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

TL;DR

This paper demonstrates that adding a local-refinement stage to tensor-network contraction significantly improves performance on Sycamore-like topologies, especially at higher bond dimensions, with the advantage increasing monotonically with bond dimension.

Contribution

It identifies a missing local-refinement step in the cotengra pipeline and shows its impact grows with bond dimension, providing a new method to enhance tensor-network contraction efficiency.

Findings

Refiner wins on all tested seeds at every bond dimension

Advantage grows approximately linearly with bond dimension

Refinement improves contraction cost on actual circuits at various depths

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.