Magic of random matrix product states
Liyuan Chen, Roy J. Garcia, Kaifeng Bu, Arthur Jaffe
Abstract
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 ${L}_{1}$-norm measure. We first relate the ${L}_{1}$ norm to the ${L}_{4}$ norm. We then employ a unitary four-design to map the ${L}_{4}$ 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 ${L}_{1}$ norm. This bound grows exponentially with respect to the qudit number $n$, indicating that the $1\mathrm{D}$ RMPS is highly magical. Our numerical results confirm that the magic grows exponentially in the qubit case.