Nonstabilizerness of quantum tensor network states is intractable in two dimensions

Abstract: Nonstabilizerness is a necessary resource for quantum systems to lie beyond the classically simulable regime. With the advent of stabilizer $α$-Rényi entropies, nonstabilizerness has also become a many-body diagnostic, complementary to entanglement. While deciding whether an arbitrary quantum state has nonstabilizerness is provably hard, such states already require a description exponentially large in the number of qubits and are thus out of reach for many-body physics. Here we instead consider 2D tensor network (TN) states, which compactly capture states obeying an entanglement area law, and ask whether they admit a simpler algorithm for the same task. In contrast to the 1D case, we prove that, even at small, constant bond dimension, computing the stabilizer entropy of 2D TN states is $# P$ hard for any integer $α\ge 2$, and deciding stabilizer membership is $\rm C_{=} P$ complete. Hence, under standard complexity assumptions, no classical or quantum algorithm with polynomial resources can solve either problem in the worst case. Both hardness results persist even when the problems are formulated up to constant additive accuracy.