Topological Complexity Revivals in M\"obius-MERA Quantum Error-Correcting Codes

Sungwoo Kang

Published Mar 7, 2026• 0 citations

Abstract

Companion studies showed that engineered code geometry constrains Krylov complexity~\cite{kang2025codespace} and that natural Hamiltonian subspaces exhibit approximate quantum error correction~\cite{kang2025natural}, but both used open boundary conditions with no topological feedback.Here we construct a scale-periodic ``M\"obius-MERA'' by threading a crosscap twist through the infrared end of a holographic MERA tensor network, creating a non-orientable loop in the renormalization-group scale direction.We extract code subspaces and measure codeness, Krylov complexity, and error annihilation at $N=8$ and $16$ physical qubits.M\"obius codes achieve $61\%$ higher codeness than standard MERA ($0.235$ vs.\ $0.146$); constrained Krylov complexity shows periodic revivals (amplitude ${\sim}\,1.8$, period ${\sim}\,11$) absent in standard MERA; transfer matrix eigenvalues $\{1, \pm i\}$ on the unit circle set the revival period; and ${\sim}\,62\%$ of weight-1 errors approximately satisfy the Knill-Laflamme conditions.Partial correlations controlling for system size confirm a codeness--revival amplitude link at $r=0.87$.Topological feedback in the renormalization-group direction creates approximate quantum error correction and periodic complexity dynamics, connecting holographic tensor network geometry to observable dynamical signatures.

DOI: 10.5281/zenodo.18897552

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arXiv:1910.06329Journal of High Energy PhysicsMar 31, 202044 citations

Topological Complexity Revivals in M\"obius-MERA Quantum Error-Correcting Codes

Sungwoo Kang

Published Mar 7, 2026• 0 citations

Publisher

Abstract

N=8

and

16

physical qubits.M"obius codes achieve

61\%

higher codeness than standard MERA (

0.235

vs.\

0.146

); constrained Krylov complexity shows periodic revivals (amplitude

{\sim}\,1.8

, period

{\sim}\,11

) absent in standard MERA; transfer matrix eigenvalues

\{1, \pm i\}

on the unit circle set the revival period; and

{\sim}\,62\%

of weight-1 errors approximately satisfy the Knill-Laflamme conditions.Partial correlations controlling for system size confirm a codeness--revival amplitude link at

r=0.87

.Topological feedback in the renormalization-group direction creates approximate quantum error correction and periodic complexity dynamics, connecting holographic tensor network geometry to observable dynamical signatures.

DOI: 10.5281/zenodo.18897552

Related papers

Learning Holographic Quantum Codes:Code Geometry Constrains Krylov Complexity Dynamics

Sungwoo Kang

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.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.

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Approximate Quantum Error Correction in Natural Subspaces:Evidence for Code Structure as a Physical Attractor

