从协方差矩阵可计算的费米子非高斯性度量
Computable fermionic non-Gaussianity from the covariance matrix
AI总结：
基于协方差矩阵的Williamson标准型，提出两类可计算的纯费米子态非高斯性度量（占据数熵和自然轨道参与熵），证明其单调性并约束非高斯门数量与经典模拟成本。
AI中文摘要：
费米子非高斯性（或费米子魔法）是费米子量子系统计算复杂性的关键资源，但目前可计算且操作上有意义的量化方法仍然有限。我们通过发展费米子非高斯性的凸资源理论，并引入两类可计算的纯费米子态度量来应对这一挑战，两者均源自协方差矩阵的Williamson标准型。第一类为占据数熵，定义为占据数的Tsallis-α熵。我们证明该族中的一个成员在高斯协议下是单调的，从而确立其为可计算的凸资源单调量，因此它下界约束了态制备所需的非高斯门数量。第二类为自然轨道参与熵，由态在自然轨道基（由协方差矩阵的特征向量定义）中的振幅平方的Rényi-α熵给出。这些度量量化了态在该基下的可压缩性，从而上界约束了正交归一高斯基下的经典模拟成本。我们分析了稳定子态和平移不变态下的这两类度量，此时它们简化并揭示出额外结构。我们还研究了代表性例子，包括随机SWAP掺杂的matchgate电路和键调制XXZ模型，突出了非高斯性在多体现象中的作用。我们的工作建立了一个可计算的费米子非高斯性的资源理论框架，统一了量子信息、凝聚态物理和量子化学中出现的概念，为研究量子多体系统的复杂性开辟了新方向，并提供了评估与量子优势相关的费米子态经典可模拟性的实用工具。
英文摘要：
Fermionic non-Gaussianity, or fermionic magic, is a key resource underlying the computational complexity of fermionic quantum systems, yet tractable and operationally meaningful ways to quantify it remain limited. We address this challenge by developing a convex resource theory of fermionic non-Gaussianity and introducing two families of computable quantities for pure fermionic states, both derived from the Williamson normal form of the covariance matrix. The first family, occupation number entropies, is defined as the Tsallis-$α$ entropy of the occupation numbers. We prove that one member of this family is monotonic under Gaussian protocols, establishing it as a computable convex resource monotone. It consequently lower bounds the number of non-Gaussian gates needed for state preparation. The second family, natural-orbital participation entropies, is given by the Rényi-$α$ entropy of the squared amplitudes of the state in the natural-orbital basis, defined by the eigenvectors of the covariance matrix. They quantify state compressibility in this basis and thus upper bound the classical simulation cost in an orthonormal Gaussian basis. We analyze both families for stabilizer and translation-invariant states, where they simplify and reveal additional structure. We further study representative examples, including random SWAP-doped matchgate circuits and the bond-modulated XXZ model, highlighting the role of non-Gaussianity in many-body phenomena. Our work establishes a resource-theoretic framework for computable fermionic non-Gaussianity that unifies notions arising across quantum information, condensed-matter physics, and quantum chemistry, opening new directions for studying the complexity of quantum many-body systems and providing practical tools to assess the classical simulability of fermionic states relevant for quantum advantage.
