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arXiv:2412.06145

Optimizing Qubit Mapping with Quasi-Orthogonal Space-Time Block Codes and Quaternion Orthogonal Designs

Valentine Nyirahafashimana, Nurisya Mohd Shah, Umair Abdul Halim, Мohamed Othman, Sharifah Kartini Said Husain

Published Dec 9, 2024• 0 citations
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Abstract

This study explores the qubit mapping through the integration of Quasi-Orthogonal Space-Time Block Codes (QOSTBCs) with Quaternion Orthogonal Designs (QODs) in quantum error correction (QEC) frameworks. QOSTBCs have gained prominence for enhancing performance and reliability in quantum computing and communication systems. These codes draw on stabilizer group formalism and QODs to boost error correction, with QOSTBCs mapping logical qubits to physical ones, refines error handling in complex channels environments. Simulations results demonstrate the effectiveness of this approach by comparing the percentage improvement under various detected and corrected error conditions for four different cases, \textbf{$Z_1$} up to \textbf{$Z_4$}. The obtained simulations and implemental results show that QOSTBCs consistently achieve a higher correction improvement percentage than stabilizer Group for \textbf{$Z_1$}, \textbf{$Z_2$}, and \textbf{$Z_4$}; QOSTBCs can correct more errors than those detected, achieving over 100\% correction rates for first two cases, which indicates their enhanced resilience and redundancy in high-error environments. While for \textbf{$Z_3$}, stabilizer consistently remains above that of QOSTBCs, reflecting its slightly better performance. These outcomes indicate that QOSTBCs are reliable in making better logarithmic efficiency and error resilience, making them a valuable asset for quantum information processing and advanced wireless communication.

DOI
10.48550/arxiv.2412.06145

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This study explores the qubit mapping through the integration of Quasi-Orthogonal Space-Time Block Codes (QOSTBCs) with Quaternion Orthogonal Designs (QODs) in quantum error correction (QEC) frameworks. QOSTBCs have gained prominence for enhancing performance and reliability in quantum computing and communication systems. These codes draw on stabilizer group formalism and QODs to boost error correction, with QOSTBCs mapping logical qubits to physical ones, refines error handling in complex channels environments. Simulations results demonstrate the effectiveness of this approach by comparing the percentage improvement under various detected and corrected error conditions for four different cases, \textbf{Z1Z_1Z1​} up to \textbf{Z4Z_4Z4​}. The obtained simulations and implemental results show that QOSTBCs consistently achieve a higher correction improvement percentage than stabilizer Group for \textbf{Z1Z_1Z1​}, \textbf{Z2Z_2Z2​}, and \textbf{Z4Z_4Z4​}; QOSTBCs can correct more errors than those detected, achieving over 100% correction rates for first two cases, which indicates their enhanced resilience and redundancy in high-error environments. While for \textbf{Z3Z_3Z3​}, stabilizer consistently remains above that of QOSTBCs, reflecting its slightly better performance. These outcomes indicate that QOSTBCs are reliable in making better logarithmic efficiency and error resilience, making them a valuable asset for quantum information processing and advanced wireless communication.
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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 XXX-gate to detect and correct errors, as well as the Hadamard HHH-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.
Quantum error correction (QEC) is a crucial aspect of realizing fault-tolerant quantum computation. However, existing QEC codes often suffer from inefficiencies, including excessive qubit overhead and vulnerability to error propagation. This paper proposes a novel approach to QEC code design leveraging topological optimization. The core idea is to optimize the topological structure of the QEC code itself, aiming to minimize redundant encoding and reduce the probability of error propagation. We introduce a framework based on topological principles to systematically explore and refine the topology of QEC codes. This optimization process focuses on identifying and eliminating unnecessary connections within the code, resulting in a more resilient and efficient error correction scheme. The theoretical analysis demonstrates that optimized topological structures can significantly improve the code's performance metrics, such as the quantum error correction threshold and the logical error rate. This work represents a fundamental shift in the design of QEC codes, moving beyond traditional approaches to incorporate the inherent structure of the code as a key element in enhancing its performance.
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