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Learning Holographic Quantum Codes:Code Geometry Constrains Krylov Complexity Dynamics

Sungwoo Kang

Published Mar 7, 2026• 0 citations
Publisher

Abstract

Quantum error-correcting codes underpin models of holographic spacetime, yet the relationship between code geometry and complexity dynamics remains unexplored. We present a unified machine learning framework that (i)~encodes stabilizer code structure via a hypergraph GNN achieving distance prediction MAE of~$0.10$, (ii)~predicts Krylov complexity trajectories with a DeepONet conditioned on GNN-derived code embeddings ($R^2 = 0.67$ vs.\ $R^2 = -10$ without), and (iii)~reveals that code geometry constrains complexity dynamics through partial correlation analysis (distance$\to$max complexity: $r_{\mathrm{partial}} = -0.60$, $p = 5.1 \times 10^{-65}$). Our key insight is that code-space projections $P = \prod_i (I + g_i)/2$ act as a dynamical filter: they mathematically annihilate operator branches that anti-commute with any stabilizer, effectively pruning the Krylov tree and making complexity growth sensitive to the code's geometric distance. We validate robustness across XXZ and random Hamiltonians while identifying that Ising symmetry decouples geometry from dynamics---a physically interpretable null result. Across 661 stabilizer codes ($n \leq 12$), within-size correlations confirm that the geometry--complexity link persists after controlling for system size and \emph{strengthens} at larger $n$ (reaching $r = -0.91$ at $n=12$). Multi-seed averaging over 10 Haar-random projections per code confirms that $99.8\%$ of trajectory variance is between-code (structural), with seed-averaged correlations strengthening to $r_{\mathrm{partial}} = -0.70$, providing computational evidence for holographic intuitions connecting code distance to constrained Krylov evolution.

DOI
10.5281/zenodo.18897501

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