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Previous work showed that the geometry of engineered quantum error-correcting codes constrains Krylov complexity dynamics~\cite{kang2025codespace}.Here we ask whether low-energy subspaces of \emph{physical} Hamiltonians spontaneously satisfy the Knill-Laflamme conditions for approximate quantum error correction.We extract two-dimensional code subspaces from the lowest eigenstates of five model families---clean and disordered Ising, Heisenberg, XXZ, and SYK $_4$ ---across 825 instances with $n=4$ -- $12$ qubits and measure a continuous ``codeness'' derived from the Knill-Laflamme discrepancy.No subspace achieves exact error correction ( $d_\mathrm{eff}=1$ universally), but continuous codeness varies substantially: SYK $_4$ averages $0.32$ while disordered Ising averages $0.09$ .Raw correlations between codeness and Krylov complexity (e.g., $r=0.61$ with saturation value) are largely confounded by system size ( $n\leftrightarrow\text{saturation}$ : $r=0.91$ ).Partial correlations controlling for $n$ reveal the genuine structure: Knill-Laflamme discrepancy survives as a predictor of full-space dynamics ( $r=-0.18$ , $p<10^{-7}$ ), codeness constrains within-subspace complexity ( $r=-0.21$ , $p<10^{-9}$ ), and a suppressed energy gap $\leftrightarrow$ Krylov dimension correlation emerges ( $r=+0.28$ , $p<10^{-16}$ ).Codeness also tracks quantum phase transitions in the Ising model.These results suggest that approximate quantum error correction is a physical attractor of interacting quantum systems, not merely an artifact of stabilizer engineering.Previous work showed that the geometry of engineered quantum error-correcting codes constrains Krylov complexity dynamics~\cite{kang2025codespace}.Here we ask whether low-energy subspaces of \emph{physical} Hamiltonians spontaneously satisfy the Knill-Laflamme conditions for approximate quantum error correction.We extract two-dimensional code subspaces from the lowest eigenstates of five model families---clean and disordered Ising, Heisenberg, XXZ, and SYK$_4$---across 825 instances with $n=4$--$12$ qubits and measure a continuous ``codeness'' derived from the Knill-Laflamme discrepancy.No subspace achieves exact error correction ($d_\mathrm{eff}=1$ universally), but continuous codeness varies substantially: SYK$_4$ averages $0.32$ while disordered Ising averages $0.09$.Raw correlations between codeness and Krylov complexity (e.g., $r=0.61$ with saturation value) are largely confounded by system size ($n\leftrightarrow\text{saturation}$: $r=0.91$).Partial correlations controlling for $n$ reveal the genuine structure: Knill-Laflamme discrepancy survives as a predictor of full-space dynamics ($r=-0.18$, $p<10^{-7}$), codeness constrains within-subspace complexity ($r=-0.21$, $p<10^{-9}$), and a suppressed energy gap$\leftrightarrow$Krylov dimension correlation emerges ($r=+0.28$, $p<10^{-16}$).Codeness also tracks quantum phase transitions in the Ising model.These results suggest that approximate quantum error correction is a physical attractor of interacting quantum systems, not merely an artifact of stabilizer engineering.

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Inverse Holographic Design: Generating Quantum Error-Correcting Codes from Target Krylov Complexity Trajectories

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Approximate Bacon-Shor code and holography

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We explicitly construct a class of holographic quantum error correction codes with non-trivial centers in the code subalgebra. Specifically, we use the Bacon-Shor codes and perfect tensors to construct a gauge code (or a stabilizer code with gauge-fixing), which we call the holographic hybrid code. This code admits a local log-depth encoding/decoding circuit, and can be represented as a holographic tensor network which satisfies an analog of the Ryu-Takayanagi formula and reproduces features of the sub-region duality. We then construct approximate versions of the holographic hybrid codes by “skewing” the code subspace, where the size of skewing is analogous to the size of the gravitational constant in holography. These approximate hybrid codes are not necessarily stabilizer codes, but they can be expressed as the superposition of holographic tensor networks that are stabilizer codes. For such constructions, different logical states, representing different bulk matter content, can “back-react” on the emergent geometry, resembling a key feature of gravity. The locality of the bulk degrees of freedom becomes subspace-dependent and approximate. Such subspace-dependence is manifest from the point of view of the “entanglement wedge” and bulk operator reconstruction from the boundary. Exact complementary error correction breaks down for certain bipartition of the boundary degrees of freedom; however, a limited, state-dependent form is preserved for particular subspaces. We also construct an example where the connected two-point correlation functions can have a power-law decay. Coupled with known constraints from holography, a weakly back-reacting bulk also forces these skewed tensor network models to the “large N limit” where they are built by concatenating a large N number of copies.We explicitly construct a class of holographic quantum error correction codes with non-trivial centers in the code subalgebra. Specifically, we use the Bacon-Shor codes and perfect tensors to construct a gauge code (or a stabilizer code with gauge-fixing), which we call the holographic hybrid code. This code admits a local log-depth encoding/decoding circuit, and can be represented as a holographic tensor network which satisfies an analog of the Ryu-Takayanagi formula and reproduces features of the sub-region duality. We then construct approximate versions of the holographic hybrid codes by “skewing” the code subspace, where the size of skewing is analogous to the size of the gravitational constant in holography. These approximate hybrid codes are not necessarily stabilizer codes, but they can be expressed as the superposition of holographic tensor networks that are stabilizer codes. For such constructions, different logical states, representing different bulk matter content, can “back-react” on the emergent geometry, resembling a key feature of gravity. The locality of the bulk degrees of freedom becomes subspace-dependent and approximate. Such subspace-dependence is manifest from the point of view of the “entanglement wedge” and bulk operator reconstruction from the boundary. Exact complementary error correction breaks down for certain bipartition of the boundary degrees of freedom; however, a limited, state-dependent form is preserved for particular subspaces. We also construct an example where the connected two-point correlation functions can have a power-law decay. Coupled with known constraints from holography, a weakly back-reacting bulk also forces these skewed tensor network models to the “large N limit” where they are built by concatenating a large N number of copies.

