SignificanceConvolution has a significant impact on many scientific disciplines, ranging from probability theory and harmonic analysis to information theory. Here, we introduce a framework to study quantum convolution in discrete-variable (DV) quantum systems. We establish a maximal entropy principle and a quantum central limit theorem for DV quantum systems. We also provide a bound on the rate of convergence in the central limit theorem that we call the magic gap. The magic gap has some similarities to the Cheeger constant in graph theory. Our convolutional framework provides an approach to study stabilizer and magic states.