随机Matchgate酉变换的精确算子复杂度度量
Exact Operator Complexity Measures from Random Matchgate Unitaries
- Freie Universität Berlin（柏林自由大学）
- Helmholtz-Zentrum Berlin für Materialien und Energie（亥姆霍兹柏林材料能源中心）
- Universität zu Köln（科隆大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究通过提出Majorana基下的自由费米子Weingarten微积分，精确计算了随机Matchgate系综中的算子纠缠、稳定子熵和OTOC，揭示了这些复杂度度量在经典可模拟电路中的饱和行为，并建立了算子纠缠与非高斯门数量的下界关系。
AI中文摘要：
Matchgate电路描述了在物理相关设置中经典可处理的动力学，同时展现出复杂量子系统的特征，如高含量的魔法（magic）。这引发了一个更广泛的问题：在平凡动力学下，典型的复杂度度量在多大程度上达到饱和？在此，我们通过研究随机Matchgate系综中不同海森堡图像复杂度度量的行为来解决这一问题。我们的主要技术贡献是提出了一种完全在Majorana基下表述的可处理且可推广的自由费米子酉Weingarten微积分。该方法专为计算算子矩而设计，并简化为一个依赖于给定算子泛函所规定的边界条件的组合问题。利用这一框架，我们推导了典型局域算子纠缠、算子稳定子熵以及高阶离时有序关联函数的闭式表达式。对于广泛的初始Majorana弦，算子纠缠和稳定子熵均遵循体积律，尽管底层电路仍然是经典可模拟的。离时有序关联函数同样可以变得指数级小，表明其晚期饱和值本身并不能证明计算难度，并构成了一个经典可处理的涌现自由独立性的例子。最后，我们将这些分离与算子资源理论联系起来，展示了超越高斯基线的算子纠缠如何为掺杂Matchgate电路中的非高斯门数量提供下界。我们的结果为计算随机Matchgate上的精确平均值提供了一个实用工具箱，同时阐明了量子资源与模拟复杂性之间的区别，以及它们对所利用表示的依赖性。
英文摘要：
Matchgate circuits describe classically tractable dynamics in a physically relevant setting while nonetheless exhibiting features characteristic of complex quantum systems, such as high amounts of magic. This invites the broader question: to what extent are typical complexity measures saturated under trivial dynamics? Here, we address this by studying the behaviour of different Heisenberg-picture complexity measures in random matchgate ensembles. Our main technical contribution is a tractable and generalisible free-fermionic unitary Weingarten calculus formulated entirely in the Majorana basis. This method is tailored to computing moments of operators, and simplifies to a combinatorics problem which depends on the boundary conditions prescribed by a given operator functional. Using this framework, we derive closed-form expressions for the typical local-operator entanglement, operator stabiliser entropy, and higher-order out-of-time-ordered correlators. For extensive initial Majorana strings, both operator entanglement and stabiliser entropy obey volume laws, even though the underlying circuits remain classically simulable. Out-of-time-ordered correlators can likewise become exponentially small, showing that their late-time saturation value alone does not witness computational hardness and constituting a classically tractable example of emergent free independence. Finally, we relate these separations to operator resource theories, showing how operator entanglement beyond the Gaussian baseline lower-bounds the number of non-Gaussian gates in doped matchgate circuits. Our results present a practical toolbox for computing exact averages over random matchgates, while clarifying the distinction between quantum resources versus simulation complexity, and their dependence on the representation being exploited.
