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arXiv:2602.20605

Quantum circuit design from a retraction-based Riemannian optimization framework

Zhijian Lai, Hantao Nie, Jiayuan Wu, Dong An

quant-phmath-phmath.OCSubmitted Feb 24, 2026• Announced Feb 25, 2026• 0 citations

Abstract

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.

Comments: 28 pages

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Abstract The Quantum Natural Gradient (QNG) method enhances optimization in variational quantum algorithms (VQAs) by incorporating geometric insights from the quantum state space through the Fubini-Study metric. In this work, we extend QNG by introducing higher-order integrators and geodesic corrections using the Riemannian Euler update rule and geodesic equations, deriving an updated rule for the Quantum Natural Gradient with Geodesic Correction (QNGGC). We also develop an efficient method for computing the Christoffel symbols necessary for these corrections, leveraging the parameter-shift rule to enable direct measurement from quantum circuits. Through theoretical analysis and practical examples, we demonstrate that QNGGC significantly improves convergence rates over standard QNG, highlighting the benefits of integrating geodesic corrections into quantum optimization processes. Our approach paves the way for more efficient quantum algorithms, leveraging the advantages of geometric methods.

Variational quantum algorithms are a promising class of algorithms that can be performed on currently available quantum computers. In most settings, the free parameters of a variational circuit are optimized using a classical optimizer that updates parameters in Euclidean geometry. Since quantum circuits are elements of the special unitary group, we can consider an alternative optimization perspective that depends on the structure of this group. In this work, we investigate a Riemannian optimization scheme over the special unitary group and we discuss its implementation on a quantum computer. We illustrate that the resulting Riemannian gradient-flow algorithm has favorable optimization properties for deep circuits and that an approximate version of this algorithm can be performed on near-term hardware. We highlight the connections of our work with previously proposed heuristics like ADAPT-VQE and show that they can be understood as variants of our algorithm.