Quantum State Designs with Clifford-Enhanced Matrix Product States

Abstract

Nonstabilizerness, or "magic,"is a critical quantum resource that, together with entanglement, characterizes the nonclassical complexity of quantum states. Here, we address the problem of quantifying the average nonstabilizerness of random matrix product states (RMPSs). RMPSs represent a generalization of random product states featuring bounded entanglement that scales logarithmically with the bond dimension χ. We demonstrate that the stabilizer Rényi entropies converge to that of Haar-random states as N/χα, where N is the system size and the α are integer exponents. This indicates that MPSs with a modest bond dimension are as magical as generic states. Subsequently, we introduce the ensemble of Clifford-enhanced matrix product states (CMPSs), built by the action of Clifford unitaries on RMPSs. Leveraging our previous result, we show that CMPSs can approximate quantum state 4-designs with arbitrary accuracy. Specifically, for a constant N, CMPSs become close to 4-designs, with a scaling as χ-2. Our findings indicate that combining Clifford unitaries with polynomially complex tensor-network states can generate highly nontrivial quantum states.