Unveiling the Stabilizer Group of a Matrix Product State

• Published in: Physical review letters 2024-07, Vol.133 (1), p.010602, Article 010602
• Main Authors: Lami, Guglielmo, Collura, Mario
• Format: Article
• Language: English
• ISSN: 0031-9007; 1079-7114; 1079-7114
• DOI: 10.1103/PhysRevLett.133.010602

We present a novel classical algorithm designed to learn the stabilizer group-namely, the group of Pauli strings for which a state is a ±1 eigenvector-of a given matrix product state (MPS). The algorithm is based on a clever and theoretically grounded biased sampling in the Pauli (or Bell) basis. Its output is a set of independent stabilizer generators whose total number is directly associated with the stabilizer nullity, notably a well-established nonstabilizer monotone. We benchmark our method on T-doped states randomly scrambled via Clifford unitary dynamics, demonstrating very accurate estimates up to highly entangled MPS with bond dimension χ∼10^{3}. Our method, thanks to a very favorable scaling O(χ^{3}), represents the first effective approach to obtain a genuine magic monotone for MPS, enabling systematic investigations of quantum many-body physics out of equilibrium.