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Quantum Error Correction. Michele Mosca . 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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Quantum error correction bit flip errors
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


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

  • Then we get (assuming )

Bits with errors

Correct information


Quantum error correction1
Quantum Error Correction correct errors

  • Then we get (assuming )


Quantum error correction2
Quantum Error Correction correct errors

  • Then we get (assuming )


Quantum error correction3
Quantum Error Correction correct errors

  • Then we get (assuming )


Quantum error correction4
Quantum Error Correction correct errors

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


Quantum error correction5
Quantum Error Correction correct errors

  • and if the state only consists of mixtures of superpositions of codewords and

    (that is, )

    then the correction procedure (call it ) will map


Main error correction theorem
Main Error Correction Theorem correct errors

  • 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.


Quantum error correction6
Quantum Error Correction correct errors

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


Quantum error correction7
Quantum Error Correction correct errors

  • Notice that if lives in a Hilbert space of dimension then the operators

    Form a basis for the set of possible operation elements


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