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

Sungwoo Kang

Published Mar 7, 2026• 0 citations
Publisher

Abstract

Recent work has established that quantum error-correcting code geometry constrains Krylov complexity dynamics: a forward pipeline mapping stabilizer codes through GNN embeddings and DeepONet predictions to complexity trajectories $C_K(t)$.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.

DOI
10.5281/zenodo.18897543

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Recent work has established that quantum error-correcting code geometry constrains Krylov complexity dynamics: a forward pipeline mapping stabilizer codes through GNN embeddings and DeepONet predictions to complexity trajectories CK(t)C_K(t)CK​(t).We invert this pipeline: given a \emph{target} complexity trajectory CKtarget(t)C_K^{\mathrm{target}}(t)CKtarget​(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 ρˉ=+0.701\bar{\rho} = +0.701ρˉ​=+0.701, best single-code correlation +0.995+0.995+0.995, and 100% of generated codes with distance d≥2d \geq 2d≥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.
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~0.100.100.10, (ii)~predicts Krylov complexity trajectories with a DeepONet conditioned on GNN-derived code embeddings (R2=0.67R^2 = 0.67R2=0.67 vs.\ R2=−10R^2 = -10R2=−10 without), and (iii)~reveals that code geometry constrains complexity dynamics through partial correlation analysis (distance→\to→max complexity: rpartial=−0.60r_{\mathrm{partial}} = -0.60rpartial​=−0.60, p=5.1×10−65p = 5.1 \times 10^{-65}p=5.1×10−65). Our key insight is that code-space projections P=∏i(I+gi)/2P = \prod_i (I + g_i)/2P=∏i​(I+gi​)/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≤12n \leq 12n≤12), within-size correlations confirm that the geometry--complexity link persists after controlling for system size and \emph{strengthens} at larger nnn (reaching r=−0.91r = -0.91r=−0.91 at n=12n=12n=12). Multi-seed averaging over 10 Haar-random projections per code confirms that 99.8%99.8\%99.8% of trajectory variance is between-code (structural), with seed-averaged correlations strengthening to rpartial=−0.70r_{\mathrm{partial}} = -0.70rpartial​=−0.70, providing computational evidence for holographic intuitions connecting code distance to constrained Krylov evolution.
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=8N=8N=8 and 161616 physical qubits.M\"obius codes achieve 61%61\%61% higher codeness than standard MERA (0.2350.2350.235 vs.\ 0.1460.1460.146); constrained Krylov complexity shows periodic revivals (amplitude ∼ 1.8{\sim}\,1.8∼1.8, period ∼ 11{\sim}\,11∼11) absent in standard MERA; transfer matrix eigenvalues {1,±i}\{1, \pm i\}{1,±i} on the unit circle set the revival period; and ∼ 62%{\sim}\,62\%∼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=0.87r=0.87r=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.
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 SYK4_44​---across 825 instances with n=4n=4n=4--121212 qubits and measure a continuous ``codeness'' derived from the Knill-Laflamme discrepancy.No subspace achieves exact error correction (deff=1d_\mathrm{eff}=1deff​=1 universally), but continuous codeness varies substantially: SYK4_44​ averages 0.320.320.32 while disordered Ising averages 0.090.090.09.Raw correlations between codeness and Krylov complexity (e.g., r=0.61r=0.61r=0.61 with saturation value) are largely confounded by system size (n↔saturationn\leftrightarrow\text{saturation}n↔saturation: r=0.91r=0.91r=0.91).Partial correlations controlling for nnn reveal the genuine structure: Knill-Laflamme discrepancy survives as a predictor of full-space dynamics (r=−0.18r=-0.18r=−0.18, p<10−7p<10^{-7}p<10−7), codeness constrains within-subspace complexity (r=−0.21r=-0.21r=−0.21, p<10−9p<10^{-9}p<10−9), and a suppressed energy gap↔\leftrightarrow↔Krylov dimension correlation emerges (r=+0.28r=+0.28r=+0.28, p<10−16p<10^{-16}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.
We formulate a bounded distance decoding strategy applicable to all stabilizer codes including both CSS and non-CSS code-families. The framework emerges out of the local Clifford equivalence between arbitrary stabilizer states and graph states. Using the graphical representation of the stabilizers and the syndromes, we constitute the bounded distance decoding as an adaptable generalization of maximum likelihood decoding, ensuring correction of all errors with weights upper bounded by a target weight. We show that strategic pruning associated with a feed-forward network structure of the graph can reduce the search space and subsequently the runtime of the designed decoder. We demonstrate satisfactory performance of the bounded distance decoder in the case of the optimal non-CSS codes up to distance d=11d=11d=11 subjected to the depolarizing error on all qubits, and near-optimal decoding for the color and the surface codes, both belonging to the CSS family, under the bit-flip errors on the qubits. We also develop an open-source library, QGDecoder, enabling the graph-aware bounded distance decoding of arbitrary stabilizer codes.
