Block encoding techniques: an explicit quantum circuit for the Heisenberg Hamiltonian

Quantum computing leverages quantum mechanics to solve problems that are intractable for classical computers. A central framework in this domain is Quantum Singular Value Transformation (QSVT), which enables efficient manipulation of matrix operations and underlies many quantum algorithms, including those for solving linear systems and simulating quantum dynamics. Block encoding is a key technique that embeds non-unitary matrices into unitary operators, making them amenable to QSVT. Among block encoding methods, the Linear Combination of Unitaries (LCU) technique is widely used, but its practical utility is limited by high gate overhead—particularly from multi-controlled operations. This thesis introduces a new formulation, FOQCS-LCU, which reduces both practical and asymptotic circuit complexity. By exploiting the structure of physically motivated Hamiltonians, we also develop efficient routines for preparing Dicke states, which are superpositions over basis states with fixed Hamming weight. We demonstrate our method by constructing explicit block encoding circuits for the Heisenberg Hamiltonian, achieving an order-of-magnitude reduction in CNOT gate count compared to standard LCU approaches. Detailed gate counts and numerical benchmarks confirm the efficiency of our technique. This work advances the feasibility of block encoding as a subroutine for large-scale quantum algorithms and supports more efficient implementations on fault-tolerant quantum devices.