Bond-dimension scaling of a local-refinement advantage over hyperoptimized tensor-network contraction on Sycamore like topologies

Abstract

We identify a missing local-refinement stage in the cotengra tensor-network contraction pipeline and show that its impact grows monotonically with bond dimension on the \emph{connectivity graph} of Sycamore-like topologies. Appending a nearest-neighbor interchange (NNI) search to the \cotengra{} output at matched 8-s wallclock yields a median \emph{predicted} cost-model gap $\Delta\fT$ at $n{=}500$ that grows monotonically and approximately linearly in $\chi$, from $\sim\!15$~bits at $\chi{=}2$ to $\sim\!116$~bits at $\chi{=}16$ (Fig.~\ref{fig:chi_sweep}), with the refiner winning on $25/25$ seeds at every tested $\chi$. Two control families -- random $3$-regular and QAOA $p{=}2$ interaction graphs -- show median $|\Delta\fT| \leq 0.71$~bits across both controls at every $\chi$, with refiner win rate falling toward chance as $\chi$ grows; the signal is topology-specific, not a generic refinement-budget effect. An ablation establishes that refinement itself, not the four-axis Pareto acceptance rule, drives the gain ($|\Delta\fT| \lesssim 0.1$ bits between scalar and Pareto arms at $\chi{=}2$). The Sycamore-circuit envelope (App.~\ref{em:sec:results:syccirc}) reports the corresponding refinement on actual random circuits at depths $m \in \{4, 6, 8, 10, 12\}$, where the refiner wins on $5/5$ instances at every depth. The advantage is therefore largest precisely in the bond-dimension regime relevant to physical contraction.