean gates can be
combined to achieve classical computation.
 He then showed that quantum circuits can compute anything that the
universal quantum computer can compute, and vice versa.
David Deutsch
 Andrew Chi-Chih Yao picked up where Deutsch left off and addressed
the complexity of quantum computation in his 1993 paper “Quantum
circuit complexity“
 Specifically, he showed that any function that can be computed in
polynomial time by a quantum Turing machine can also be computed by
a quantum circuit of polynomial size.
Andrew Chi-Chih
 This finding allowed researchers to focus on quantum circuits, which are
easier than quantum Turing machines.
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MODELS OF COMPUTATION
 Also in 1993, Ethan Bernstein and Umesh Vazirani presented
”Quantum complexity theory", in which they described a universal
quantum Turing machine that can eficiently simulate any
quantum Turing machine.
Umesh
Vazirani
Ethan Bernstein
Peter Shor
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COMPUTING CRYPTOGRAPHY AND INFORMATION
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Quantum Mechanic Review
Quantum Computation
Quantum Algorithms and
Implementations
Quantum Cryptography
Quantum Information
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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QUANTUM GATES
 In 1995, a cluster of articles examined which sets of quantum gates are
adequate for quantum computation so that is, which sets of gates are
sufficient for creating any given quantum circuit.
Artur Ekert
 Adriano Barenco et al. “Elementary gates for quantum computation“,
showed that any quantum circuit can be constructed using nothing more
than quantum gates on one qubit and controlled exclusive-OR gates on
two qubits.
 David DiVincenzo, “Two-bit gates are universal for quantum
computation“ proved that two-qubit quantum gates are adequate.
 Adriano Barenco, David Deutsch, and Artur Ekert showed that quantum
controlled-NOT gates and one-qubit gates are together adequate;
David DiVincenzo
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QUANTUM ALGORITHMS AND IMPLEMENTATIONS
 In 1992, David Deutsch and Richard Jozsa coauthored “Rapid solution of problems by
quantum computation".
 They presented an algorithm that determines whether a function f is constant over all
inputs or balanced.
 The Deutsch-Jozsa algorithm was the first quantum algorithm to run faster than its
classical counterparts.
Richard Jozsa
 L. Chuang et al. detailed how they used bulk nuclear magnetic resonance techniques
to implement a simplifed version of the Deutsch-Jozsa algorithm.
 In Quantum complexity theory“, Bernstein and Vazirani were the first to identify a
problem that can be solved in polynomial time by a quantum algorithm but requires
superpolynomial time classically.
 Daniel R. Simon introduced a problem that a quantum algorithm can solve
exponentially faster than any known classical algorithm.
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QUANTUM ALGORITHMS AND IMPLEMENTATIONS
 This research inspired Peter W. Shor, who then invented two quantum algorithms
that outshone all others:
polynomial-time algorithms for finding prime factors and discrete logarithms,
problems widely believed to require exponential time on classical computers.
 Simon and Shor both presented their discoveries at the 1994, “Algorithms for
quantum computation: Discrete logarithms and factoring“.
Isaac L. Chuang
 Shor's factorization algorithm in particular heightened excitement and even
generated anxiety about the power and promise of quantum computing.
 To pose a practical threat to RSA cryptography, Shor's algorithm must be
implemented on quantum computers that can hold and manipulate large
numbers, and these do not exist yet.
 Isaac L. Chuang and his research team made headlines when they factored the
number 15 on a quantum computer with seven qubits.
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BQP, BPP, PSPACE, P, NP AND ….
Factoring
Problem
Quantum
Polynomial
Algorithms
PSPACE
NP
P
BPP
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
BQP
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QUANTUM ALGORITHMS AND IMPLEMENTATIONS
 Lov K. Grover's algorithm for searching an unordered list, described in both “A fast
quantum mechanical algorithm for database search" and “Quantum mechanics
helps in searching for a needle in a haystack“,
Lov K. Grover
 Unlike Shor's algorithm, Grover's algorithm solves a problem for which there are
polynomial-time classical algorithms; however, Grover's algorithm does it
quadratically faster than classical algorithms can.
 In 2003, Peter W. Shor addressed this stagnation in a short article called “Why
haven't more quantum algorithms been found?“Shor oered several possible
explanations,
1. The possibility that computer scientists have not yet developed intuitions for
quantum behavior.
2. The article should be required reading for all computer science students,
whose intuitionsare still being formed.
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CANDIDATES FOR QUANTUM COMPUTERS
1. Superconductor-based quantum computers (including SQUID-based quantum computers)
2. Ion trap-based quantum computers
3. "Nuclear magnetic resonance on molecules in solution"-based
4. “Quantum dot on surface"-based
5. “Laser acting on floating ions (in vacuum)"-based (Ion trapping)
6. "Cavity quantum electrodynamics" (CQED)-based
7. Molecular magnet-based
8. Fullerene-based ESR quantum computer
9. Solid state NMR Kane quantum computer
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COMPUTING CRYPTOGRAPHY AND INFORMATION
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MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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QUANTUM COMPUTING PROBLEMS
1. Current technology
≈ 40 Qubit operating machine needed to rival current classical equivalents.
