Quantum Error Correction

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Quantum Error Correction

• Michele Mosca

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Quantum Error Correction Bit Flip Errors

• Suppose the environment will effect error
• (i.e. operation )
• on our quantum computer when the environment
  is in state

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Analysis of 6-qubit system which uses syndrome
bits to correct errors

• Then we get (assuming )

Bits with errors
Correct information
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Quantum Error Correction

• Then we get (assuming )

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Quantum Error Correction

• Then we get (assuming )

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Quantum Error Correction

• Then we get (assuming )

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Quantum Error Correction

• More generally, if the error effected on the
  system in state is of the form

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Quantum Error Correction

• and if the state only consists of mixtures of
  superpositions of codewords and
• (that is, )
• then the correction procedure (call it )
  will map

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Main Error Correction Theorem

• Theorem 10.2 Suppose C is a quantum code and
  is the error-correction operation constructed
  in the proof of Theorem 10.1 to recover from a
  noise process with operation elements
  . Suppose is a quantum operation with
  elements which are linear combinations of
  the . Then the error correction operation
  also corrects the effects of the noise
  process on the code C.

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Quantum Error Correction

• E.g. the error correction procedure that we
  have described for the 3-qubit code will correct
  any combination of error operators

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Quantum Error Correction

• Notice that if lives in a Hilbert space of
  dimension then the operators
• Form a basis for the set of possible operation
  elements