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, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mover accent="true"> <a:mi>H</a:mi> <a:mo>̂</a:mo> </a:mover> <a:mrow> <a:mo>(</a:mo> <a:mi>t</a:mi> <a:mo>)</a:mo> </a:mrow> <a:mo>=</a:mo> <a:msubsup> <a:mo>∑</a:mo> <a:mrow> <a:mi>l</a:mi> <a:mo>=</a:mo> <a:mn>1</a:mn> </a:mrow> <a:mi>L</a:mi> </a:msubsup> <a:msub> <a:mi>η</a:mi> <a:mi>l</a:mi> </a:msub> <a:mrow> <a:mo>(</a:mo> <a:mi>t</a:mi> <a:mo>)</a:mo> </a:mrow> <a:msub> <a:mover accent="true"> <a:mi>g</a:mi> <a:mo>̂</a:mo> </a:mover> <a:mi>l</a:mi> </a:msub> </a:mrow> </a:math> . The Wei-Norman framework provides a powerful analytical method for computing the exact time-evolution operator in a factorized form, <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mover accent="true"> <d:mi>U</d:mi> <d:mo>̂</d:mo> </d:mover> <d:mrow> <d:mo>(</d:mo> <d:mi>t</d:mi> <d:mo>)</d:mo> </d:mrow> <d:mo>=</d:mo> <d:msubsup> <d:mo>∏</d:mo> <d:mrow> <d:mi>l</d:mi> <d:mo>=</d:mo> <d:mn>1</d:mn> </d:mrow> <d:mi>L</d:mi> </d:msubsup> <d:msup> <d:mi>e</d:mi> <d:mrow> <d:msub> <d:mi mathvariant="normal">Λ</d:mi> <d:mi>l</d:mi> </d:msub> <d:mrow> <d:mo>(</d:mo> <d:mi>t</d:mi> <d:mo>)</d:mo> </d:mrow> <d:msub> <d:mover accent="true"> <d:mi>g</d:mi> <d:mo>̂</d:mo> </d:mover> <d:mi>l</d:mi> </d:msub> </d:mrow> </d:msup> </d:mrow> </d:math> . 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 <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:msub> <h:mi mathvariant="normal">Λ</h:mi> <h:mi>l</h:mi> </h:msub> <h:mrow> <h:mo>(</h:mo> <h:mi>t</h:mi> <h:mo>)</h:mo> </h:mrow> </h:mrow> </h:math> from the Hamiltonian ones <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:msub> <j:mi>η</j:mi> <j:mi>l</j:mi> </j:msub> <j:mrow> <j:mo>(</j:mo> <j:mi>t</j:mi> <j:mo>)</j:mo> </j:mrow> </j:mrow> </j:math> . Although robust and formally extendable, applying the method by hand becomes increasingly difficult or even intractable for high-order systems ( <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:mi>L</k:mi> <k:mo>≥</k:mo> <k:mn>6</k:mn> </k:mrow> </k:math> ). 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 <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:mi mathvariant="fraktur">su</l:mi> <l:mo>(</l:mo> <l:mn>1</l:mn> <l:mo>,</l:mo> <l:mn>1</l:mn> <l:mo>)</l:mo> </l:mrow> <l:mo>,</l:mo> <l:mo> </l:mo> <l:mrow> <l:mi mathvariant="fraktur">su</l:mi> <l:mo>(</l:mo> <l:mn>2</l:mn> <l:mo>)</l:mo> </l:mrow> <l:mo>,</l:mo> <l:mo> </l:mo> <l:mrow> <l:mi mathvariant="fraktur">sl</l:mi> <l:mo>(</l:mo> <l:mn>2</l:mn> <l:mo>)</l:mo> </l:mrow> </l:math> , and <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:mrow> <p:mi mathvariant="fraktur">so</p:mi> <p:mo>(</p:mo> <p:mn>2</p:mn> <p:mo>,</p:mo> <p:mn>1</p:mn> <p:mo>)</p:mo> </p:mrow> </p:math> , 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 <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"> <r:mrow> <r:mi mathvariant="italic">SU</r:mi> </r:mrow> <r:mrow> <r:mo>(</r:mo> <r:mi>N</r:mi> <r:mo>)</r:mo> </r:mrow> </r:math> using a generic Cartan-Weyl basis for its algebra, and demonstrate its versatility by deriving the Wei-Norman equations for <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"> <t:mrow> <t:mi mathvariant="fraktur">su</t:mi> <t:mo>(</t:mo> <t:mn>2</t:mn> <t:mo>)</t:mo> </t:mrow> <t:mo>,</t:mo> <t:mo> </t:mo> <t:mrow> <t:mi mathvariant="fraktur">su</t:mi> <t:mo>(</t:mo> <t:mn>3</t:mn> <t:mo>)</t:mo> </t:mrow> </t:math> , and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML"> <w:mrow> <w:mi mathvariant="fraktur">su</w:mi> <w:mo>(</w:mo> <w:mn>4</w:mn> <w:mo>)</w:mo> </w:mrow> </w:math> , 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 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) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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".