发表机构
Yau Mathematical Sciences Center, Tsinghua University; Qiuzhen College, Tsinghua University; Department of Mathematics, Tsinghua University; Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（清华大学丘成桐数学科学中心; 清华大学求真学院; 清华大学数学系; 北京雁栖湖应用数学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明费米子和玻色子非高斯态的张量幂的高斯秩在任意固定误差下至少指数增长，为基于分解的经典模拟设置了根本性限制。
AI 中文摘要
经典模拟的能力与局限是理解量子计算优势的核心问题。一种领先的模拟范式基于将态相干分解为经典可处理的自由态，其中分解秩决定了模拟复杂度。证明该数的强下界是一个众所周知困难且数学上丰富的问题，例如量子比特稳定子秩问题。本文研究了玻色子和费米子系统中该问题的高斯版本，并建立了高斯秩的鲁棒指数下界。特别地，我们证明对于有限模式上的每个纯非高斯态（在费米子情形中具有确定宇称），其张量幂的近似边界高斯秩在任意固定范数误差低于1时至少呈指数增长。我们的证明结合了费米子的约化到四模式与马约拉纳谱界，以及玻色子的高斯后选择与基于熵的秩界。作为具体例子，我们推导了四模式费米子GHZ态和玻色子单光子态的显式指数下界。玻色子结果不需要对目标态的能量作任何假设。我们的结果表明，非高斯性普遍导致指数级高斯分解复杂度，为基于分解的玻色子和费米子系统经典模拟设置了根本性限制。
英文摘要
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.
