Magic of random matrix product states

Magic, or nonstabilizerness, characterizes how far away a state is from the stabilizer states, making it an important resource in quantum computing, under the formalism of the Gotteman-Knill theorem. In this paper, we study the magic of the one-dimensional (1D) random matrix product states (RMPSs) using the L₁-norm measure. We first relate the L₁ norm to the L₄ norm. We then employ a unitary four-design to map the L₄ norm to a 24-component statistical physics model. By evaluating partition functions of the model, we obtain a lower bound on the expectation values of the L₁ norm. This bound grows exponentially with respect to the qudit number n, indicating that the 1D RMPS is highly magical. Our numerical results confirm that the magic grows exponentially in the qubit case.