2. Errors
 Decoherence - the tendency of a quantum computer to decay from a given
quantum state into an incoherent state as it interacts with the
environment.
Interactions are unavoidable and induce breakdown of information
stored in the quantum computer resulting in computation errors.
 Error rates are typically proportional to the ratio of operating time to
decoherence time
operations must be completed much quicker than the decoherence
time.
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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Quantum Mechanic Review
Quantum Computation
Quantum Algorithms and
Implementations
Quantum Cryptography
Quantum Information
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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QUANTUM CRYPTOGRAPHY
 Shor's factorization algorithm has yet to be implemented on more than a few
qubits.
 But if the efficient factorization of large numbers becomes possible, RSA
cryptography will need to be replaced by a new form of cryptography,
Gilles Brassard
 The cryptographic method in question is quantum key distribution, which was
introduced in 1984 by Charles H. Bennett and Gilles Brassard in “Quantum
cryptography: Public key distribution and coin tossing“ called BB84.
 In short, quantum key distribution is secure not because messages are
encrypted in some difficult-to-decrypt way but rather because eavesdroppers
cannot intercept messages undetected, regardless of computational resources.
Charles H.
Bennett
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QUANTUM CRYPTOGRAPHY
 Quantum cryptography describes the use of quantum mechanical effects (in
particular quantum communication and quantum computation) to perform
cryptographic tasks or to break cryptographic systems.
 quantum key distribution and (hypothetical) use of quantum computers,
 For example, quantum mechanics guarantees that measuring quantum data
disturbs that data; this can be used to detect eavesdropping in quantum key
distribution.
1. Position-based quantum cryptography
2. Quantum commitment
3. Quantum key distribution
4. Post-quantum cryptography
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
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POST-QUANTUM CRYPTOGRAPHY
 In a predictive sense, quantum computers may become a technological reality; it
is therefore important to study cryptographic schemes that are (supposedly)
secure even against adversaries with access to a quantum computer.
 The need for post-quantum cryptography arises from the fact that many popular
encryption and signature schemes
RSA and its variants, and schemes based on elliptic curves, can be broken
using Shor's algorithm for factoring and computing discrete logarithms on a
quantum computer.
 Examples for schemes that are, as of today's knowledge, secure against quantum
adversaries are
1. McEliece schemes
2. Lattice-based schemes
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
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QUANTUM KEY DISTRIBUTION
 The most well known and developed application of quantum cryptography is quantum key
distribution (QKD)
 QKD describes the process of using quantum communication to establish a shared key between
two parties (usually called Alice and Bob) without a third party (Eve) learning anything about that
key, even if Eve can eavesdrop on all communication between Alice and Bob.
 This is achieved by Alice encoding the bits of the key as quantum data and sending them to
Bob,
 If Eve tries to learn these bits, the messages will be disturbed and Alice and Bob will notice.
 The security of QKD can be proven mathematically without imposing any restrictions on the
abilities of an eavesdropper, something not possible with classical key distribution.
 Eve should not be able to impersonate Alice or Bob as otherwise a man-in-the-middle attack
would be possible.
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QUANTUM KEY DISTRIBUTION NETWORKS
1.In 2004, the world's first bank transfer using quantum key distribution was carried in Vienna,
Austria.
2. Swiss company Id Quantique was used in the Swiss canton (state) of Geneva to transmit
ballot results to the capitol in the national election occurring on October 21, 2007
3.The DARPA Quantum network, a 10-node quantum key distribution network, has been running
since 2004 in Massachusetts, USA.
4.The world's first computer network protected by quantum key distribution was implemented in
October 2008, at a scientific conference in Vienna. SECOQC (Secure Communication Based on
Quantum Cryptography) used 200 km of standard fibre optic cable to interconnect six
locations across Vienna and the town of St Poelten located 69 km to the west.
5.SwissQuantum network, installed in the Geneva metropolitan area in March 2009, was to
validate the reliability and robustness of QKD in continuous operation over a long time period
in a field environment. ran stably for nearly 2 years until the completion of the project in
January 2011, confirming the viability of QKD as a commercial encryption technology.
6.Tokyo QKD Network, UQCC2010, the network involves an international collaboration between 7
partners: NEC, Mitsubishi Electric, NTT and NICT from Japan, Toshiba (UK), Id Quantique
(Switzerland)
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
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QUANTUM COMMITMENT
1. Following the discovery of quantum key distribution and its unconditional security,
researchers tried to achieve other cryptographic tasks with unconditional security.
2. A commitment scheme allows a party Alice to fix a certain value (to "commit") in such a way
that Alice cannot change that value while at the same time ensuring that the recipient Bob
cannot learn anything about that value until Alice decides to reveal it.
