Correction of Memory Errors

In this section I will outline the idea of memory error correction as put forward by Peter Shor and Andrew Steane (Shor 1995, Steane 1996). In classical bits the only kind of error which can happen is a hit flip which changes the logical value from 0 to 1 and vice versa. In qubits a second type of error can happen which we shall call phase errors. A bit flip error in a qubit will give rise to the following 

After a phase error the states representing the logical values 0 and 1 remain unchanged. However, the relative sign between the two basis states will be changed  

If the qubit is prepared in one of the basis states 0) or 1) such a phase error will have no measurable effect. On the other hand, if the qubit is in a superposition state the state after the phase error can be orthogonal to the state before the error as in the following example  

In reality errors take not place as pure bit-flip errors or pure phase errors. However all possible errors in a qubit will be a combination of these two kinds of errors. 

In the theory of classical error correcting codes the strategy is to introduce redundancy (Sloane and MacWilliams 1977). The most trivial example was already given in Eq, (6.23). The triple repetition code has the capacity to store one classical bit and can correct for a single error. This simple strategy does not work in the general quantum case. 

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