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

TL;DR

This paper identifies a class of fermionic quantum states for which key quantum simulation tasks can be efficiently approximated classically, clarifying the boundary of quantum advantage in fermionic systems.

Contribution

It introduces a classical approximation method for certain non-Gaussian fermionic states, extending classical simulability to specific quantum chemistry and quantum simulation tasks.

Findings

Efficient classical estimators for transition amplitudes and correlators in non-Gaussian fermionic states.

Constructed a classical benchmark for trapped-ion quantum experiments.

Demonstrated classical simulability of core overlap calculations in quantum chemistry.

Abstract

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. We prove that for block-product 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 efficient additive-error Pfaffian-kernel estimators. Ultimately, these results sharpen the boundary of quantum advantage, establishing that the paired-electron scaffold is dequantized and clarifying where quantum resources are indispensable.