Quantum Error Correction for the Toric Code

In this thesis, the toric code error correction process, one method to check a quantum memory storage for errors and correcting them, is analysed theoretically. The focus was set on the determination of the accuracy threshold, the upper limit pcrit of the probability with which the manifestation of errors can be allowed during consecutive measurements for errors, so that the non-erroneous state of the quantum memory storage can be restored. To do this, an analogy between the toric code error correction model (TCECM) and the random-bond Ising model (RBIM) was used. In this analogy, the probability of error creation pEC in the TCECM is equivalent to the probability to have an antiferromagnetic bond p AFB in the RBIM. The behaviour of the TCECM with increasing pEC corresponds to the behaviour of the RBIM along the so called Nishimori line with increasing p AFB . The two phases of the TCECM, where the non-erroneous state can be restored respectively cannot, correspond to the ferromagnetic respectively paramagnetic phase of the RBIM. In the RBIM, the transition between these two phases along the Nishimori line occurs at the so called Nishimori point. Thus, by numerically simulating the RBIM along this Nishimori line and determining the position of the phase transition, the accuracy threshold pcrit of the TCECM can be determined. The two different variants of the algorithm that were used yielded pcrit=0.1081±0.0007 respectively pcrit=0.108±0.001 as a result for this accuracy threshold. These values deviate slightly from reference values found in the literature. The reason for this remains somewhat unclear, but likely sources were determined and propositions made on how to get rid of them by improving the method. Table of content