Quantum Error Correction Michele Mosca Quantum Error Correction

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 ) Bits with errors Correct information

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 map ) will

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 the operators Form a basis for the set of possible operation elements