Symdyn: An automated algebraic solution for high-order quantum systems
Bibliographic record
Abstract
Many important quantum systems are characterized by Hamiltonians expressible as a linear combination of time-independent generators of a finite Lie algebra, H ̂ ( t ) = ∑ l = 1 L η l ( t ) g ̂ l . The Wei-Norman framework provides a powerful analytical method for computing the exact time-evolution operator in a factorized form, U ̂ ( t ) = ∏ l = 1 L e Λ l ( t ) g ̂ l . By exploiting the underlying algebraic structure, the method yields a system of differential equations—the Wei-Norman equations—which determine the time-evolution operator coefficients Λ l ( t ) from the Hamiltonian ones η l ( t ) . Although robust and formally extendable, applying the method by hand becomes increasingly difficult or even intractable for high-order systems ( L ≥ 6 ). In this work, we introduce , a Python library that automates the application of this method. Given the commutators between generators, the library symbolically derives the Wei-Norman equations and efficiently computes nested commutators, similarity transformations, and the differential equations that yield Baker-Campbell-Hausdorff-like relations. We validate the library by recovering known results for quantum systems associated with the Lie algebras su ( 1 , 1 ) , su ( 2 ) , sl ( 2 ) , and so ( 2 , 1 ) , and further demonstrate its capabilities by deriving the Wei-Norman equations for a system of two coupled time-dependent harmonic oscillators involving 11 generators. Additionally, we specialize the library to handle the Lie group SU ( N ) using a generic Cartan-Weyl basis for its algebra, and demonstrate its versatility by deriving the Wei-Norman equations for su ( 2 ) , su ( 3 ) , and su ( 4 ) , including the explicit construction of a set of quantum gates for universal quantum computation.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.026 | 0.006 |
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".