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Quantum Error Correction

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

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  1. Quantum Error Correction Michele Mosca

  2. 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

  3. Analysis of 6-qubit system which uses syndrome bits to correct errors • Then we get (assuming ) Bits with errors Correct information

  4. Quantum Error Correction • Then we get (assuming )

  5. Quantum Error Correction • Then we get (assuming )

  6. Quantum Error Correction • Then we get (assuming )

  7. Quantum Error Correction • More generally, if the error effected on the system in state is of the form

  8. 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

  9. 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.

  10. 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

  11. 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

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