Robust exponential lower bounds for fermionic and bosonic Gaussian ranks
Abstract
The power and limitations of classical simulation are central to understanding quantum computational advantages. A leading simulation paradigm is based on coherent decomposition into classically tractable free states where the decomposition rank determines the simulation complexity. Proving strong lower bounds on this number is a notoriously difficult and mathematically rich problem, as exemplified by the qubit stabilizer rank problem. Here, we study the Gaussian version of this problem in both bosonic and fermionic systems and establish robust exponential lower bounds on Gaussian rank. In particular, we prove that for every pure non-Gaussian state on finitely many modes, with definite parity in the fermionic case, the approximate border Gaussian rank of its tensor powers grows at least exponentially at any fixed norm error below one. Our proofs combine reduction to four modes with Majorana spectral bounds for fermions, and Gaussian postselection with an entropy-based rank bound for bosons. As concrete examples, we derive explicit exponential lower bounds for the four-mode fermionic GHZ state and the bosonic single-photon state. The bosonic results require no assumption on the energy of the target state. Our results show that non-Gaussianity universally entails exponential Gaussian decomposition complexity, setting fundamental limitations on decomposition-based classical simulation of bosonic and fermionic systems.
Related
- Exponential lower bounds on the fermionic Gaussian rank of magic states and the bosonic coherent state rank of Fock states
- Testing bosonic and fermionic Gaussian states with a constant number of copies
- Entangled measurements are necessary for optimal tomography of mixed fermionic Gaussian states and of bosonic Gaussian states near the vacuum
- Optimal testing of fermionic and bosonic Gaussian states
- Exponential de Finetti Theorems for Fermionic Gaussian States
BibTeX
@misc{wei2026robust,
title = {{Robust exponential lower bounds for fermionic and bosonic Gaussian ranks}},
author = {Fuchuan Wei and Kong-Wing Wu and Zhengwei Liu and Zi-Wen Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02172},
eprint = {2610.02172},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02172},
}
RIS
TY - UNPB
TI - Robust exponential lower bounds for fermionic and bosonic Gaussian ranks
AU - Fuchuan Wei
AU - Kong-Wing Wu
AU - Zhengwei Liu
AU - Zi-Wen Liu
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02172
UR - https://arxiv.org/abs/2610.02172
AB - The power and limitations of classical simulation are central to understanding quantum computational advantages. A leading simulation paradigm is based on coherent decomposition into classically tractable free states where the decomposition rank determines the simulation complexity. Proving strong lower bounds on this number is a notoriously difficult and mathematically rich problem, as exemplified by the qubit stabilizer rank problem. Here, we study the Gaussian version of this problem in both bosonic and fermionic systems and establish robust exponential lower bounds on Gaussian rank. In particular, we prove that for every pure non-Gaussian state on finitely many modes, with definite parity in the fermionic case, the approximate border Gaussian rank of its tensor powers grows at least exponentially at any fixed norm error below one. Our proofs combine reduction to four modes with Majorana spectral bounds for fermions, and Gaussian postselection with an entropy-based rank bound for bosons. As concrete examples, we derive explicit exponential lower bounds for the four-mode fermionic GHZ state and the bosonic single-photon state. The bosonic results require no assumption on the energy of the target state. Our results show that non-Gaussianity universally entails exponential Gaussian decomposition complexity, setting fundamental limitations on decomposition-based classical simulation of bosonic and fermionic systems.
ER -