Quantum error correction from complexity in Brownian SYK
Bibliographic record
Abstract
A bstract We study the robustness of quantum error correction in a one-parameter ensemble of codes generated by the Brownian SYK model, where the parameter quantifies the encoding complexity. The robustness of error correction by a quantum code is upper bounded by the “mutual purity” of a certain entangled state between the code subspace and environment in the isometric extension of the error channel, where the mutual purity of a density matrix ρ AB is the difference $$ {\mathcal{F}}_{\rho}\left(A:B\right)\equiv \textrm{Tr}\ {\rho}_{AB}^2-\textrm{Tr}\ {\rho}_A^2\ \textrm{Tr}\ {\rho}_B^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>ρ</mml:mi> </mml:msub> <mml:mfenced> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>:</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:mfenced> <mml:mo>≡</mml:mo> <mml:mi>Tr</mml:mi> <mml:mspace/> <mml:msubsup> <mml:mi>ρ</mml:mi> <mml:mi>AB</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>−</mml:mo> <mml:mi>Tr</mml:mi> <mml:mspace/> <mml:msubsup> <mml:mi>ρ</mml:mi> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mspace/> <mml:mi>Tr</mml:mi> <mml:mspace/> <mml:msubsup> <mml:mi>ρ</mml:mi> <mml:mi>B</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> . We show that when the encoding complexity is small, the mutual purity is O (1) for the erasure of a small number of qubits (i.e., the encoding is fragile). However, this quantity decays exponentially, becoming O (1/ N ) for O (log N ) encoding complexity. Further, at polynomial encoding complexity, the mutual purity saturates to a plateau of O ( e − N ). We also find a hierarchy of complexity scales associated to a tower of subleading contributions to the mutual purity that quantitatively, but not qualitatively, adjust our error correction bound as encoding complexity increases. In the AdS/CFT context, our results suggest that any portion of the entanglement wedge of a general boundary subregion A with sufficiently high encoding complexity is robustly protected against low-rank errors acting on A with no prior access to the encoding map. From the bulk point of view, we expect such bulk degrees of freedom to be causally inaccessible from the region A despite being encoded in it.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".