Symposium on Advanced Topics in Information Technology

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SymposiumonAdvanced Topics in Information
Technology

• 20/1/2009

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An Overview of Developments in Quantum
Computation and Communication By Prof M.
Adeeb R. Ghonaimy Professor Emeritus, Faculty of
Engineering Ain Shams University Cairo,
Egypt 20 January 2009
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CONTENTS

• INTRODUCTION
• CIRCUIT MODEL FOR QUANTUM COMPUTATION
• Quantum Bits and Gates
• Quantum Circuits
• Quantum Algorithms
• Evolution of Quantum Entanglement
• Implementation Considerations for Quantum
  Computers
• III. QUANTUM COMMUNICATION
• Quantum Error Correction
• Quantum Cryptography
• Quantum Information Theory
• The Quantum Internet
• IV. CONCLUSIONS

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I. Introduction

• The Road to Quantum Computation
• In 1936, Turing introduced a precise mathematical
  formulation of the concept of algorithm (Turing
  machine). The Universal Turing Machine is an
  abstract model of a computer.
• In 1985, Deutsch considered computing devices
  based upon the principles of quantum mechanics,
  and ultimately proposing a Universal Quantum
  Computer.
• In 1994, Peter Shor demonstrated that two
  important problems-the problem of finding the
  prime factors of an integer. and the discrete
  logarithm problem-could be solved efficiently on
  a quantum computer.
• In 1995 Grover showed that the problem of
  searching some unstructured search space could
  also be sped up on a quantum computer.
• In 2003, Raussendorf and Briegel proposed a
  cluster-state model for quantum computation
  (One-way or measurement-only quantum computing).
• In 2005, the Zeilinger group gave a
  proof-of-concept experimental verification for
  the cluster-state model

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The Road to Quantum Communication

• In 1948, Claude Shannon laid the foundations for
  the modern theory of information and
  communication
• Shannon's noiseless channel coding theorem,
  quantifies the physical resources required to
  store the output from an information source.
• The noisy channel coding theorem, quantifies how
  much information it is possible to reliably
  transmit through a noisy communication channel.
• In 1995, Ben Schumacher provided an analog to
  Shannon's noiseless channel coding theorem, and
  in the process defined the 'quantum bit' or
  'qubit' as a tangible physical resource.
• No complete analog to Shannon's noisy channel
  coding theorem is yet known for quantum
  information.
• Bennett and Wiesner in 1992 explained how to
  transmit two classical bits while only
  transmitting one quantum bit, a result called
  superdense coding.
• Quantum error-correcting codes started in 1996.

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• Quantum Cryptography
• An important field that has now reached the
  commercial state is quantum cryptography and in
  particular, Quantum Key Distribution (QKD).
• In 1984 Bennett and Brassard proposed a protocol
  using quantum mechanics to distribute keys
  between two locations without any possibility of
  compromise.
• Quantum Networks
• Quantum networks are also being developed. An
  experimental network is already operational in
  Cambridge, Mass. and is operated by BBN within a
  research project supported by DARPA with the name
  Quantum Network.
• Quantum Entanglement
• One of the basic concepts in quantum mechanics
  that is used in many of the above applications is
  quantum entanglement.
• This is a fundamental resource of Nature, of
  comparable importance to energy, information,
  entropy, or any other fundamental resource.
• Quantum Internet
• Use of entanglement and quantum memory to
  implement teleportation.

