Quantum Gate Error Correction

In this section I will discuss a recently proposed error correction scheme designed particularly for the ion trap QC (Cirac et al. 1996). The scheme corrects for an important source of errors during the execution of 2-bit quantum gates. Because the scheme is not intended to correct for the most general error it can be implemented efficiently with regard to time and memory overhead. It is likely that this scheme can be tested as soon as a prototype ion trap QC is available. 

We will consider a simple model for the errors in the CM phonon mode. This model is not necessarily realistic, but it allows for a simple explanation of the scheme. Moreover, the scheme can be generalized to more realistic errors. In the simplest error model only decay of the CM phonon mode is included. If an error takes place the motional quantum number is decreased by one. In particular, the first excited motional state 1)cm decays to the ground state 0)cm7 whereas the ground state amplitude vanishes. Thus after an error in the CM phonon mode a superposition of the two energetically lowest states undergo the following transformation (phonon 

6 Quantum Computers First Steps Towards a Realization 

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A physical mechanism to perform reliable measurements on the qubits. This is neccessary for several reasons. Firstly, we need to read out the final result after a quantum computation. Secondly, the error correction schemes discussed in Sect. 6.8.1 requires to perform state measurements during the computation and apply quantum gates conditioned on the outcome of these measurements. Finally, in order to initialize the quantum register before the computation it is necessary to bring the qubits of the QC into a well-defined state by measurement of all qubits. Note that measurements require a strong coupling of the quantum system to the environment. This is in contrast to the first requirement. From these two requirements it follows that we need a means to switch on and off the interaction of any arbitrary qubit to the environment deliberately. 

Fig. 6.14 Error correction of single bit flips. A qubit is encoded redundantly in three. An error is detected by computing the error syndrome in a ancillary quantum system. The syndrome is measured and an appropriate quantum gate is applied to the code word to reconstruct the original state. A similar procedure is applied for correcting general errors.

Fig. 6.15 Error correction of quantum gates. After the erroneous gate an appropriate measurement is perfomed to determine if an error has taken place or not and the initial state I in) is reconstructed if neccessary.

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