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<p>Advances in quantum computing over the last two decades have required sophisticated mathematical frameworks to deepen the understanding of quantum algorithms. In this review, we introduce the theory of Lie groups and their algebras to analyze two fundamental problems in quantum computing as done in some recent works, e.g., Nielsen (Quantum Inf. Comput. <b>6</b>(3), 213–262 <CitationRef CitationID="CR1">2006</CitationRef>) and Cerezo et al. (Nat. Rev. Phys. <b>3</b>(9), 625–644 <CitationRef CitationID="CR2">2021</CitationRef>). Firstly, we describe the geometric formulation of quantum computational complexity, given by the length of the shortest path on the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13538_2025_1923_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">\(SU(2^n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>U</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> manifold with respect to a right-invariant Finsler metric. Secondly, we deal with the barren plateau phenomenon in variational quantum algorithms (VQAs), where we use the dynamical Lie algebra (DLA) to identify algebraic sources of untrainability.</p>

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Lie Groups for Quantum Complexity and Barren Plateau Theory

  • 期刊论文

  • 发表日期：

    2025年10月20日

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  • P. A. S. de Alcântara,
  • Gabriel Audi,
  • Leandro Morais

摘要

Advances in quantum computing over the last two decades have required sophisticated mathematical frameworks to deepen the understanding of quantum algorithms. In this review, we introduce the theory of Lie groups and their algebras to analyze two fundamental problems in quantum computing as done in some recent works, e.g., Nielsen (Quantum Inf. Comput. 6(3), 213–262 2006) and Cerezo et al. (Nat. Rev. Phys. 3(9), 625–644 2021). Firstly, we describe the geometric formulation of quantum computational complexity, given by the length of the shortest path on the \(SU(2^n)\) S U ( 2 n ) manifold with respect to a right-invariant Finsler metric. Secondly, we deal with the barren plateau phenomenon in variational quantum algorithms (VQAs), where we use the dynamical Lie algebra (DLA) to identify algebraic sources of untrainability.