AI 中文总结
挑战随机二分法控制量子计算等基本计算边界的观点，提出消失几何相位（VGP）能更充分刻画边界。构造特殊哈密顿量，证明相关问题复杂度，确定VGP识别情况，表明计算边界应理解为几何相位结构间的边界。
AI 中文摘要
我们对随机二进制控制量子计算和量子系统经典模拟中基本计算边界的观点提出挑战。我们认为，哈密顿量跃迁图上的几何条件——消失几何相位（VGP），能更充分地刻画这些边界。为区分VGP与随机性质，我们构造了形式上难以随机化的VGP 3 - 局部哈密顿量。在输入哈密顿量具有VGP的前提下，我们证明了局部哈密顿量问题是$\mathsf{StoqMA}$ - 完全的，且无挫折变体在同一前提下属于$\mathsf{MA}$。我们还用此结果论证了对于任何通过逃离$\mathsf{StoqMA}$ regime来证明的绝热优势，非VGP是必要的。此外，我们确定了能在多项式时间内识别VGP性质的自然设置，同时表明对于几何局部哈密顿量，一般情况下VGP的识别是$\mathsf{PSPACE}$ - 完全的。我们的结果表明，传统上归因于随机性质的计算边界$\mathsf{MA} \subseteq \mathsf{StoqMA} \subseteq \mathsf{QMA}$，更好地理解为消失和非消失几何相位结构之间的边界。
英文摘要
We challenge the notion that a stoquastic binary governs fundamental computational boundaries in quantum computing and classical simulation of quantum systems. We argue that vanishing geometric phase (VGP), a geometric condition on the Hamiltonian's transition graph, more adequately captures these boundaries. To distinguish VGP from stoquasticity, we construct VGP 3-local Hamiltonians that are formally hard to stoquastize, yet belong to a family admitting polynomial-time recognition of the VGP property. Without constructing a stoquastizing unitary, we prove that the local Hamiltonian problem is $\mathsf{StoqMA}$-complete under the promise that the input Hamiltonian has VGP, and that a frustration-free variant is in $\mathsf{MA}$ under the same promise. We use this result to argue that non-VGP is necessary for any claimed adiabatic advantage justified by escaping the $\mathsf{StoqMA}$ regime. Further, we identify natural settings where the VGP property can be recognized in polynomial time. In contrast, we show that recognition of VGP is $\mathsf{PSPACE}$-complete in general for geometrically local Hamiltonians. Our results show that the computational boundaries $\mathsf{MA} \subseteq \mathsf{StoqMA} \subseteq \mathsf{QMA}$ traditionally attributed to stoquasticity are better understood as boundaries between vanishing and non-vanishing geometric phase structure.
Comments33 pages, 1 figure
