This paper introduces a new quantum computing framework integrated with a two-step compressed sensing technique, applied to a joint channel estimation and user identification problem. We propose a variational quantum circuit (VQC) design as a new denoising solution. For a practical grant-free communications system having correlated device activities, variational quantum parameters for Pauli rotation gates in the proposed VQC system are optimized to facilitate to the non-linear estimation. Numerical results show that the VQC method can outperform modern compressed sensing techniques using an element-wise denoiser.