Universal Adapters between Quantum Low-Density Parity Check Codes
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Bibliographic record
Abstract
We propose the repetition code adapter as a way to perform joint logical Pauli measurements within a quantum low-density parity check (LDPC) codeblock or between separate such codeblocks, thus providing a flexible tool for fault-tolerant computation with quantum LDPC codes. This adapter is universal in the sense that it works regardless of the LDPC codes involved and the logical Paulis being measured. The construction achieves joint logical Pauli measurement of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>t</a:mi> </a:math> weight <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>O</c:mi> <c:mo stretchy="false">(</c:mo> <c:mi>d</c:mi> <c:mo stretchy="false">)</c:mo> </c:math> operators using <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>O</g:mi> <g:mo stretchy="false">(</g:mo> <g:mi>d</g:mi> <g:mo stretchy="false">)</g:mo> </g:math> time and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mrow> <k:mover> <k:mi>O</k:mi> <k:mo stretchy="false">~</k:mo> </k:mover> </k:mrow> <k:mo stretchy="false">(</k:mo> <k:mi>t</k:mi> <k:mi>d</k:mi> <k:mo stretchy="false">)</k:mo> </k:math> additional qubits and checks, up to a factor polylogarithmic in <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:mi>d</p:mi> </p:math> . As a special case, for some geometrically local codes in fixed <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:mi>D</r:mi> <r:mo>≥</r:mo> <r:mn>2</r:mn> </r:math> dimensions, only <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi>O</t:mi> <t:mo stretchy="false">(</t:mo> <t:mi>t</t:mi> <t:mi>d</t:mi> <t:mo stretchy="false">)</t:mo> </t:math> additional qubits and checks are required instead. By extending the adapter in the case <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:mi>t</x:mi> <x:mo>=</x:mo> <x:mn>2</x:mn> </x:math> , we also construct a toric code adapter that uses <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:mi>O</z:mi> <z:mo stretchy="false">(</z:mo> <z:msup> <z:mi>d</z:mi> <z:mn>2</z:mn> </z:msup> <z:mo stretchy="false">)</z:mo> </z:math> additional qubits and checks to perform addressable logical gates on arbitrary LDPC codes via Dehn twists. To obtain these results, we develop a more general form of graph edge expansion and the <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"> <db:mrow> <db:mi mathvariant="sans-serif">S</db:mi> <db:mi mathvariant="sans-serif">k</db:mi> <db:mi mathvariant="sans-serif">i</db:mi> <db:mi mathvariant="sans-serif">p</db:mi> <db:mi mathvariant="sans-serif">T</db:mi> <db:mi mathvariant="sans-serif">r</db:mi> <db:mi mathvariant="sans-serif">e</db:mi> <db:mi mathvariant="sans-serif">e</db:mi> </db:mrow> </db:math> algorithm, which ensures a sparse transformation between different weight-2 check bases for the classical repetition code.
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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 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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