Efficiently learning fermionic unitaries with few non-Gaussian gates

Fermionic Gaussian unitaries are known to be efficiently learnable and simulatable. In this paper, we present a learning algorithm that learns an -mode circuit containing  parity-preserving non-Gaussian gates. While circuits with  are unlikely to be efficiently learnable, for constant , we present a polynomial-time algorithm for learning the description of the unknown fermionic circuit within a small diamond-distance error. Building on work that studies the state-learning version of this problem, our approach relies on learning approximate Gaussian unitaries that transform the circuit into one that acts non-trivially only on a constant number of Majorana operators. Our result also holds for the case where we have a qubit implementation of the fermionic unitary.