Symdyn: An automated algebraic solution for high-order quantum systems

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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