标题： 测量费米子中的非高斯魔术：卷积、熵以及Wick定理和Matchgate恒等式的破坏 显示英文标题

标题： Measuring Non-Gaussian Magic in Fermions: Convolution, Entropy, and the Violation of Wick's Theorem and the Matchgate Identity

摘要： 经典难以模拟的量子态，或称“魔法态”，是实现量子优势的前提，突显了经典可解问题与量子可解问题之间的明显区别。 经典可模拟的态，如稳定态上的 Clifford 电路、自由玻色态、自由费米子和匹配门电路，在某种意义上都是高斯态。 虽然自由玻色子和费米子来源于二次哈密顿量，但近期的研究表明，玻色系统和多维子系统在卷积下会趋近于高斯态和稳定态。 在本工作中，我们同样为费米子确定了卷积，并找到了纯费米态中非高斯魔法的有效度量方法。 我们证明了费米系统的一个中心极限定理，从而得出三种自然的态高斯化的概念（1）具有相同协方差矩阵的高斯态，（2）卷积的不动点，（3）相对熵意义下最近的高斯态）是相同的。 随后，我们利用 Wicks 定理的破坏和匹配门恒等式来量化非高斯魔法，除此之外还使用了 SWAP 测试。 显示英文摘要

摘要： Classically hard to simulate quantum states, or "magic states", are prerequisites to quantum advantage, highlighting an apparent separation between classically and quantumly tractable problems. Classically simulable states such as Clifford circuits on stabilizer states, free bosonic states, free fermions, and matchgate circuits are all in some sense Gaussian. While free bosons and fermions arise from quadratic Hamiltonians, recent works have demonstrated that bosonic and qudit systems converge to Gaussians and stabilizers under convolution. In this work, we similarly identify convolution for fermions and find efficient measures of non-Gaussian magic in pure fermionic states. We demonstrate that three natural notions for the Gaussification of a state, (1) the Gaussian state with the same covariance matrix, (2) the fixed point of convolution, and (3) the closest Gaussian in relative entropy, coincide by proving a central limit theorem for fermionic systems. We then utilize the violation of Wick's theorem and the matchgate identity to quantify non-Gaussian magic in addition to a SWAP test.