Research Projects

The quantum realm is inherently fragile, where quantum states are susceptible to decoherence and operational errors caused by environmental noise and imperfect quantum gates. Effective quantum error-correcting codes (QECCs) are essential to protect quantum information, enabling quantum systems to perform complex computations with high fidelity. This proposal advances four interrelated frameworks to deliver more efficient, scalable, and fault-tolerant quantum systems. 1. Quantum Locally Recoverable Codes (QLRCs): Building on the CSS-based constructions, we will optimize rate-distance locality trade offs, aiming to approach Singleton-like bounds. We will compare these trade-offs against QLDPC families and explore alternative constructions via Hermitian inner products. Entanglement-assisted QLRCs will be defined, leveraging classical LRCs without the dual containing constraint to achieve locality. Finally, we will design quantum (r,δ)-LRCs based on cyclic classical codes, establishing explicit dual containing conditions for optimal locality. 2. Quantum Quasi Cyclic LDPC Codes (QQLDPCs): Quasi cyclic QLDPCs offer sparse parity checks and favorable decoding complexity. We will analyze Tanner graph trapping sets and girth properties—extending beyond existing girth 8 and girth 12 constructions—to identify structures that degrade performance. By systematically characterizing small trapping sets and optimizing girth across diverse QLDPC families, we will propose new code constructions with enhanced decoding thresholds and resilience under realistic noise models. 3. CSS T Codes over Finite Fields: CSS T codes admit transversal T gates, which are crucial for fault-tolerant logical operations. We will first derive a comprehensive algebraic characterization of CSS T codes over arbitrary finite fields, clarifying necessary Clifford transversal conditions. Leveraging quasi-cyclic classical code frameworks, we will then construct asymptotically good CSS T families that meet or approach the Gilbert–Varshamov bound. 4. Automorphism-Based Synthesis: Code automorphisms—symmetries preserving stabilizer structure—can simplify decoding and logical gate implementation. We will classify stabilizer codes whose Clifford automorphism groups are isomorphic to large permutation groups (e.g., S_n, A_n, Mathieu groups), and study codes with k-transitive or cyclic automorphism actions. For codes with trivial or cyclic automorphism groups, we will investigate the existence of families achieving arbitrarily large distances. By unifying group theory, combinatorial graph analysis, algebraic geometry approach, and quantum computation and information techniques, this research will yield: optimal local recovery code families, automorphism group characterizations, and trade-off bounds. Together, these advances will deepen foundational understanding and deliver practical tools, accelerating progress toward robust, resource-efficient, and fault-tolerant quantum systems.