Quantum Computing

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Quantum Computing

• Steven Filsell

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Dissertation Title

• Quantum devices. This dissertation will consider
  the application of quantum device technology in
  computing technology. The basic theory behind
  quantum computation will be investigated to see
  what makes quantum computers so different from
  their counterparts. This dissertation will then
  consider other fields involving quantum
  information such as Quantum Cryptography, Quantum
  Communication and Quantum Error Correction.

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Why Quantum Computers?

• Imagine the following problem-
• Factorising a number N into its prime
  factors (eg 51688 may be decomposed as 23 x 7 x
  13 x 71)
• How long will this take?
• This depends on how the number of steps to
  complete the algorithm scales with the size of
  the input N
• Best known factorising algorithm for conventional
  computers runs in exp(64/9)1/3(ln N)1/3(lnln
  N)2/3 steps

Steven Filsell Matriculation Number 9636002
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Why Quantum Computers?

• The last algorithm scales exponentially with
  input size log N
• The following table shows typical times for
  conventional computers

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Why Quantum Computers?

• An algorithm for quantum computers has been
  developed which runs in log(N)2? steps (? is
  very small)
• Roughly a quadratic
• Factorising a 1000 digit number requires a few
  million steps and not 1025 years

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How do they work?

• Easy to just consider one bit
• In classical counterparts
• Logic 0 or Logic 1
• True or False
• In quantum systems
• atom is in both states at once
• Quantum two state
• called quantum bit or qubit

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An Experiment to prove this

• Consider the mirror to be half-silvered
• Which path does the photon take
• reflected or transmitted
• Is it a random process?
• Turns out that the photon takes the two paths at
  once

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Register

• Consider a register composed of three physical
  bits
• A classical register can store at any moment of
  time only on of the different numbers
• A quantum register composed of three qubits at
  any moment of time can store all eight numbers
• in general L qubits can store 2L numbers at once

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Register

• Quantum computing generates a massive parallel
  computation in one piece of quantum hardware
• This means that quantum computer can in one
  computational step perform the same mathematical
  operation on 2L different input numbers encoded
  in coherent superpositions of L qubits
• Classical computers would repeat the same
  computation 2L times

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Summary

• Quantum Computer
• very fast
• do not slow down too much as number of bits
  increase
• but ..
• No working models exist yet!

• Classical Computer
• Require exponentially more time
• Can require more registers in parallel
• slow down dramatically as number of bits increase