Learning the stabilizer group of a Matrix Product State

TL;DR

This paper introduces a classical algorithm to efficiently learn the stabilizer group of a Matrix Product State, enabling analysis of quantum states' properties with high accuracy and scalability.

Contribution

The authors develop the first effective method to determine the stabilizer group of MPS, leveraging biased sampling in the Pauli basis with favorable scaling.

Findings

Accurately estimates stabilizer groups for highly-entangled MPS.

Demonstrates effectiveness on T-doped states scrambled via Clifford dynamics.

Scales as O(χ^3), suitable for large bond dimensions.

Abstract

We present a novel classical algorithm designed to learn the stabilizer group -- namely the group of Pauli strings for which a state is a $\pm 1$ 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 $\chi\sim 10^3$. Our method, thanks to a very favourable scaling $\mathcal{O}(\chi^3)$, represents the first effective approach to obtain a genuine magic monotone for MPS, enabling systematic investigations of quantum many-body physics out-of-equilibrium.