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II. CIRCUIT MODEL FOR QUANTUM COMPUTATION A-
QUANTUM BITS AND GATES
A-1 Single Qubits Just as a
classical bit has a state-either 0 or 1-a quantum
bit or qubit also has a state. Two possible
states for a qubit are the states 0gt and 1gt.
The difference between bits and qubits is that a
qubit can be in a state other than 0gt or 1gt.
It is possible to form linear combinations of
states, called superpositions. ?gt a0gt
ß1gt (1) The numbers a and ß are complex
numbers. Thus, the state of a qubit is a vector
in two-dimensional complex vector space. The
special states 0gt and 1gt are known as
computational basis states and form an
orthonormal basis for this vector space.
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It is possible to examine a bit to determine
whether it is in the state 0 or 1, but we cannot
examine a qubit to determine its quantum state,
that is the values a and ß.
When we measure a qubit we get either the result
0, with probability a2, or the result 1, with
probability ß2, where a2ß2 1. For
example, a qubit in the state
?
gives the result 0 fifty per cent of the time,
and the result 1 fifty per cent of the time. This
state is sometimes denoted by gt.
A qubit could be realized as the two different
polarizations of a photon, or as the two states
of an electron orbiting a single atom.
In the atom model, the electron can exist in
either the "ground' or "excited" states, 0gt and
1gt respectively. By shining light on the atom
with appropriate energy and for an appropriate
length of time, it is possible to move the
electron from the 0gt to 1gt state and vice
versa. It is also possible by reducing the time
we shine the light, an electron initially at
state 0gt can be moved halfway between 0gt and
1gt, into the gt state.
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A-3 Single Qubit Gates A quantum NOT gate acts
linearly on the state, a0gt ß1gt (6) to the
corresponding state in which the role of 0gt and
1gt have been interchanged, a1gt
ß0gt (7) It is possible to represent the
quantum NOT gate by a matrix,
Normalization requires that a2 ß2 1 X
must be a unitary matrix The unitarity
constraint is the only constraint on quantum
gates.
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Two more gates are Z gate given by the matrix,
and the Hadamard gate given by the matrix,
Fig. (2) summarizes the operation of the above
three quantum gates
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A-4 Multiple Qubit Gates Any multiple qubit
gate may be composed from CNOT and single qubit
gates.
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B- QUANTUM CIRCUITS Quantum circuits may be
formed from a number of quantum gates. However,
there are restrictions
Feedback from one part of the quantum circuit to
another is not allowed. FANIN (bitwise OR of
inputs) is not allowed. FANOUT is not allowed (no
cloning).
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Quantum Teleportation

• Suppose Alice and Bob generated an EPR pair, each
  taking one qubit.
• They were then separated, and Alice now wants to
  deliver a qubit to Bob.
• She does not know the state of the qubit, and
  moreover can only send classical information to
  Bob.

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The state to be teleported is ?gt a0gt ß1gt,
where a and ß are unknown. The state input into
the circuit is ?0gt ?gtß00gt (11)
(1/ )a0gt (00gt 11gt) ß1gt
(00 11gt) Where the first two bits belong to
Alice and the third bit belongs to Bob.

If Bob receives 00 over the classical channel,
then he has the qubit, if he receives 01 he can
apply the X gate, if 10 the Z gate, and if 11 the
X and then the Z gate.
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C- QUANTUM ALGORITHMS Quantum Parallelism is a
fundamental feature of many quantum algorithms.
Fig. (6)a shows a quantum circuit for evaluating
f(0) and f(1) simultaneously. The output state
is given by,
0gt 1gt
x
x
v2
Uf
?gt
y ? f (x)
y
0gt
Fig. (6)a Quantum circuit for evaluating f(0) and
f(1) simultaneously.
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E- Implementation Considerations

• E-1 Trapped Ions

Qubit (one per ion)-a pair of ions internal
states. Phonon (vibrational states) Decoherence
time in the thousands of seconds. It starts to be
difficult after 10 qubits
E-2 Nuclear Magnetic Resonance
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• E-3 Solid State (Bruce Kane)

• Bury an array of phosphorous atoms in silicon,
  and overlay it with an insulating layer,on top of
  which a similar array of electrodes.
• Nuclei spins can be flipped using radio waves.
• Address a single nucleus using electrode
  voltage
• Superconductivity.
• Quantum dots.

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E-5 Conditions for Quantum Computation
E-6 Architectural Considerations
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E-7 D-Wave
E-8 Deep Web Search
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III QUANTUM COMMUNICATION
A- QUANTUM ERROR-CORRECTION A simple means of
protecting the bit against the effects of noise
is to replace the bit with three copies of
itself 0 ? 000 (18) 1 ? 111 (19) If p is
not too high, it is possible to use a majority
voting decoding scheme to get the correct bit at
the receiver. This scheme fails if two or more
bits were flipped. The type of code used in this
case is called a repetition code.
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• To protect quantum states against the effects of
  noise we would like to develop quantum
  error-correcting codes based upon similar
  principles.

• However, there are some important differences

• It is forbidden according to the no cloning
  theorem to duplicate quantum states. Also, it is
  not possible to measure and compare the three
  quantum states at the receiver.
• A continuum of different errors may occur on a
  single qubit.
• 3. Measurement destroys quantum information.

