Block encoding of sparse matrices is a foundational primitive underlying quantum algorithms such as Quantum Singular Value Transformation (QSVT), Hamiltonian simulation, and quantum linear system solvers. However, translating abstract block-encoding constructions into efficient, hardware-aware gate-level circuits—especially for arbitrary sparse matrices—remains a significant challenge. In this work, we present a unified framework that bridges this gap by addressing key practical obstacles including multi-controlled gate overhead, amplitude reordering, and limited hardware connectivity. Our approach introduces a novel connection to combinatorial optimization for systematic assignment of control qubits, enabling nearest-neighbor implementations, along with coherent permutation operators that preserve quantum superposition while enabling efficient amplitude reordering. These techniques yield explicit, depth-efficient gate-level block encodings for arbitrary sparse and structured matrices. Building on this framework, we develop a systematic pathway for solving differential equations within the quantum linear systems paradigm by combining block encoding with QSVT. We demonstrate the approach on tridiagonal linear systems and extend it to applications in computational fluid dynamics, including the heat equation with mixed boundary conditions and the nonlinear Burgers’ equation. Through a detailed scaling analysis of the heat equation, we show how spatial discretization impacts the minimum singular value and, consequently, the polynomial degree and circuit depth required for QSVT, identifying depth as a primary bottleneck. For Burgers’ equation, we illustrate how Carleman-linearized nonlinear dynamics can be efficiently block encoded and solved within the same framework. Together, these results highlight both the practical feasibility and current limitations of quantum linear system methods, emphasizing the need for depth-reduction strategies, efficient estimation of minimum singular values, and rigorous benchmarking against classical methods. Our work advances block encoding from a theoretical abstraction toward a practical tool for large-scale quantum simulation and scientific computing.