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<p>Stochastic gradient descent (SGD) is a widely used optimization technique in classical machine learning and the Variational Quantum Eigensolver (VQE). In VQE implementations on quantum hardware, measurement shot noise is inevitable. We analyze how this noise affects optimization dynamics, especially escape from saddle points in non-convex loss landscapes. Our simulations show that the escape time scales as a power law with respect to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\eta /N_s\)</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\eta\)</EquationSource> </InlineEquation> is the learning rate and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(N_s\)</EquationSource> </InlineEquation> is the number of measurements. Through SGD analysis, we provide theoretical insight into how measurement noise facilitates escape. In particular, we demonstrate that a continuous-time approximation via stochastic differential equations (SDE) accurately captures the transient escape dynamics. This suggests that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\eta /N_s\)</EquationSource> </InlineEquation> represents effective noise strength, indicating that increasing <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\eta\)</EquationSource> </InlineEquation> or decreasing <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(N_s\)</EquationSource> </InlineEquation> has similar effects. While concerns exist about the SDE’s validity in stationary regimes, our findings clarify its applicability to transient behavior. Our work improves understanding of the role of measurement noise in VQE optimization.</p>

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Impact of measurement noise on escaping saddles in variational quantum algorithms

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月17日

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  • Eriko Kaminishi,
  • Takashi Mori,
  • Michihiko Sugawara,
  • Naoki Yamamoto

摘要

Stochastic gradient descent (SGD) is a widely used optimization technique in classical machine learning and the Variational Quantum Eigensolver (VQE). In VQE implementations on quantum hardware, measurement shot noise is inevitable. We analyze how this noise affects optimization dynamics, especially escape from saddle points in non-convex loss landscapes. Our simulations show that the escape time scales as a power law with respect to \(\eta /N_s\) , where \(\eta\) is the learning rate and \(N_s\) is the number of measurements. Through SGD analysis, we provide theoretical insight into how measurement noise facilitates escape. In particular, we demonstrate that a continuous-time approximation via stochastic differential equations (SDE) accurately captures the transient escape dynamics. This suggests that \(\eta /N_s\) represents effective noise strength, indicating that increasing \(\eta\) or decreasing \(N_s\) has similar effects. While concerns exist about the SDE’s validity in stationary regimes, our findings clarify its applicability to transient behavior. Our work improves understanding of the role of measurement noise in VQE optimization.