Quantum Error Correction Michele Mosca

Quantum Error Correction: Bit Flip Errors l 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 l Then we get (assuming ) Correct information Bits with errors

Quantum Error Correction l Then we get (assuming )

Quantum Error Correction l Then we get (assuming )

Quantum Error Correction l Then we get (assuming )

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

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