Noise is a major obstacle in the development of practical schemes for quantum computation and communication. Similar to the case of classical communication, this noise can be protected against by employing a code, which provides a means for encoding quantum states prior to transmission and allows for errors to be inferred, and hopefully corrected, by a decoder at the receiver. Unfortunately, designing good codes and decoders is typically a difficult problem. This thesis focuses on developing low-complexity heuristic approaches to three such problems: the design of modified belief propagation decoders for quantum low-density parity-check codes, the design of stabilizer codes for asymmetric channels, and the design of codeword stabilized codes. Quantum low-density parity-check codes are stabilizer codes with low-weight generators. Such codes permit low-complexity decoding via the use of belief propagation, which is an iterative message passing algorithm that takes place on a factor graph defined by the code. However, the performance of such a decoder is limited both by code structure and the degenerate nature of quantum errors. To overcome these limitations, at least in part, a number of modifications to belief propagation are developed. Central among these is the augmented decoder, which in the case of a decoding error, iteratively reattempts decoding using modified factor graph. This heuristic modification simply involves the duplication of a randomly selected subset of the graph’s check nodes, which are in one-to-one correspondence with the code’s stabilizer generators. Across a range of codes, it is shown that the decoders developed perform as well as or better than other modified decoders presented in literature. For a number of channels of physical interest, phase-flip errors occur far more frequently than bit-flip errors. When transmitting across these so-called asymmetric channels, the decoding error rate can be minimized by tailoring the code used to the channel. However, assessing the performance of codes on a given channel is made difficult by the #P-completeness of optimal decoding. To address this complexity, it is shown that the decoding error rate can be accurately approximated using only a small fraction of the possible errors caused by the channel. This approximation is then used to identify a number of cyclic stabilizer codes that perform well on two different asymmetric channels. To further build on this, a heuristic is demonstrated for assessing code performance based on the decoding error rate of an associated classical code. The complexity of calculating this classical error rate is relatively low, and it is shown that it can be used as the basis for a hill-climbing search algorithm. Such searches have yielded a large number of highly performant codes satisfying various structure constraints. The family of codeword stabilized codes encompasses both the stabilizer codes as well as many of the best known nonadditive codes. Constructing a standard form codeword stabilized code is a matter of selecting a simple undirected graph and a binary classical code. This makes designing optimal codes difficult as the number of possible graphs grows exponentially with code length, and the clique search required to construct the classical code is NP-hard. To address the exponential growth of the search space, a heuristic is developed for assessing graphs. This heuristic is then employed by a genetic algorithm that also makes use of a novel crossover operation based on spectral bisection, which is show to be superior to more standard crossover operations. With a graph selected, it is demonstrated that the complexity of the clique search required to construct the associated classical code can be mitigated through the use of a heuristic clique finding algorithm. A number of best known codes are presented that have been found using this approach.
Quantum error correction and the use of quantum error correction codes are likely to be essential for the realization of practical quantum computing. Because the error models of quantum devices vary widely, quantum codes that are tailored for a particular error model may have much better performance. In this work, we present a novel evolutionary algorithm that searches for an optimal stabilizer code for a given error model, number of physical qubits, and number of encoded qubits. We demonstrate an efficient representation of stabilizer codes as binary strings, which allows for random generation of valid stabilizer codes as well as mutation and crossing of codes. Our algorithm finds stabilizer codes whose distance closely matches the best-known-distance codes of Grassl (2007) forn≤20n \leq 20n≤20physical qubits. We perform a search for optimal distance Calderbank–Steane–Shor codes and compare their distance to the best known codes. Finally, we show that the algorithm can be used to optimize stabilizer codes for biased error models, demonstrating a significant improvement in the undetectable error rate for[[12,1]]2[[12,1]]_{2}[[12,1]]2​codes versus the best-known-distance code with the same parameters. As part of this work, we also introduce an evolutionary algorithm QDistEvol for finding the distance of quantum error correction codes.