Geometric Analysis of Variational Quantum Eigensolver
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916053077983232 |
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| author | Qin, Zhen |
| author_facet | Qin, Zhen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27795 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric Analysis of Variational Quantum Eigensolver Qin, Zhen Quantum Physics Signal Processing Optimization and Control 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. |
| title | Geometric Analysis of Variational Quantum Eigensolver |
| topic | Quantum Physics Signal Processing Optimization and Control |
| url | https://arxiv.org/abs/2605.27795 |