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One-loop universality of holographic codes

Xi Dong, Marolf, Donald

Recent work showed holographic error correcting codes to have simple universal features at O(1/G). In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order (O(1) corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in G, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.Recent work showed holographic error correcting codes to have simple universal features at O(1/G). In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order (O(1) corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in G, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.

Jan 1, 202039 citations

One-loop universality of holographic codes

Xi Dong, Donald Marolf

Abstract Recent work showed holographic error correcting codes to have simple universal features at O(1/G). In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order (O(1) corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in G, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.Abstract Recent work showed holographic error correcting codes to have simple universal features at O(1/G). In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order (O(1) corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in G, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.

arXiv:1910.06329Journal of High Energy PhysicsMar 31, 202044 citations

N=8

and

16

physical qubits.M"obius codes achieve

61\%

higher codeness than standard MERA (

0.235

vs.\

0.146

); constrained Krylov complexity shows periodic revivals (amplitude

{\sim}\,1.8

, period

{\sim}\,11

) absent in standard MERA; transfer matrix eigenvalues

\{1, \pm i\}

on the unit circle set the revival period; and

{\sim}\,62\%

of weight-1 errors approximately satisfy the Knill-Laflamme conditions.Partial correlations controlling for system size confirm a codeness--revival amplitude link at

r=0.87

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

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.

_4

---across 825 instances with

n=4

12

qubits and measure a continuous ``codeness'' derived from the Knill-Laflamme discrepancy.No subspace achieves exact error correction (

d_\mathrm{eff}=1

universally), but continuous codeness varies substantially: SYK

_4

averages

0.32

while disordered Ising averages

0.09

.Raw correlations between codeness and Krylov complexity (e.g.,

r=0.61

with saturation value) are largely confounded by system size (

n\leftrightarrow\text{saturation}

r=0.91

).Partial correlations controlling for

n

reveal the genuine structure: Knill-Laflamme discrepancy survives as a predictor of full-space dynamics (

r=-0.18

p<10^{-7}

), codeness constrains within-subspace complexity (

r=-0.21

p<10^{-9}

), and a suppressed energy gap

\leftrightarrow

Krylov dimension correlation emerges (

r=+0.28

p<10^{-16}

).Codeness also tracks quantum phase transitions in the Ising model.These results suggest that approximate quantum error correction is a physical attractor of interacting quantum systems, not merely an artifact of stabilizer engineering.

C_K(t)

.We invert this pipeline: given a \emph{target} complexity trajectory

C_K^{\mathrm{target}}(t)

, we generate stabilizer codes whose actual dynamics match the target.Our three-stage approach combines (i)~gradient-based embedding optimization through a frozen DeepONet to find the 128-dimensional code embedding that best reproduces the target trajectory, (ii)~a Transformer-based Graph VAE decoder with Gumbel-Softmax discretization and physics-informed losses (commutativity, overlap penalty, minimum weight) that maps embeddings to stabilizer code structures, and (iii)~a commutativity repair and validation pipeline with distance-aware filtering.Across six target trajectories spanning Lloyd-bound saturation, power-law growth, and linear-then-saturate profiles, our method achieves 87\% code validity, mean trajectory correlation

\bar{\rho} = +0.701

, best single-code correlation

+0.995

, and 100\% of generated codes with distance

d \geq 2

.Ablation studies confirm that each component---overlap penalty, multi-strategy repair, distance filtering, and canonical code injection---contributes to the final performance.This work demonstrates the first \emph{inverse holographic design} pipeline: specifying desired dynamical behavior and automatically generating the quantum code that realizes it, opening pathways toward complexity-aware quantum code engineering.