Time optimal problems on Lie groups and applications to quantum control
Bibliographic record
Abstract
In this paper we introduce a natural compactification of a left (right) invariant affine control system on a semi-simple Lie group $ G $ in which the control functions belong to the Lie algebra of a compact Lie subgroup $ K $ of $ G $ and we investigate conditions under which the time optimal solutions of this compactified system are "approximately" time optimal for the original system. The basic ideas go back to the papers of R.W. Brockett and his collaborators in their studies of time optimal transfer in quantum control ([ 1 ], [ 2 ]). We showed that every affine system can be decomposed into two natural systems that we call horizontal and vertical. The horizontal system admits a convex extension whose reachable sets are compact and hence posess time-optimal solutions. We then obtained an explicit formula for the time-optimal solutions of this convexified system defined by the symmetric Riemannian pair $ (G, K) $ under the assumption that the Lie algebra generated by the control vector fields is equal to the Lie algebra of $ K $. In the second part of the paper we applied our results to the quantum systems known as Icing $ n $-chains (introduced in [ 2 ]). We showed that the two-spin chains conform to the theory in the first part of the paper but that the three-spin chains show new phenomena that take it outside of the above theory. In particular, we showed that the solutions for the symmetric three-spin chains studied by ([ 3 ], [ 4 ]) are solvable in terms of elliptic functions with the solutions completely different from the ones encountered in the two-spin chains.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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".