Quantum computing

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Quantum computing

• Alex Karassev

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Quantum Computer

• Quantum computer uses properties of elementary
  particle that are predicted by quantum mechanics
• Usual computers information is stored in bits
• Quantum Computers information is stored in
  qubits
• Theoretical part of quantum computing is
  developed substantially
• Practical implementation is still a big problem

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What is a quantum computer good for?

• Many practical problems require too much time if
  we attempt to solve them on usual computers
• It takes more then the age of the Universe to
  factor a 1000-digits number into primes!
• The increase of processor speed slowed down
  because of limitations of existing technologies
• Theoretically, quantum computers can provide
  "truly" parallel computations and operate with
  huge data sets

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Probability questions

• How many times (in average) do we need to toss a
  coin to get a tail?
• How many times (in average) do we need to roll a
  die to get a six?
• Loaded die alter a die so that the probability
  of getting 6 is 1/2.

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Quantum computers and probability

• When the quantum computer gives you the result of
  computation, this result is correct only with
  certain probability
• Quantum algorithms are designed to "shift" the
  probability towards correct result
• Running the same algorithm sufficiently many
  times you get the correct result with high
  probability, assuming that we can verify whether
  the result is correct or not
• The number of repetition is much smaller then for
  usual computers

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Short History

• 1970-? the beginning of quantum information
  theory
• 1980 Yuri Manin set forward the idea of quantum
  computations
• 1981 Richard Feynman proposed to use quantum
  computing to model quantum systems. He also
  describe theoretical model of quantum computer
• 1985 David Deutsch described first universal
  quantum computer
• 1994 Peter Shor developed the first algorithm
  for quantum computer (factorization into primes)

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Short History

• 1996 Lov Grover developed an algorithm for
  search in unsorted database
• 1998 the first quantum computers on two qubits,
  based on NMR (Oxford IBM, MIT, Stanford)
• 2000 quantum computer on 7 qubits, based on NMR
  (Los-Alamos)
• 2001 15 3 x 5 on 7- qubit quantum comp. by IBM
• 2005-2006 experiments with photons quantum
  dots fullerenes and nanotubes as "particle
  traps"
• 2007 D-Wave announced the creation of a quantum
  computer on 16 qubits

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Quantum system

• Quantum system is a system of elementary
  particles (photons, electrons, or nucleus)
  governed by the laws of quantum mechanics
• Parameters of the system may include positions of
  particles, momentum, energy, spin, polarization
• The quantum system can be characterized by its
  state that is responsible for the parameters
• The state can change under external influence
• fields, laser impulses etc.
• measurements

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Some quantum mechanics

• Superposition if a system can be in either of
  two states, it also can be in superposition of
  them
• Some parameters of elementary particles are
  discrete (energy, spin, polarization of photons)
• Changes are reversible
• The parameters are undetermined before
  measurements
• The original state is destroyed after measurement
• No Cloning Theorem it is impossible to create a
  copy of unknown state
• Quantum entanglement and quantum teleportation

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Qubit

• Qubit is a unit of quantum information
• In general, one qubit simultaneously "contains"
  two classical bits
• Qubit can be viewed as a quantum state of one
  particle (photon or electron)
• Qubit can be modeled using polarization, spin, or
  energy level
• Qubit can be measured
• As the result of measurement, we get one
  classical bit 0 or 1

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A model of qubit
or

• a0 ? a1 are complex numbers such that a02 a1
  2 1
• ?gt is a superposition of basis states 0gt ? 1gt
• The choice of basis states is not unique
• The measurement of ?gt resultsin 0 with
  probability a02 and in 1 with probability a12
  
• After the measurement the qubit collapses into
  the basis state that corresponds to the result

1/4
Example
3/4
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Several qubits

• The system of n qubits "contain" 2n classical
  bits (basis states)
• Thus the potential of a quantum computer grows
  exponentially
• We can measure individual qubits in the
  multi-qubit system
• For example, in a two-qubit system we can measure
  the state of first or second qubit, or both
• The results of measurement are probabilistic
• After the measurement the system collapses in the
  corresponding state

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Example two qubits
Let's measure the first bit
1
0
result
probability
The coefficients changes so that the ratio is the
same
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Independent qubits

• A system of two independent qubits(two
  non-interacting particles)

