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 ⟨ M2(m)⟩=N−log2[⟨ 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.
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BibTeX
@misc{heath2026typical,
title = {{Typical Superpositions are Magical}},
author = {Joshuah T. Heath and Andrew Urilyon and Damien Quinn and Graham Kells},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22476},
eprint = {2610.22476},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22476},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Typical Superpositions are Magical
AU - Joshuah T. Heath
AU - Andrew Urilyon
AU - Damien Quinn
AU - Graham Kells
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22476
UR - https://arxiv.org/abs/2610.22476
AB - 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.
ER -