Phase transition in stabilizer entropy and efficient purity estimation
Lorenzo Leone, Salvatore F. E. Oliviero, Gianluca Esposito, Alioscia Hamma
Abstract
Stabilizer entropy (SE) quantifies the spread of a state in the basis of Pauli operators. It is a computationally tractable measure of nonstabilizerness and thus a useful resource for quantum computation. SE can be moved around a quantum system, effectively purifying a subsystem from its complex features. We show that there is a phase transition in the residual subsystem SE as a function of the density of non-Clifford resources. This phase transition has important operational consequences: it marks the onset of a subsystem purity estimation protocol that requires $poly(n)exp(t)$ many queries to a circuit containing $t$ non-Clifford gates that prepares the state from a stabilizer state. Then, for $t=O({log}_{2}n)$, it estimates the purity with polynomial resources, and, for highly entangled states, attains an exponential speed-up over the known state-of-the-art algorithms.