Quantum State Designs with Clifford Enhanced Matrix Product States

Nonstabilizerness, or `magic', is a critical quantum resource that, together with entanglement, characterizes the non-classical complexity of quantum states. Here, we address the problem of quantifying the average nonstabilizerness of random Matrix Product States (RMPS). RMPS represent a generalization of random product states featuring bounded entanglement that scales logarithmically with the bond dimension. We demonstrate that the 2-Stabilizer R\'enyi Entropy converges to that of Haar random states as N/², where N is the system size. This indicates that MPS with a modest bond dimension are as magical as generic states. Subsequently, we introduce the ensemble of Clifford enhanced Matrix Product States (CMPS), built by the action of Clifford unitaries on RMPS. Leveraging our previous result, we show that CMPS can approximate 4-spherical designs with arbitrary accuracy. Specifically, for a constant N, CMPS become close to 4-designs with a scaling as ^-2. Our findings indicate that combining Clifford unitaries with polynomially complex tensor network states can generate highly non-trivial quantum states.