A simple example:bit flip code(2) Each of the three qubits is passed through an independent copy of the bit flip channel We can use the following measurement to do error-detection or syndrome diagnosis P≡000)(000+I111)(111 no error P=100)(100+011)(011 bit flip on qubit one P=010)(010+101)(101 bit flip on qubit two P 001)(001+110)(110 bit flip on qubit three. Note that this is a valid measurement If only one qubit is flipped,then the outcome is deterministic,and the measurement does not change the state(does not allow us to infer anything about the value of a or b) 0 Using the value of the error syndrome,we can recover the initial state
Each of the three qubits is passed through an independent copy of the bit flip channel We can use the following measurement to do error-detection or syndrome diagnosis ◦ Note that this is a valid measurement ◦ If only one qubit is flipped, then the outcome is deterministic, and the measurement does not change the state (does not allow us to infer anything about the value of a or b) ◦ Using the value of the error syndrome, we can recover the initial state
A simple example:bit flip code(3) We have a different way to do syndrome measurement 。First measure the observable 2,Z2(Z⑧Z⑧I) Then measure the observable Z223 ÷Note that ZiZ2=(00)(00+|11)(11)⑧I-(01)01+l10)(10)⑧I Thus measuring 2122 can compare the first two qubits Neither measurement gives any information about the amplitudes a and b,and thus neither measurement destroys the states that we wish to preserve
We have a different way to do syndrome measurement ◦ First measure the observable ( ) ◦ Then measure the observable Note that ◦ Thus measuring can compare the first two qubits Neither measurement gives any information about the amplitudes a and b, and thus neither measurement destroys the states that we wish to preserve
Phase flip code(1) Phase flip channel:leaves the qubits untouched with probability 1-p, and flips the phase())with probability p The state a0)+b1)is taken to a0)-b1) 。Therefore,,the error takes+〉=(l0)+l1)/W2to-〉≡(l0)-I)/W2 and vice versa ÷We can use0r〉≡|++)and |1L〉≡|---）as the logical states. All the operations needed for encoding,error-detection,and recovery-are performed just as for the bit flip channel
Phase flip channel: leaves the qubits untouched with probability , and flips the phase ( ) with probability ◦ The state is taken to ◦ Therefore, the error takes to and vice versa We can use and as the logical states. All the operations needed for encoding, error-detection, and recovery – are performed just as for the bit flip channel