Classical simulation of free-fermionic dynamics and quantum chemistry with magic input

Establishing the precise computational boundary between classically tractable fermionic systems and those capable of genuine quantum advantage is a central challenge in quantum simulation. While injecting non-Gaussian "magic" inputs into free-fermion circuits is widely expected to generate intractable complexity, we identify a physically motivated intermediate regime. Supported by rigorous bounds and numerical evidence, we show that for a class of paired non-Gaussian fermionic states, essential quantum simulation primitives - transition amplitudes, overlaps, and arbitrary-weight number correlators - can be efficiently approximated to additive error under free-fermionic dynamics. This tractability stems from an algebraic reduction that compresses exponentially large multiparticle interference into a single coefficient of a multivariate Pfaffian polynomial. Because these classical estimators match the intrinsic O\(1/sqrt{K}\) statistical uncertainty of quantum hardware utilizing K measurement shots, they constitute a practical benchmark. Building on this foundation, we construct an additive-error estimator for high-weight Wilson observables in the noninteracting quench of recent trapped-ion experiments, providing a rigorous classical benchmark. Extending this to quantum chemistry, we demonstrate that core overlap-based subroutines for antisymmetrized products of strongly orthogonal geminals admit exact Pfaffian reductions. Ultimately, these results sharpen the boundary of quantum advantage, establishing that the paired-electron scaffold is effectively dequantized and clarifying exactly where quantum resources are indispensable.