The fermionic linear optical extent is multiplicative for 4 qubit parity eigenstates

The Fermionic linear optical (FLO) extent is a quantity that serves two roles, firstly it serves as a measure of the "quantumness" (or non-classicality) of quantum circuits. Secondly it controls the runtime of a class of classical simulation algorithms, which are state-of-the-art for simulating quantum circuits formed mostly of FLO unitaries and promoted to universality by the addition of "magic states". It is therefore interesting to understand the scaling behaviour of the extent as magic states are added to a circuit. In this work we solve this problem for the case of 4-qubit parity eigenstates. We show that the FLO extent of a tensor product of any pure state and a 4 qubit parity eigenstate is the product of the extents of the two tensor factors. Applying this result recursively one proves a conjecture that the extent is multiplicative for arbitrary tensor products of 4 qubit magic states.