Quantum lattice Boltzmann methods (QLBM) provide a promising framework for solving kinetic formulations of transport and fluid flow equations on quantum computers. However, a major bottleneck in near-term implementations is the extraction of macroscopic quantities, such as fluid density and velocity, which is commonly performed via quantum state tomography and scales exponentially with the number of qubits. This severely limits the practical applicability of QLBM on noisy intermediate-scale quantum (NISQ) devices. In this work, we apply a measurement-efficient QLBM formulation for the one-dimensional advection--diffusion equation on a D1Q3 lattice. The macroscopic density field is obtained directly from quantum measurement statistics using sampler-based expectation values, avoiding full density-matrix reconstruction. Whereas quantum state tomography of an n-qubit QLBM state requires 3^n distinct measurement circuits, the sampler-based approach extracts macroscopic observables from a single computational-basis measurement circuit. As a result, the measurement cost scales only with the number of shots needed to achieve a desired statistical accuracy. We consider the evolution of a Gaussian density hill under periodic boundary conditions and implement the QLBM collision and streaming operators as quantum circuits using Qiskit. Accuracy and efficiency are evaluated by comparing sampler-based measurements with full state tomography on the Aer simulator, and by running the same circuits on a 156-qubit IBM Heron processor. The results show that sampler-based reconstruction faithfully reproduces the expected advection–diffusion dynamics while achieving exponential reductions in measurement cost. On real hardware, physically consistent transport behavior is observed despite device noise, with errors decreasing systematically as the number of shots increases. These findings demonstrate that sampler-based measurement enables scalable QLBM simulations on NISQ hardware and provides a practical pathway toward quantum-accelerated computational mechanics.