Limits on the storage of quantum information in a volume of space
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Bibliographic record
Abstract
We study the fundamental limits on the reliable storage of quantum information in lattices of qubits by deriving tradeoff bounds for approximate quantum error correcting codes. We introduce a notion of local approximate correctability and code distance, and give a number of equivalent formulations thereof, generalizing various exact error-correction criteria. Our tradeoff bounds relate the number of physical qubits <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>, the number of encoded qubits <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>, the code distance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>, the accuracy parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B4;</mml:mi></mml:math> that quantifies how well the erasure channel can be reversed, and the locality parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x2113;</mml:mi></mml:math> that specifies the length scale at which the recovery operation can be done. In a regime where the recovery is successful to accuracy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B4;</mml:mi></mml:math> that is exponentially small in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x2113;</mml:mi></mml:math>, which is the case for perturbations of local commuting projector codes, our bound reads <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi><mml:msup><mml:mi>d</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mfrac><mml:mn>2</mml:mn><mml:mrow><mml:mi>D</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mo>&#x2264;</mml:mo><mml:mi>O</mml:mi><mml:mrow class="MJX-TeXAtom-OPEN"><mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo></mml:mrow><mml:mi>n</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>log</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mi>n</mml:mi><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>D</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mrow class="MJX-TeXAtom-CLOSE"><mml:mo maxsize="1.2em" minsize="1.2em">)</mml:mo></mml:mrow></mml:math> for codes on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math>-dimensional lattices of Euclidean metric. We also find that the code distance of any local approximate code cannot exceed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mrow class="MJX-TeXAtom-OPEN"><mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo></mml:mrow><mml:mi>&#x2113;</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">(</mml:mo><mml:mi>D</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mrow class="MJX-TeXAtom-CLOSE"><mml:mo maxsize="1.2em" minsize="1.2em">)</mml:mo></mml:mrow></mml:math> if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B4;</mml:mi><mml:mo>&#x2264;</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>&#x2113;</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>. As a corollary of our formulation of correctability in terms of logical operator avoidance, we show that the code distance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math> and the size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>d</mml:mi><mml:mo stretchy="false">&#x007E;</mml:mo></mml:mover></mml:mrow></mml:math> of a minimal region that can support all approximate logical operators satisfies <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>d</mml:mi><mml:mo stretchy="false">&#x007E;</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>D</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mo>&#x2264;</mml:mo><mml:mi>O</mml:mi><mml:mrow class="MJX-TeXAtom-OPEN"><mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo></mml:mrow><mml:mi>n</mml:mi><mml:msup><mml:mi>&#x2113;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mfrac><mml:mi>D</mml:mi><mml:mrow><mml:mi>D</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mrow class="MJX-TeXAtom-CLOSE"><mml:mo maxsize="1.2em" minsize="1.2em">)</mml:mo></mml:mrow></mml:math>, where the logical operators are accurate up to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mrow class="MJX-TeXAtom-OPEN"><mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mi>&#x03B4;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow class="MJX-TeXAtom-CLOSE"><mml:mo maxsize="1.2em" minsize="1.2em">)</mml:mo></mml:mrow></mml:math> in operator norm. Finally, we prove that for two-dimensional systems if logical operators can be approximated by operators supported on constant-width flexible strings, then the dimension of the code space must be bounded. This supports one of the assumptions of algebraic anyon theories, that there exist only finitely many anyon types.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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 it