Local equivalence, surface-code states, and matroids
Bibliographic record
Abstract
Recently, Ji et al. disproved the local-unitary--local Clifford (LU-LC) conjecture and showed that the local unitary (LU) and local Clifford (LC) equivalence classes of the stabilizer states are not always the same. Despite the fact that this settles the LU-LC conjecture, a sufficient condition for stabilizer states that violate the LU-LC conjecture is not known. In this paper, we investigate further the properties of stabilizer states with respect to local equivalence. Our first result shows that there exist infinitely many stabilizer states that violate the LU-LC conjecture. In particular, we show that for all numbers of qubits $n\ensuremath{\geqslant}28$, there exist distance-two stabilizer states which are counterexamples to the LU-LC conjecture. We prove that, for all odd $n\ensuremath{\geqslant}195$, there exist stabilizer states with distance greater than two that are LU equivalent but not LC equivalent. Two important classes of stabilizer states that are of great interest in quantum computation are the cluster states and stabilizer states of the surface codes. We show that, under some minimal restrictions, both these classes of states preclude any counterexamples. In this context, we also show that the associated surface codes do not have any encoded non-Clifford transversal gates. We characterize the Calderbank-Shor-Steane (CSS) surface-code states in terms of a class of minor closed binary matroids. In addition to making a connection to an important open problem in binary matroid theory, this characterization does in some cases provide an efficient test for CSS states that are not counterexamples.
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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 | 0.000 |
| 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 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".