Efficient LCU block encodings through Dicke states preparation

TL;DR

This paper introduces a compact, efficient method for constructing block encodings of Hamiltonians using Dicke states, significantly reducing gate overhead and enabling practical quantum simulations of structured matrices.

Contribution

We develop FOQCS-LCU, a novel LCU formulation requiring fewer ancilla qubits and lower gate counts, with explicit Dicke state preparation routines and constant-depth SELECT oracles.

Findings

Over an order-of-magnitude reduction in CNOT count compared to traditional methods

Explicit block encoding circuits for Heisenberg and spin glass Hamiltonians

Demonstrated practical efficiency through numerical benchmarks

Abstract

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. To address this, we introduce the Fast One-Qubit-Controlled Select LCU (FOQCS-LCU), a compact LCU formulation that requires only a linear number of ancilla qubits and is explicitly decomposed into one- and two-qubit gates. By exploiting the underlying Hamiltonian structure, we design a parametrized family of 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 representative spin models such as the Heisenberg and spin glass Hamiltonians and provide detailed, 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.