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

Sungwoo Kang

Published Mar 7, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.18897527

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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 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 develop a theory of approximate quantum error correction (QEC) based on the error-set model, complemented by general methods for code construction. Exact QEC has a powerful error-set structure: by the Knill-Laflamme conditions, a code correcting a given error set automatically protects against every channel whose Kraus operators lie in their linear span. This linearity gives rise to code distance, the equivalence between erasures and general errors, and a theory of asymptotically good codes. A longstanding view has been that these features do not extend to AQEC, leaving the theory essentially channel-by-channel. We show instead that, although full Knill--Laflamme linearity fails, a restricted form survives and suffices to extend all three structural features to the approximate setting. Specifically, a common error-set criterion governs families of channels whose Kraus operators are linear combinations of a given error set and whose coefficient matrices satisfy a spectral constraint. Using the Bény-Oreshkov worst-case and Petz average-case frameworks, we derive uniform fidelity guarantees for these families in terms of two new code parameters--the \emph{environment-leakage distance}, controlling worst-case performance, and the \emph{Knill-Laflamme Hellinger distance}, characterizing the average-case performance of Petz recovery. To demonstrate the scope of this model, we develop partition-based constructions across diverse quantum systems and geometries, placing exact and approximate correction on equal footing. These constructions lead to a metric--error alignment hierarchy for Hilbert spaces, metrics, and error families, which in turn characterizes the resulting recovery guarantees. They yield the first known asymptotically good code families for fermionic systems, one-dimensional Rydberg-blockaded systems, and deletion errors, and extend to other physical platforms.
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.
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.
Quantum error correction acts as an entropy pump, transferring noise-induced uncertainty from a protected quantum system into syndrome information stored in an auxiliary memory. Repeated operation requires this memory to be cleared which unavoidably contributes to the energetic cost of error correction. Here, we characterise this contribution for subspace quantum error-correcting codes and identify how it depends on the joint structure of the code, the noise, and the representation of the retained syndrome information. Starting from the Knill--Laflamme conditions, we construct an effective syndrome state whose von Neumann entropy sets a lower bound on the ideal work required to maintain a reusable syndrome register. Projective syndrome readout generally generates additional entropy, and we quantify the resulting gap through measurement inefficiency. We then specialise to stabiliser codes under independent local Pauli noise and analyse two classical levels of syndrome representation. At the level of abstract error labels, degeneracies among single-qubit errors reduce the leading-order entropy of processed recovery labels. At the parity-check level, lower-weight checks reduce the marginal entropy generated by individual measurement outcomes in the low-noise regime. We identify the additional burden associated with retaining and separately erasing these outcomes as a bit-level inefficiency, and illustrate both costs for the five-qubit, Steane, generalised Shor, and rotated surface codes. Our results establish a hierarchy of syndrome-memory energetic costs and identify the code, noise, and measurement structures that control the ideal thermodynamic burden of subspace quantum error correction.
The Knill-Laflamme conditions distinguish exact quantum error correction codes, and they have played a critical role in the discovery of state-of-the-art codes. However, the family of exact codes is a very restrictive one and does not necessarily contain the best-performing codes. Therefore, it is desirable to develop a generalized and quantitative performance metric. In this Letter, we derive the near-optimal channel fidelity, a concise and optimization-free metric for arbitrary codes and noise. The metric provides a narrow two-sided bound to the optimal code performance, and it can be evaluated with exactly the same input required by the Knill-Laflamme conditions. We demonstrate the numerical advantage of the near-optimal channel fidelity through multiple qubit code and oscillator code examples. Compared to conventional optimization-based approaches, the reduced computational cost enables us to simulate systems with previously inaccessible sizes, such as oscillators encoding hundreds of average excitations. Moreover, we analytically derive the near-optimal performance for the thermodynamic code and the Gottesman-Kitaev-Preskill code. In particular, the Gottesman-Kitaev-Preskill code's performance under excitation loss improves monotonically with its energy and converges to an asymptotic limit at infinite energy, which is distinct from other oscillator codes.
Quantum error correction (QEC) is essential for protecting quantum information against noise, yet understanding the structure of the Knill-Laflamme (KL) coefficients λijλ_{ij}λij​ from the condition PEi†EjP=λijPPE_i^\dagger E_j P = λ_{ij} PPEi†​Ej​P=λij​P remains challenging, particularly for nonadditive codes. In this work, we introduce the signature vector λ⃗(P)\vecλ(P)λ(P), composed of the off-diagonal KL coefficients λijλ_{ij}λij​, where each coefficient corresponds to equivalence classes of errors counted only once. We define its Euclidean norm λ∗(P)λ^*(P)λ∗(P) as a scalar measure representing the total strength of error correlations within the code subspace defined by the projector PPP. We parameterize PPP on a Stiefel manifold and formulate an optimization problem based on the KL conditions to systematically explore possible values of λ∗λ^*λ∗. Moreover, we show that, for ((n,K,d))((n,K,d))((n,K,d)) codes, λ∗λ^*λ∗ is invariant under local unitary transformations. Applying our approach to the ((6,2,3))((6, 2, 3))((6,2,3)) quantum code, we find that λmin∗=0.6λ^*_{\text{min}} = \sqrt{0.6}λmin∗​=0.6​ and λmax∗=1λ^*_{\text{max}} = 1λmax∗​=1, with λ∗=1λ^* = 1λ∗=1 corresponding to a known degenerate stabilizer code. We construct continuous families of new nonadditive codes parameterized by vectors in R5\mathbb{R}^5R5, with λ∗λ^*λ∗ varying over the interval [0.6,1][\sqrt{0.6}, 1][0.6​,1]. For the ((7,2,3))((7, 2, 3))((7,2,3)) code, we identify λmin∗=0λ^*_{\text{min}} = 0λmin∗​=0 (corresponding to the non-degenerate Steane code) and λmax∗=7λ^*_{\text{max}} = \sqrt{7}λmax∗​=7​ (corresponding to the permutation-invariant code by Pollatsek and Ruskai), and we demonstrate continuous paths connecting these extremes via cyclic codes characterized solely by λ∗λ^*λ∗. Our findings provide new insights into the structure of quantum codes, advance the theoretical foundations of QEC, and open new avenues for investigating intricate relationships between code subspaces and error correlations.