Typical Superpositions are Magical

Abstract

The non-stabilizerness, or "magic", of a quantum state is a common metric for the hardness of simulation on a classical computer, and is often quantified by the stabilizer Rényi entropy (SRE). We show how random superpositions of classically simulable states saturate the SRE to the symmetry-restricted Haar-random maximum. For any large-$N$ random pure-state ensemble satisfying the conditions of Schur's lemma, the average SRE of the $m$-component superposition $⟨M_2(m)⟩=N-{\log }_2[⟨W(m)⟩]$ is governed by an average witness $⟨W(m)⟩$ that universally decomposes into a Haar floor and an ensemble-specific excess. As the number of components $m$ increases, the excess is suppressed, resulting in a logarithmic scaling of the SRE before plateauing at the Haar-valued maximum. We derive a general closed form for $⟨W(m)⟩$ in the thermodynamic limit, and numerically verify our result for random stabilizer states and random Gaussian states.

arXiv commentMain text: 7 pages, 4 figures. Supplemental Material: 25 pages, 3 figures