超越键规范：加权图态处的精确张量网络切空间
Beyond Bond Gauge: Exact Tensor-Network Tangent Spaces at Weighted Graph States
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中文总结 AI 辅助
该研究精确刻画了加权图态张量网络的切空间，区分了键规范变化与缺失物理方向，证明在多种平铺上移除冗余与补充方向是独立操作。
中文摘要 AI 辅助
对于普通PEPS，基本定理在有限层面解决了虚拟键规范是否穷尽张量网络的表示自由度。我们在固定表示下从微分两侧提出一阶问题：哪些张量变化在不作为键规范的情况下仍使态不变，以及哪些物理方向无法被任何变化达到？我们针对物理维数与键维数相等且$q\ge2$的加权图态（WGS）张量网络精确回答这两个问题，其中包括基于测量量子计算的图态资源。将单张量变化除以非零振幅，可将其转化为其顶点闭邻域上的任意函数。重叠邻域给出完整核。其非键部分在周期正多边形平铺上由三角形、无弦正方形和共享边菱形生成。在所述周期条件下，所研究的全部三十九个族中的一般规范完备性将此零空间识别为WGS秩下降：三十三个族在每一个$q\ge2$下成立，六个族在$q=2$下成立。每个邻域之外的支持给出缺失方向：在十一种阿基米德平铺上的量子比特图态中，每个最近邻$XX$和$YY$方向均缺失，包括蜂窝状平铺，其中非键核为空。在大蜂窝和方形环面上，小的非零相干$XX$误差使同一图上的键二PEPS集合保持不变。因此，移除冗余参数和补充缺失方向是两种不同的操作。
英文摘要
For normal PEPS, the fundamental theorem settles at the finite level whether virtual-bond gauge exhausts a tensor network's representation freedom. We ask the first-order question at a fixed representation, from both sides of the differential: which tensor variations leave the state unchanged without being bond gauges, and which physical directions can no variation reach? We answer both exactly for weighted graph state (WGS) tensor networks with equal physical and bond dimension $q\ge2$, which include the graph-state resources of measurement-based quantum computation. Dividing a single-tensor variation by the nonvanishing amplitude turns it into an arbitrary function on the closed neighborhood of its vertex. Overlapping neighborhoods give the complete kernel. Its non-bond part is generated on periodic regular-polygon tilings by triangles, chordless squares, and edge-sharing diamonds. Under stated period conditions, generic gauge completeness on all thirty-nine families studied identifies this nullity as a WGS rank drop: for thirty-three families at every $q\ge2$, and for six at $q=2$. Supports outside every neighborhood give the missing directions: at qubit graph states on the eleven Archimedean tilings every nearest-neighbor $XX$ and $YY$ direction is missing, including on honeycomb, where the non-bond kernel is empty. On large honeycomb and square tori, small nonzero coherent $XX$ errors leave the bond-two PEPS set on the same graph. Removing redundant parameters and supplying missing directions are therefore distinct operations.
发表机构
- Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）
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