1. Quantum Computing Lecture 22 Michele Mosca

2. Correcting Phase Errors l Suppose the environment effects error on our quantum computer, where This is a description of errors in phase because we use powers of operator Z

3. Quantum Error Correction l We can encode l Consider error termacting on the logical 0 gives Z error in upper bit Such error arriving in decoder is shown next slide

4. Quantum Error Correction error Please observe repetitions of these patterns

5. Equivalently, cancelling pairs of H inside the diagram we get Final circuit for correcting phase errors

6. Quantum Error Correction l If the error effected on the system in state is of the form

7. Quantum Error Correction l and if the state only consists of mixtures of superpositions of codewords and then the correction procedure (call it ) will map

8. Correcting both phase errors and bit flip errors l Consider the codewords of Shor’s code l We can easily correct any single X- error in one of the 3 three-bit parts l We can then also correct a single Z- error on one of the 9 qubits. l This means we can also correct Y-errors on one of the 9 qubits

9. Quantum Error Correction 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. Correcting any error l Since any error operator E k can be written as a linear combination of I,X,Z and Y, then the same procedure will correct ANY error acting on just 1 of the 9 qubits. l If where is a quantum operator whose operator terms are correctable with correction operator, then

11. Correcting any error l Theorem 10.1 (Quantum Error Correction Conditions) Let C be a quantum code, and let P be the projector onto C. Suppose is a quantum operation with operation elements A necessary and sufficient condition for the existence of an error-correction operation correcting on C is that for some Hermitian matrix of complex numbers. l (no mention of efficiency)

12. Degenerate Codes l Consider the 9-qubit code. l A single Z-error on the first qubit of a codeword produces the same outcome as a single Z-error on either the 2 nd or 3 rd qubit. l The correction procedure will correct these errors regardless l A degenerate code is one where two correctable errors produce the same effect on the codewords (this is impossible with classical codes).

13. Quantum Hamming Bound l Any non-degenerate quantum error correcting code that encodes k logical qubits into n qubits and can correct errors on up to t qubits must have l If t=k=1, we get (there exists a 5- qubit code that accomplishes this)