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Quantum Error CorrectionPowerPoint Presentation

Quantum Error Correction

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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

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

- Then we get (assuming )

Bits with errors

Correct information

Quantum Error Correction correct errors

- Then we get (assuming )

Quantum Error Correction correct errors

- Then we get (assuming )

Quantum Error Correction correct errors

- Then we get (assuming )

Quantum Error Correction correct errors

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

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 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 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 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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