This report offers a comprehensive analysis of the evolving landscape of quantum algorithm software specifically tailored for condensed matter physics. It examines fundamental quantum algorithms such as Variational Quantum Eigensolver (VQE), Quantum Phase Estimation (QPE), Quantum Annealing (QA), Quantum Approximate Optimization Algorithm (QAOA), and Quantum Machine Learning (QML) as applied to key condensed matter problems including strongly correlated systems, topological phases, and quantum magnetism. This review details leading software development kits (SDKs) like Qiskit, Cirq, PennyLane, and Q\#, and profiles key academic, commercial, and governmental initiatives driving innovation in this domain. Furthermore, it assesses current challenges, including hardware limitations, algorithmic scalability, and error mitigation, and explores future trajectories, anticipating new algorithmic breakthroughs, software enhancements, and the impact of next-generation quantum hardware. The central theme emphasizes the critical role of a co-design approach, where algorithms, software, and hardware evolve in tandem, and highlights the necessity of standardized benchmarks to accelerate progress towards leveraging quantum computation for transformative discoveries in condensed matter physics.

Recent advances in Kitaev materials have highlighted their potential to host Majorana fermions without or high-temperature of superconductivity. In this research, we propose $Li_2IrO_3$ as a promising High-temperature superconducting platform supporting Majorana edge modes due to its strong spin-orbit coupling, honeycomb lattice structure, and proximity to a quantum spin liquid (QSL) phase. A theoretical and numerical framework based on the Kitaev-Heisenberg Hamiltonian is developed to model spin interactions in $Li_2IrO_3$. Here, the existence of topological zero-energy states is demonstrated, and their signatures in the edge-localized spectral weight are identified. A device concept based on this material is also proposed with potential industrial applications in spintronics, magnetic field sensing, and topological quantum memory.