Transforming qubits via quasi-geometric approaches

We develop a theory based on quasi-geometric (QG) approach to transform a small number of qubits into a larger number of error-correcting qubits by considering four different cases. More precisely, we use the 2-dimensional quasi-orthogonal complete complementary codes (2D-QOCCCSs) and quasi-cyclic asymmetric quantum error-correcting codes (AQECCs) via quasigroup and group theory properties. We integrate the Pauli $X$-gate to detect and correct errors, as well as the Hadamard $H$-gate to superpose the initial and final qubits in the quantum circuit diagram. We compare the numerical results to analyze the success, consistency, and performance of the corrected errors through bar graphs for 2D-QOCCCs and AQECCs according to their characteristics. The difficulty in generating additional sets of results and counts for AQECCs arises because mapping a smaller initial number of qubits to a larger final number is necessary to correct more errors. For AQECCs, the number of errors that can be corrected must be equal to or less than the initial number of qubits. High error correction performance is observed when mapping 1-qubit state to 29-qubits to correct 5 errors using 2D-QOCCCs. Similarly, transforming 1-qubit to 13-qubits using AQECCs also shows high performance, successfully correcting 2 errors. The results show that our theory has the advantage of providing a basis for refining and optimizing these codes in future quantum computing applications due to its high performance in error correction.