Geometric Analysis of Variational Quantum Eigensolver

• URL: http://arxiv.org/abs/2605.27795v1
• Date: Wed, 27 May 2026 00:31:18 GMT
• Title: Geometric Analysis of Variational Quantum Eigensolver
• Abstract summary: Variational Quantum Eigensolver (VQE) is a fundamental algorithm in quantum computing.<n>In this paper, we establish a geometric analysis of VQE in terms of optimization landscape, initialization guarantee, and noise robustness.
• Score: 8.743290038942542
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Variational Quantum Eigensolver (VQE) is a fundamental algorithm in quantum computing, yet a coherent geometric characterization of VQE remains missing due to fragmented analyses across fixed-ansatz and adaptive-circuit formulations. In this paper, we establish a geometric analysis of VQE in terms of optimization landscape, initialization guarantee, and noise robustness. First, we study the optimization landscape via an ansatz-free product-unitary formulation over the unitary group, unifying both paradigms. For the single-unitary case, we establish linear convergence of Riemannian gradient descent (RGD) and prove the strict saddle property. For the product-unitary case, we show the convergence rate deteriorates polynomially with circuit depth, providing a geometric explanation of the barren plateau phenomenon. Second, we prove that small-angle random Pauli-rotation circuits satisfy the required initialization conditions with high probability. Third, we show that RGD retains linear convergence under finite-shot measurements, and that coefficient-adaptive allocation achieves strictly lower statistical error than uniform sampling under a fixed measurement budget.

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