In this thesis we numerically investigate the performance of logical measurements on the toric quantum error correcting code using the ancilla construction of Cohen et al.[1], both with and without the morphing circuit design technique. We simulate the measurement of the logical Z1 observable on the toric code under circuit-level noise using the STIM package, exploring the effects of the number of ancilla layers and QEC rounds on the logical measurement and idle error rates. We find that for the toric code, a single dual layer performs best and that additional layers only increase the measurement error rate without improving the idle error rate. Regarding the number of QEC rounds, we find that performing at least as many rounds as the code distance is necessary to ensure the measurement error rate scales with the code distance rather than the weight of low-order measurement errors over different QEC rounds. Beyond this threshold, additional rounds improve the measurement error rate at the cost of a higher idle error rate, presenting a tunable trade-off. Finally, we compare the morphing and standard versions of the circuit. Despite the morphing version using half as many physical qubits, it achieves better logical measurement error rates and similar idle error rates relative to the standard circuit, leading us to conclude that the morphing version outperforms its standard counterpart.

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In this paper we will naturally extend the concept of Fourier analysis to functions on arbitrary groups. We will generalise the idea of a convolution and try to find a formula for Fourier coefficients in such way that the coefficients of the convolution can easily be calculated. In the first section we will start off in familiar territory as we work our way through the Abelian groups. On the cyclic groups the comparison with the torus and the Fourier series is easily made and this enables us to easily copy the functions from the Fourier series and use them on our group. We then expand this idea by comparing the other groups to Fourier series on multiple variables. Here we can again copy the functions over and after some calculations we end up with our desired theorems. Then we will continue working on groups in general but sadly for the non-Abelian groups the idea of comparing it to the Fourier series does not work. To remedy this problem we will introduce representations, homomorphisms between the group and invertible matrices. After introducing the concept of a representation we will show some remarkable theorems from Representation theory, such as Maschke’s theorem and Schur’s lemma. With the help of these theorems we can find the irreducible representations, whose matrix entries from an orthogonal basis. These representations are what we will use to transform the convolution into matrix multiplication. In the last chapter we will go into more specifics on the representations of the symmetric group. The representations on this group can be found with the help of the Young tableaux. Among these tableaux we will find the Specht Modules, on which the group action of Sn action will give rise to the irreducible representations. To conclude we will show how to turn these irreducible representations of the symmetric group into matrices. ... Read more