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

TL;DR

This paper proves that the fermionic linear optical extent is multiplicative for tensor products of 4-qubit parity eigenstates, providing insights into the quantumness measure and classical simulation complexity of quantum circuits.

Contribution

It establishes the multiplicativity of the FLO extent for tensor products of 4-qubit parity eigenstates, confirming a key conjecture in quantum circuit complexity.

Findings

FLO extent of a tensor product of states equals the product of their extents.

Proves the multiplicativity conjecture for 4-qubit magic states.

Provides a tool for analyzing the classical simulation of quantum circuits.

Abstract

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.