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Entangled states
There is no qubitsa0 0gt a1 1gtb0 0gt b1
1gt s.t. the state
The value ofsecond bit with100 probability
01gt
1
0
measure the first bit
1
10gt
0
could be represented asa0b0 00gt a0 b1 01gt
a1 b0 10gt a1 b1 11gt
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Examples
Maximally entangled states (Bell's basis)
Is the following state entangled?
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Quantum Teleportation
Entangled qubits A and B
qubit with unknown statethat Alice wants to
send to Bob
Now Bob knowsthe state of B
makes ? and C entangled
Communication channel (e.g. phone)
makes B into C
some transformations
Now Bob has qubit C
measures C
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Operations on bits

• NOT NOT(0) 1, NOT(1)0
• OR 0 OR 0 0, 1 OR 0 0 OR 1 1 OR 1 1
• AND 0 AND 0 1 AND 0 0 AND 1 0, 1 AND 1
  1
• XOR (addition modulo two)0 ? 0 1 ? 1 0, 0
  ? 1 1 ? 0 1
• What is NOT ( x OR y)?
• What is NOT (x AND y)?
• NOT (x OR y) NOT (x) AND NOT (y)
• NOT (x AND y) NOT (x) OR NOT (y)

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Classical and quantum computation

• Operations AND and OR are not invertible even
  if we know the value of one of two bits and the
  result of the operation we still cannot restore
  the value of the other bit
• Example suppose x AND y 0 and y 0
• what is x?
• Because of the laws of quantum mechanics quantum
  computations must be invertible (since the
  changes of the quantum system are reversible)
• Are there such operations?
• Yes! E.g. XOR (addition modulo two)

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Linearity and parallel computations

• Example let F be a quantum operation that
  correspond to a function f(x,y) (x',y'). Then
• Thus one application of F gives a system that
  contains the results of f on all inputs!
• It is enough to know the results on basis states
• Matrix representation
• Invertibility

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Some matrices

• A matrix is a table of numbers, e.g.
• We can multiply matrices by vectors
• Moreover, we even can multiply matrices!

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Operations on one qubit

• Quantum NOTNOT( a0 0gt a1 1gt) a0 1gt a1
  0gt
• Hadamard gateH( a0 0gt a1 1gt) 1/v2 (a0
  a1)0gt (a0 - a1)0gt

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Two qubits controlled NOT (CNOT)
CNOT (x,y) (x, x XOR y) (x, x?y)
0?01?10, 0?11?01
CNOT( a000gta101gta210gta311gt )
a000gta101gta311gta210gt
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How quantum computer works

• The routine
• Initialization (e.g. all qubits are in state 0gt
• Quantum computations
• Reading of the result (measurement)
• "Ideal" quantum computer
• must be universal (capable of performing
  arbitrary quantum operations with given
  precision)
• must be scalable
• must be able to exchange data

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Quantum algorithms

• Shor's algorithm
• Factorization into primes
• Work in polynomial time with respect to the
  number of digits in the representation of an
  integer
• Can be used to break RSA encryption
• Grover's algorithm
• Database search
• "Brute force" about N operations where N is the
  number of records in the database
• Grover's algorithm about operations

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Problems

• Decoherence
• Quantum system is extremely sensitive to external
  environment, so it should be safely isolated
• It is hard to achieve the decoherence time that
  is more than the algorithm running time
• Error correction (requires more qubits!)
• Physical implementation of computations
• New quantum algorithms to solve more problems
• Entangled states for data transfer

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Practical Implementations

• The use of nucleus spins and NMR
• Electrons spins and quantum dots
• Energy level of ions and ion traps
• Use of superconductivity
• Adiabatic quantum computers

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D-Wave quantum computer Orion

• January 19, 2007 D-Wave Systems (Burnaby,
  British Columbia) announced a creation of a
  prototype of commercial quantum computer, called
  Orion
• According to D-Wave, adiabatic quantum computer
  Orion uses 16 qubits and can solve quite complex
  practical problems (e.g. search a database and
  solve Sudoku puzzle)
• Unfortunately, D-Wave did not disclose any
  technical details of their computer
• This caused a significant criticism among
  specialists
• Recently, the company received 17 millions
  investments

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Homework

• Is the following state entangled?
• What happens if we apply twice
• negation?
• Hadamard gate?

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Thank You!

• http//www.nipissingu.ca/numeric
• http//www.nipissingu.ca/faculty/alexandk/popular/
  popular.html