  We present NFNet, a PyTorch-based framework for polynomial-time simulation of
large-scale, continuously controlled quantum systems, supporting parallel
matrix computation and auto-differentiation of network parameters. It is based
on the non-interacting Fermionic formalism that relates the Matchgates by
Valiant to a physical analogy of non-interacting Fermions in one dimension as
introduced by Terhal and DiVincenzo. Given an input bit string
$\boldsymbol{x}$, NFNet computes the probability
$p(\boldsymbol{y}|\boldsymbol{x})=\langle x|U_{\theta}^\dagger \Pi_y U_\theta
|x\rangle$ of observing the bit string $\boldsymbol{y}$, which can be a sub or
full-system measurement on the evolved quantum state
$U_{\mathbf{\theta}}|x\rangle$, where $\mathbf{\theta}$ is the set of
continuous rotation parameters, and the unitary $U_{\mathbf{\theta}}$'s
underlying Hamiltonians are not restricted to nearest-neighbor interactions. We
first review the mathematical formulation of the Matchgate to Fermionic mapping
with additional matrix decomposition derivations, and then show that on top of
the pair-wise circuit gates documented in Terhal and DiVincenzo, the Fermionic
formalism can also simulate evolutions whose Hamiltonians are sums of arbitrary
two-Fermion-mode interactions. We then document the design philosophy of NFNet,
its software structure, and demonstrate its usage in various quantum system
simulation, benchmarking, and quantum learning tasks involving 512+ qubits. As
NFNet is both an efficient large-scale quantum simulator, and a
quantum-inspired classical computing network structure, many more exciting
topics are worth exploring, such as its connection to recurrent neural
networks, discrete generative learning and discrete normalizing flow. NFNet
source code can be found at https://github.com/BILLYZZ/NFNet.
