Learning Stabilizer Structure of Quantum States

We consider the task of learning a structured stabilizer decomposition of an arbitrary n-qubit quantum state |ψ⟩: for every ε > 0, output a succinctly describable state |φ⟩ with stabilizer-rank poly(1/ε) such that |ψ⟩=|φ⟩+|φ′⟩ where |φ′⟩ has stabilizer fidelity at most ε. We firstly show the existence of such decompositions using the inverse theorem for the Gowers-3 norm of quantum states that was recently established by our prior work [AD, STOC’25].