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In this paper, we reformulate the non-convex $\ell_q$-norm minimization problem with $q\in(0,1)$ into a 2-step problem, which consists of one convex and one non-convex subproblems, and propose a novel iterative algorithm called QISTA ($\ell_q$-ISTA) to solve the $\left(\ell_q\right)$-problem. By taking advantage of deep learning in accelerating optimization algorithms, together with the speedup strategy that using the momentum from all previous layers in the network, we propose a learning-based method, called QISTA-Net-s, to solve the sparse signal reconstruction problem. Extensive experimental comparisons demonstrate that the QISTA-Net-s yield better reconstruction qualities than state-of-the-art $\ell_1$-norm optimization (plus learning) algorithms even if the original sparse signal is noisy. On the other hand, based on the network architecture associated with QISTA, with considering the use of convolution layers, we proposed the QISTA-Net-n for solving the image CS problem, and the performance of the reconstruction still outperforms most of the state-of-the-art natural images reconstruction methods. QISTA-Net-n is designed in unfolding QISTA and adding the convolutional operator as the dictionary. This makes QISTA-Net-s interpretable. We provide complete experimental results that QISTA-Net-s and QISTA-Net-n contribute the better reconstruction performance than the competing.