当 Matchgate 基坍缩失败时：Qutrit 三分法与无界精确宽度

中文摘要 AI 辅助

全息算法通过将有限域的每个值编码到若干布尔 matchgate 线中来解决平面计数问题。基坍缩（base collapse）询问是否每个精确表示都能被压缩为每条边上的线数，该线数仅由域大小限定。Chen（STOC 2016）和 Xia（STOC 2016）在满秩及相关结构假设下建立了广泛的正面坍缩定理。这些工作共同留下了一个开放问题：当多个局部缺陷签名必须共享一个编码时，仅依赖域的坍缩界限是否仍然成立。我们否定地解决了这个开放问题，且已经在三态（qutrit）域上实现。对于每个 $a\ge4$，我们构造了一个最大元数 $a$ 的五标签整数值 qutrit 语言，它具有精确的有理 matchgate 表示，然而任何精确等价的保标签表示都需要公共宽度 $\Omega(2^{a/2}/a)$，即使其域、基、张量和复权重都可以改变。该构造避免了通常的局部退化，因此障碍是真正同时发生的。代数几何随后驱动了一个穷尽的三分法来解释坍缩的边界：小工具闭包要么分离成射线，要么是高斯可移动的并坍缩到至多宽度三，要么被限制在一个刚性平面加射线旗中，该旗包含我们的无界族。纯旋量几何在可移动情况下强制压缩，而 matchgate 簇的维数界限和 Zariski 回避将精确整数数据置于所有低宽度代数像之外。因此，一个几何框架同时解释了坍缩为何发生以及为何可能无界失败。

英文摘要

Holographic algorithms solve planar counting problems by encoding each value of a finite domain into several Boolean matchgate wires. Base collapse asks whether every exact representation can be compressed to a number of wires per edge bounded only by the domain size. Chen (STOC 2016) and Xia (STOC 2016) established broad positive collapse theorems under full-rank and related structural hypotheses. Together, these works left open whether a domain-only collapse bound survives when several locally deficient signatures must share one encoding. We resolve this open problem negatively, already on a three-state (qutrit) domain. For every $a\ge4$, we construct a five-label integer-valued qutrit language of maximum arity $a$ with an exact rational matchgate presentation, yet every exactly equivalent label-preserving presentation requires common width $Ω(2^{a/2}/a)$, even if its domain, base, tensors, and complex weights may all change. The construction avoids the usual local degeneracies, so the obstruction is genuinely simultaneous. Algebraic geometry then drives an exhaustive trichotomy explaining the boundary of collapse: the gadget closure either separates into rays, is Gaussian-mobile and collapses to width at most three, or is confined to a rigid plane-plus-ray flag containing our unbounded family. Pure-spinor geometry forces compression in the mobile case, while dimension bounds for matchgate varieties and Zariski avoidance place exact integer data outside every low-width algebraic image. Thus one geometric framework explains both why collapse occurs and why it can fail without bound.

发表机构

• Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所）
• University of Regensburg（雷根斯堡大学）

机构由 AI 辅助整理，请以论文原文为准。