Generic Schmidt-Rank Width of Fixed Weighted Phase Networks: Maximum-Matching Width, Template Diversity, and Single-Parameter Global Saturation

Abstract

This work studies the global multipartite entanglement width generated by fixed pairwise weighted-phase networks acting on full-support product inputs. It develops the global-width consequences of the fixed-cut weighted-phase theory established in the preceding study, Schmidt-Rank Capacity and Spectral Robustness in Fixed Weighted Phase Networks; the corresponding public research record is available at DOI 10.5281/zenodo.21858429. For a graph $G=(V,E)$ with a fixed real edge-weight template $W=(W_e)_{e\in E}$, we consider the commuting weighted-phase evolution $$U_W(t) = \exp\left( i t \sum_{e=\{u,v\}\in E} W_e n_u n_v \right),$$ with a single scalar evolution parameter $t$. For every bipartition $A\vert{}A^c$, let $\nu_G(A)$ be the maximum matching number of the crossing graph $G[A,A^c]$. The main result globalizes the fixed-cut matching law. There exists an open, dense, full-Lebesgue-measure class of fixed templates $W$ such that, outside one locally finite exceptional set of evolution times, the same scalar $t$ simultaneously saturates the matching capacity of every nontrivial bipartition: $$\log_2 \operatorname{SR}_{A\vert{}A^c}\bigl(\vert{}\Psi_W(t)\rangle\bigr) = \nu_G(A)$$ for all nontrivial cuts $A\vert{}A^c$. Consequently, $$\chi_{\mathrm{wd}}\bigl(\vert{}\Psi_W(t)\rangle\bigr) = \operatorname{mmw}(G)$$ for generic fixed weighted-phase networks, where $\chi_{\mathrm{wd}}$ is Schmidt-rank width and $\operatorname{mmw}(G)$ is the maximum matching width of the interaction graph. This provides a continuous weighted counterpart to the standard graph-state relation between Schmidt-rank width and rank-width. For complete support $K_n$, $$\operatorname{mmw}(K_n) = \left\lceil\frac{n}{3}\right\rceil,$$ so almost every nonuniform fixed template attains $$\chi_{\mathrm{wd}} = \left\lceil\frac{n}{3}\right\rceil,$$ which is the largest Schmidt-rank width permitted for any $n$-qubit pure state. Uniform complete-graph templates behave very differently. Their exact width is $$\chi_{\mathrm{wd}}^{\mathrm{unif}} = \log_2 \min\left\{ \left\lceil\frac{n}{3}\right\rceil+1,\, \operatorname{ord}(e^{i\theta}) \right\},$$ with $\operatorname{ord}(e^{i\theta})=\infty$ allowed. Thus the same complete interaction support and the same one-dimensional run-time control exhibit a sharp template hierarchy: $$\text{Clifford} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=1,$$ $$\text{generic uniform} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=\Theta(\log n),$$ $$\text{generic nonuniform} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=\Theta(n).$$ This separation is not restricted to dense complete graphs. Since $$\operatorname{rw}(G) \le \operatorname{mmw}(G),$$ graph families with linear rank-width transfer directly to fixed weighted-phase families with linear generic Schmidt-rank width. In particular, known deterministic bounded-degree high-rank-width constructions yield sparse $O(n)$-coupler networks with generic Schmidt-rank width $\Theta(n)$. The width separation has an exact tensor-network consequence. For generic uniform complete templates, an explicit subcubic tree tensor network exists with optimal maximal bond dimension $$D_{\mathrm{unif}} = \left\lceil\frac{n}{3}\right\rceil+1 = \Theta(n),$$ whereas generic nonuniform complete templates require $$D_{\mathrm{gen}} = 2^{\left\lceil n/3\right\rceil} = 2^{\Theta(n)}.$$ Thus fixed template diversity, rather than run-time control dimension alone, can change exact global tree-tensor complexity from polynomial to exponential while the interaction support and the number of run-time control parameters remain unchanged. The computational interpretation is deliberately limited. Superlogarithmic Schmidt-rank width lies outside the logarithmic-width hypothesis underlying the standard low-width exact-TTN simulation guarantee, and linear width is a particularly strong realization of this separation. However, large or linear Schmidt-rank width is neither asserted to be necessary nor sufficient for universal measurement-based quantum computation, hard classical simulation in general, fault tolerance, or quantum advantage. Version v0.2r2 is the frozen theorem-core release. It preserves the mathematical results of v0.2r1 while finalizing the claim boundary concerning tensor-network simulation, measurement-based quantum computation, and computational advantage.