Generalization bounds in hybrid quantum-classical machine learning models

Hybrid quantum-classical models aim to harness the strengths of both quantum computing and classical machine learning, but their practical potential remains poorly understood. In this work, we develop a unified mathematical framework for analyzing generalization in hybrid models, offering insight into how these systems learn from data. We establish a generalization bound of the form <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mover accent="true"> <a:mi mathvariant="script">O</a:mi> <a:mo>̃</a:mo> </a:mover> <a:mrow> <a:mo>(</a:mo> <a:mfrac> <a:msup> <a:mi>α</a:mi> <a:mi>k</a:mi> </a:msup> <a:msqrt> <a:mi>N</a:mi> </a:msqrt> </a:mfrac> <a:mrow> <a:mo>(</a:mo> <a:msup> <a:mi>k</a:mi> <a:mfrac> <a:mn>3</a:mn> <a:mn>2</a:mn> </a:mfrac> </a:msup> <a:msqrt> <a:mrow> <a:mi>m</a:mi> <a:mi>n</a:mi> </a:mrow> </a:msqrt> <a:mo>+</a:mo> <a:msqrt> <a:mrow> <a:mi>T</a:mi> <a:mo form="prefix">log</a:mo> <a:mi>T</a:mi> </a:mrow> </a:msqrt> <a:mo>)</a:mo> </a:mrow> <a:mo>)</a:mo> </a:mrow> </a:mrow> </a:math> for <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mi>N</e:mi> </e:math> training data points, <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mi>T</f:mi> </f:math> trainable quantum gates, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mi>n</g:mi> </g:math> -dimensional quantum circuit output, and <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mi>k</h:mi> </h:math> bounded linear layers <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:mrow> <i:mo>∥</i:mo> </i:mrow> <i:msub> <i:mi>F</i:mi> <i:mi>i</i:mi> </i:msub> <i:msub> <i:mrow> <i:mo>∥</i:mo> </i:mrow> <i:mi>F</i:mi> </i:msub> <i:mo>≤</i:mo> <i:mi>α</i:mi> </i:mrow> </i:math> , where <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:mi>i</j:mi> <j:mo>=</j:mo> <j:mn>1</j:mn> <j:mo>,</j:mo> <j:mo>⋯</j:mo> <j:mo>,</j:mo> <j:mi>k</j:mi> </j:mrow> </j:math> and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:mi>F</k:mi> <k:mo>∈</k:mo> <k:msup> <k:mi mathvariant="double-struck">R</k:mi> <k:mrow> <k:mi>m</k:mi> <k:mo>×</k:mo> <k:mi>n</k:mi> </k:mrow> </k:msup> </k:mrow> </k:math> interspersed with activation functions. This generalization bound decomposes into quantum and classical contributions, providing a theoretical framework to separate their influence and clarifying their interaction. Alongside the bound, we highlight conceptual limitations of applying classical statistical learning theory in the hybrid setting and suggest promising directions for future theoretical work.