从规范层丛量子局部可测试码构造的NLTM哈密顿量

中文摘要 AI 辅助

理解低能量子态的复杂性是量子复杂性理论和多体物理学的核心问题。在此，我们证明了无低能平凡魔法（NLTM）猜想，构造了一族局部哈密顿量，使得正能量密度窗口内的每个态都需要从任意稳定子态（可能高度纠缠）制备的电路深度为Ω(log n)。值得注意的是，我们的结果是内在的，即相同的哈密顿量族和能量密度阈值适用于任意有界局部维数的qudit稳定子输入。我们的构造基于非阿贝尔规范层丛码，这些码是利用良好量子局部可测试码上的杯积结构获得的。我们建立了相关哈密顿量的常数算子可靠性，使其受保护的逻辑结构能够约束所有能量密度足够低的态。通过排除基于浅层稳定子的经典见证，我们的结果显著加强了NLTS，并推进了对与量子PCP相关的低能复杂性的理解。

英文摘要

Understanding the complexity of low-energy quantum states is a central problem in quantum complexity theory and many-body physics. Here we prove the no low-energy trivial magic (NLTM) conjecture, constructing a family of local Hamiltonians for which every state within a positive energy density window requires circuit depth $Ω(\log n)$ to prepare from arbitrary stabilizer state that may themselves be highly entangled. Notably, our result is intrinsic in the sense that the same Hamiltonian family and energy density threshold apply to arbitrary qudit stabilizer inputs with any bounded local dimensions. Our construction builds on non-Abelian gauged sheaf codes obtained using the cup product structure on good quantum locally testable codes. We establish constant operator soundness for the associated Hamiltonians, enabling their protected logical structure to constrain all states at sufficiently low energy density. By excluding shallow stabilizer-based classical witnesses, Our result substantially strengthens NLTS and advances our understanding of low-energy complexity relevant to quantum PCP.

发表机构

• Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）

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