Quantum Computing Lecture 22 Michele Mosca

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

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

Quantum Error Correction error Please observe repetitions of these patterns

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

Quantum Error Correction l 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 then the correction procedure (call it ) will map

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

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.

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

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)

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).

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)