Quantum Generalized Reed-Solomon codes concatenated with random rate one inner stabilizer codes asymptotically attain the Quantum Gilbert-Varshamov bound
Bibliographic record
Abstract
AbstractA good quantum code corrects for a linear number of errors. It has been shownthat random codes attains the QGVB with a relative distance H −1q 2 ((1−R)/2) wherethe code is q-ary and R is the rate of the code (number of encoded systems / blocklength). However,randomcodeshavelittlestructure. Inthispaper,westudyafamilyofconcatenatedq-arystabilizercodes. Eachoftheinnercodesisarandomrate1q-arystabilizer code of block length n. The outer code is a quantum MDS code with blocklengthq n andalphabetsizeq n ,anarbitraryrateR≤1,anddistanceN(1−R)/2+1thatmeets the Quantum Singleton bound. Fixing the outer code rate and letting n grow,the concatenated stabilizer code has a distance that almost surely attains the QGVB.This partially generalizes Thommesen’s result, where heshowed thatthedistanceof aconcatenated code with a Reed-Solomon outer code and random inner linear codes ofarbitraryrateattainstheGilbert-Varshamovboundalmostsurely. 1 Introduction A family of q-ary quantum codes of increasing block length is defined to be good if the ratioof its distance to its block length approaches a non-zero constant. Designing good quantumcodes is highly nontrivial, just as it is in the classical case. The quantum Gilbert-Varshamovbound (QGVB) is a lower bound on an achievable relative distance of a quantum code ofa fixed rate. Explicit constructions of good quantum codes for q ≤ 7 have been studied[1, 2, 3], but they all fail to satisfy the QGVB. The QGVB bound is attainable for the familyof all random quantum codes [4], the family of random stabilizer codes [5], and the familyof random nondegenerate stabilizer codes [6], and the family of random degenerate stabilizercodes [6]. We show that concatenated quantum codes, with a quantum outer code havinga known structure and being efficiently decodable, and randomly chosen independent innerquantum codes also attains the QGVB. Thus our family of random quantum codes has morestructure than previously studied examples.In this paper, we generalize a special case of Thommesen’s result [7] to the quantumcase. He showed that a code from the family of binary concatenated codes made with a1
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 source (direct Gemma or distilled Codex), 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".