Efficient LCU block encodings through Dicke states preparation

With the Quantum Singular Value Transformation (QSVT) emerging as a unifying framework for diverse quantum speedups, the efficient construction of block encodings—their fundamental input model—has become increasingly crucial. However, devising explicit block encoding circuits remains a significant challenge. A widely adopted strategy is the Linear Combination of Unitaries (LCU) method. While general, its practical utility is often limited by substantial gate overhead. We introduce the Fast One-Qubit-Controlled Select LCU (FOQCS-LCU), a compact LCU formulation that requires only a linear number of ancillas and is explicitly decomposed into 1- and 2-qubit gates. By exploiting the underlying Hamiltonian structure, we design efficient Dicke state preparation routines, enabling systematic realization of the state preparation oracle at substantially reduced gate cost. The check matrix formalism further yields a constant-depth SELECT oracle, implemented as two fully parallelizable layers of singly controlled Pauli gates. We construct explicit block encoding circuits for spin models such as Heisenberg and spin glass Hamiltonians and provide non-asymptotic gate counts. Our numerical benchmarks confirm the efficiency of the FOQCS-LCU approach, illustrating over an order-of-magnitude reduction in CNOT count compared to conventional LCU. This framework opens a pathway toward practical, low-depth block encodings for a broad class of structured matrices beyond those considered here.