  We describe a matrix product state (MPS) extension for the Fermionic Quantum
Emulator (FQE) software library. We discuss the theory behind symmetry adapted
matrix product states for approximating many-body wavefunctions of spin-1/2
fermions, and we present an open-source, MPS-enabled implementation of the FQE
interface (MPS-FQE). The software uses the open-source pyblock3 and block2
libraries for most elementary tensor operations, and it can largely be used as
a drop-in replacement for FQE that allows for more efficient, but approximate,
emulation of larger fermionic circuits. Finally, we show several applications
relevant to both near-term and fault-tolerant quantum algorithms where
approximate emulation of larger systems is expected to be useful:
characterization of state preparation strategies for quantum phase estimation,
the testing of different variational quantum eigensolver Ans\"atze, the
numerical evaluation of Trotter errors, and the simulation of general quantum
dynamics problems. In all these examples, approximate emulation with MPS-FQE
allows us to treat systems that are significantly larger than those accessible
with a full statevector emulator.