3. In the quantum setting, they would be particularly useful
4. Crépeau and Kilian showed that from a commitment and a quantum channel, one can
construct an unconditionally secure protocol for performing so-called oblivious transfer.
5. Unfortunately, early quantum commitment protocols were shown to be flawed.
6. In fact, Mayers showed that (unconditionally secure) quantum commitment is impossible: a
computationally unlimited attacker can break any quantum commitment protocol.
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POSITION-BASED QUANTUM CRYPTOGRAPHY
1. The goal of position-based quantum cryptography is to use the geographical location of a
player as its (only) credential.
2.
For example, one wants to send a message to a player at a specified position with the
guarantee that it can only be read if the receiving party is located at that particular position.
3. It has been shown by Chandran et al. that position-verification using classical protocols is
impossible against colluding adversaries(who control all positions except the prover's claimed
position).
4. Under the name of 'quantum tagging', the first position-based quantum schemes have been
investigated in 2002 by Kent. A US-patent.
5. After several other quantum protocols for position verification have been suggested in 2010.
6. Buhrman et al. were able to show a general impossibility result:: using an enormous amount of
quantum entanglement, colluding adversaries are always able to make it look to the verifiers
as if they were at the claimed position.
7.
However, this result does not exclude the possibility of practical schemes in the bounded- or
noisy-quantum-storage model (see above).
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QUANTUM CRYPTOGRAPHY MANUFACTURERS
 MagiQ Technologies http://www.magiqtech.com/
 id Quantique (http://www.idquantique.com/),
 Smart Quantum (http://www.smartquantum.com/).
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STEMPUNK BB84
C. Bennett, F. Bessette, G. Brassard, L. Savail, J. Smolin
J. Cryptology 5, 3 (1992)
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•
Larger distances (up to 144km demonstrated) to test for satellite – earth links
Munich/Vienna/Bristol: T. Schmitt-Manderbach et al., Phys. Rev. Lett. 98,
010504 (2007)
•
Larger key rates: VCSEL lasers, detectors with better timing resolution, high clock
rate ~Mbit/sec key rate (detector limited) NIST Gaithersburg: J.C. Bienfang et al.
Optics Express 12, 2011 (2004
QUANTUM CRYPTOGRAPHY
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
COMPUTING CRYPTOGRAPHY AND INFORMATION
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Quantum Mechanic Review
Quantum Computation
Quantum Algorithms and
Implementations
Quantum Cryptography
Quantum Information
MEYSAM MADANI - AN INTRODUCTION TO QUANTUM
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QUANTUM INFORMATION
 Secure channels of communication are of course crucial, but security is not the
only consideration in the transfer of information.
 Accordingly, quantum cryptography is just one of several topics in the burgeoning
field of quantum information.
 Other topics include quantum error correction, fault-tolerant quantum
computation, quantum data compression, and quantum teleportation.
 Information needs to be protected not just from eavesdroppers but also from
errors caused by channel noise, implementation flaws, and, in the quantum
case, decoherence.
 Peter W. Shor, a trailblazer not just of quantum algorithms but also of quantum
error correction and fault-tolerant quantum computation, was the first to describe
a quantum error-correcting method.
 In his 1995 article “Scheme for reducing decoherence in quantum computer
memory“ he demonstrated that encoding each qubit of information into nine
qubits could provide some protection against decoherence.
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QUANTUM INFORMATION
 At almost the same time but without knowledge of Shor's article, Andrew
M. Steane wrote “Error correcting codes in quantum theory“which
achieved similar results.
 Very shortly thereafter, Shor and A.R. Calderbank presented improved
results in “Good quantum error-correcting codes exist“
Andrew M. Steane
 Error is not the only thing information theorists strive to reduce; they also
seek to reduce the space required to represent information.
 In this 1948 paper, Shannon showed that it is possible, up to a certain
limit, to compress data without loss of information.
 Almost 50 years later, Benjamin Schumacher developed a quantum
version of Shannon's theorem.
Benjamin
Schumacher
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QUANTUM INFORMATION
 Not everything in quantum information theory has a precedent
in classical information theory.
 In 1993, Charles H. Bennett et al. dazzled the scientific c
community and delighted science fiction fans by showing that
quantum teleportation is theoretically possible.
Dirk
Bouwmeester
Claude E. Shannon
 They described how an unknown quantum state could be
disassembled and then reconstructed perfectly in another
location.
 The first researchers to verify this method of teleportation
experimentally were Dik Bouwmeester et al., who reported
their achievement in 1997 in “Experimental quantum
teleportation"
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REFERENCES
1. Historical Bibliography of Quantum Computing, Jill Cirasella, Brooklyn College
Library.
2. Timeline of quantum computing, Wikipedia,
3. Quantum cryptography, Wikipedia,
4. Quantum key distribution, Wikipedia,
5. MagiQ Technologies, http://www.magiqtech.com
6. Practical Quantum cryptography and possible attacks, Alex Ling, et. Al, NUS,.
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