1. Quantum Error Correction Michele Mosca

2. 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

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

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

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

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

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

8. 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

9. 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.

10. 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

11. 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