  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.
