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Paper 1
Bond-dimension scaling of a local-refinement advantage over hyperoptimized tensor-network contraction on Sycamore like topologies
Rubén Darío Guerrero
- Year
- 2026
- Journal
- arXiv preprint
- DOI
- arXiv:2604.25532
- arXiv
- 2604.25532
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 $Δ\fT$ at $n{=}500$ that grows monotonically and approximately linearly in $χ$, from $\sim\!15$~bits at $χ{=}2$ to $\sim\!116$~bits at $χ{=}16$ (Fig.~\ref{fig:chi_sweep}), with the refiner winning on $25/25$ seeds at every tested $χ$. Two control families -- random $3$-regular and QAOA $p{=}2$ interaction graphs -- show median $|Δ\fT| \leq 0.71$~bits across both controls at every $χ$, with refiner win rate falling toward chance as $χ$ 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 ($|Δ\fT| \lesssim 0.1$ bits between scalar and Pareto arms at $χ{=}2$). 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.
Open paperPaper 2
Critical properties of the Anderson transition in random graphs: two-parameter scaling theory, Kosterlitz-Thouless type flow and many-body localization
Ignacio García-Mata, John Martin, Olivier Giraud, Bertrand Georgeot, Rémy Dubertrand, Gabriel Lemarié
- Year
- 2022
- Journal
- arXiv preprint
- DOI
- arXiv:2209.04337
- arXiv
- 2209.04337
The Anderson transition in random graphs has raised great interest, partly because of its analogy with the many-body localization (MBL) transition. Unlike the latter, many results for random graphs are now well established, in particular the existence and precise value of a critical disorder separating a localized from an ergodic delocalized phase. However, the renormalization group flow and the nature of the transition are not well understood. In turn, recent works on the MBL transition have made the remarkable prediction that the flow is of Kosterlitz-Thouless type. Here we show that the Anderson transition on graphs displays the same type of flow. Our work attests to the importance of rare branches along which wave functions have a much larger localization length $ξ_\parallel$ than the one in the transverse direction, $ξ_\perp$. Importantly, these two lengths have different critical behaviors: $ξ_\parallel$ diverges with a critical exponent $ν_\parallel=1$, while $ξ_\perp$ reaches a finite universal value ${ξ_\perp^c}$ at the transition point $W_c$. Indeed, $ξ_\perp^{-1} \approx {ξ_\perp^c}^{-1} + ξ^{-1}$, with $ξ\sim (W-W_c)^{-ν_\perp}$ associated with a new critical exponent $ν_\perp = 1/2$, where $\exp( ξ)$ controls finite-size effects. The delocalized phase inherits the strongly non-ergodic properties of the critical regime at short scales, but is ergodic at large scales, with a unique critical exponent $ν=1/2$. This shows a very strong analogy with the MBL transition: the behavior of $ξ_\perp$ is identical to that recently predicted for the typical localization length of MBL in a phenomenological renormalization group flow. We demonstrate these important properties for a smallworld complex network model and show the universality of our results by considering different network parameters and different key observables of Anderson localization.
Open paper