1. Quantum ComputingLecture 22 Michele Mosca

2. Correcting Phase Errors • 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 • We can encode • Consider error term acting 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 • If the error effected on the system in state is of the form

7. Quantum Error Correction • 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 • Consider the codewords of Shor’s code • We can easily correct any single X- errorin one of the 3 three-bit parts • We can then also correct a single Z- erroron one of the 9 qubits. • This means we can also correct Y-errorson one of the 9 qubits

9. Quantum Error Correction • 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 • Since any error operator Ek 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. • If where is a quantum operator whose operator terms are correctable with correction operator , then

11. Correcting any error • 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. • (no mention of efficiency)

12. Degenerate Codes • Consider the 9-qubit code. • A single Z-error on the first qubit of a codeword produces the same outcome as a single Z-error on either the 2nd or 3rd qubit. • The correction procedure will correct these errors regardless • 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 • 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 • If t=k=1, we get (there exists a 5-qubit code that accomplishes this)