Bibliographic record
Abstract
We show that using qutrits rather than qubits leads to a substantial reduction in the overhead cost associated with an approach to fault-tolerant quantum computing known as magic state distillation. We construct a family of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>9</mml:mn><mml:mi>m</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">]</mml:mo><mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mn>3</mml:mn></mml:msub></mml:math> triorthogonal qutrit error-correcting codes for any positive integers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi><mml:mo>&#x2264;</mml:mo><mml:mn>3</mml:mn><mml:mi>m</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>2</mml:mn></mml:math> that are suitable for magic state distillation. In magic state distillation, the number of ancillae required to produce a magic state with target error rate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03F5;</mml:mi></mml:math> is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>log</mml:mi><mml:mi>&#x03B3;</mml:mi></mml:msup><mml:mo>&#x2061;</mml:mo><mml:msup><mml:mi>&#x03F5;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>, where the yield parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B3;</mml:mi></mml:math> characterizes the overhead cost. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn><mml:mi>m</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>2</mml:mn></mml:math>, our codes have <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B3;</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>log</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mn>6</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>m</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:math>, which tends to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn></mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi><mml:mo stretchy="false">&#x2192;</mml:mo><mml:mi mathvariant="normal">&#x221E;</mml:mi></mml:math>. Moreover, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>20</mml:mn><mml:mo>,</mml:mo><mml:mn>7</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">]</mml:mo><mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mn>3</mml:mn></mml:msub></mml:math> qutrit code that arises from our construction when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math> already has a yield parameter of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1.51</mml:mn></mml:math> which outperforms all known qubit triorthogonal codes of size less than a few hundred qubits.
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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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.002 |
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".