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• Encode the single qubit state a0gt b1gt in
  three qubits as a000gt b111gt.
• A simple two stage error-correction quantum
  scheme is
• Error-detection or syndrome diagnosis
• P0 000gtlt000 111gtlt111 No error (20)
• P1 100gtlt100 011gtlt011 bit flip on qubit
  one (21)
• P2 010gtlt010 101gtlt101 bit flip on qubit
  two (22)
• P3 001gtlt001 110gtlt110 bit flip on qubit
  three (23)
• (2) Recovery.For example, for error syndrome 1
  we flip qubit 1 to recover the original state,
  and so on.

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A more interesting noisy quantum channel is the
phase flip error model for a single qubit, i. e.
the operator Z is applied to the qubit. It is
possible to show that the encoding circuit in
that case will be as shown in Fig. (12).
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Fig. (13) Encoding circuit for Shor nine qubit
code
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B- Quantum CryptographyQuantum Key
DistributionEvolution of QKD
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B- Quantum Cryptography(Continued)Quantum Key
DistributionEvolution of QKD
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Summary of Some Commercial QKD Systems
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C- Quantum Information Theory

• Shannon entropy measures uncertainty associated
  with classical probability distribution.
• In quantum information theory, Von Neumann
  entropy is used.
• S(?) - tr(?log ?) (24)
• ? is the density operator or density matrix.
• For a quantum system in a number of states ?igt,
  with respective probabilities Pi
• ? ?Pi ?igtlt ?i (25)
• Trace is equal to 1.

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• Another important aspect of quantum states is
  that they cannot only be copied but also they
  cannot be perfectly distinguished.
• This is quantified by the Holevo bound. Suppose,
  for example, that Alice prepares a state ?x where
  x0, . . , n with probability P0, . . ., Pn. Bob
  performs a measurement on that state with outcome
  Y.
• The Holevo bound states that for any such
  measurement
• H(XY) ? S(?) - ?xPx S(?x). (26)
• Where ? ?xPx?x, and H(XY) is the mutual
  information of X and Y.
• The Holevo bound is thus an upper bound on the
  accessible information.

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• The corresponding theorem to Shannon's noiseless
  channel coding theorem in the quantum domain is
  given by Schumacher's theorem which states that
  the number of compressed qubits are given by
• nqubits S(?Px?x) (27)
• The analog of Shannon's noisy channel coding
  theorem for sending classical information over a
  quantum channel is given partially by
  Holevo-Schumacher-Westmoreland theorem.

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D- The Quantum Internet

• Joint project between MIT and Northwestern
  university in which an architecture for the
  Quantum Internet is proposed Llyod, 2004.
• It connects distant quantum computers via
  teleportation with complete Bell-state
  measurements.
• Nonlinear optics was used to produce a stream of
  polarization entangled photon pairs (signal and
  idler states) that go along optical fibers.
• At the end there is a pair of quantum memories
  comprised of Rubidium (87Rb) atoms trapped inside
  a high Q optical cavity.
• The Bell-observable measurements are made in
  Alices memory, and the teleportation completing
  transformation is made Bobs memory.

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IV- CONCLUSIONS

• Optical methods using off-the shelf components
  are being used in Quantum Key Distribution.
  Special equipment is also being developed, e. g.
  single photon generation and detection, together
  with investigating the use of quantum
  entanglement. Prospects for generating pure
  random bit sequences are investigated using laser
  technology especially with the availability now
  of all-silicon lasers on a chip.
• Secure quantum networks have reached the
  prototype status and an experimental network is
  already operational. Financial institutions and
  national security organizations will benefit
  greatly from such developments.
• Although some techniques, like NMR and ion traps,
  were used to implement some prototype simple
  quantum computers, it is expected that
  solid-state approaches have more chances of
  implementing scalable quantum computers. Some
  proposals are being investigated with the
  possibility of using superconductivity in the
  implementation. Decoherence is a major problem
  that has to be dealt with.

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IV- CONCLUSIONS (Continued)

• Recently, Cluster state quantum computing was
  proposed in 2003, with a proof of concept
  performed in 2005.
• Architectural considerations for designing
  fault-tolerant quantum computers are also
  investigated. Quantum arithmetic units for
  implementing quantum operations and quantum
  memory modules are being proposed together with
  other units for scheduling the different quantum
  operations.
• Quantum Internet basic components are being
  developed.