Stabilizer testing and central limit theorem

In this paper we introduce the concept of stabilizer convolution and demonstrate that the only convolutions that can be used for stability testing are stabilizer convolutions. Moreover, using any stabilizer convolution yields similarly good results. Through the Choi-Jamiokowski isomorphism, this method can also be applied to Clifford testing for quantum gates. Leveraging the invariance of circuit magic classes of mean states under stabilizer self-convolution, we formulate a central limit theorem for circuit magic classes of quantum states. Moreover, employing the Choi-Jamiokowski correspondence, we generalize this finding to establish a central limit theorem for circuit magic classes of completely positive and trace-preserving channels.