Quantum circuit design from a retraction-based Riemannian optimization framework
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912923018854400 |
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| author | Lai, Zhijian Nie, Hantao Wu, Jiayuan An, Dong |
| author_facet | Lai, Zhijian Nie, Hantao Wu, Jiayuan An, Dong |
| contents | Designing quantum circuits for ground state preparation is a fundamental task in quantum information science. However, standard Variational Quantum Algorithms (VQAs) are often constrained by limited ansatz expressivity and difficult optimization landscapes. To address these issues, we adopt a geometric perspective, formulating the problem as the minimization of an energy cost function directly over the unitary group. We establish a retraction-based Riemannian optimization framework for this setting, ensuring that all algorithmic procedures are implementable on quantum hardware. Within this framework, we unify existing randomized gradient approaches under a Riemannian Random Subspace Gradient Projection (RRSGP) method. While recent geometric approaches have predominantly focused on such first-order gradient descent techniques, efficient second-order methods remain unexplored. To bridge this gap, we derive explicit expressions for the Riemannian Hessian and show that it can be estimated directly on quantum hardware via parameter-shift rules. Building on this, we propose the Riemannian Random Subspace Newton (RRSN) method, a scalable second-order algorithm that constructs a Newton system from measurement data. Numerical simulations indicate that RRSN achieves quadratic convergence, yielding high-precision ground states in significantly fewer iterations compared to both existing first-order approaches and standard VQA baselines. Ultimately, this work provides a systematic foundation for applying a broader class of efficient Riemannian algorithms to quantum circuit design. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20605 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantum circuit design from a retraction-based Riemannian optimization framework Lai, Zhijian Nie, Hantao Wu, Jiayuan An, Dong Quantum Physics Mathematical Physics Optimization and Control 81P68, 90C26, 65K10 Designing quantum circuits for ground state preparation is a fundamental task in quantum information science. However, standard Variational Quantum Algorithms (VQAs) are often constrained by limited ansatz expressivity and difficult optimization landscapes. To address these issues, we adopt a geometric perspective, formulating the problem as the minimization of an energy cost function directly over the unitary group. We establish a retraction-based Riemannian optimization framework for this setting, ensuring that all algorithmic procedures are implementable on quantum hardware. Within this framework, we unify existing randomized gradient approaches under a Riemannian Random Subspace Gradient Projection (RRSGP) method. While recent geometric approaches have predominantly focused on such first-order gradient descent techniques, efficient second-order methods remain unexplored. To bridge this gap, we derive explicit expressions for the Riemannian Hessian and show that it can be estimated directly on quantum hardware via parameter-shift rules. Building on this, we propose the Riemannian Random Subspace Newton (RRSN) method, a scalable second-order algorithm that constructs a Newton system from measurement data. Numerical simulations indicate that RRSN achieves quadratic convergence, yielding high-precision ground states in significantly fewer iterations compared to both existing first-order approaches and standard VQA baselines. Ultimately, this work provides a systematic foundation for applying a broader class of efficient Riemannian algorithms to quantum circuit design. |
| title | Quantum circuit design from a retraction-based Riemannian optimization framework |
| topic | Quantum Physics Mathematical Physics Optimization and Control 81P68, 90C26, 65K10 |
| url | https://arxiv.org/abs/2602.20605 |