Quantum circuit design from a retraction-based Riemannian optimization framework

• URL: http://arxiv.org/abs/2602.20605v1
• Date: Tue, 24 Feb 2026 06:54:32 GMT
• Title: Quantum circuit design from a retraction-based Riemannian optimization framework
• Authors: Zhijian Lai, Hantao Nie, Jiayuan Wu, Dong An,
• Abstract summary: Design of quantum circuits for ground state preparation is a fundamental task in quantum information science.<n>We adopt a geometric perspective, formulating the problem as the minimization of an energy cost function directly over the unitary group.<n>We propose a scalable second-order algorithm that constructs a Newton system from measurement data.
• Score: 4.049905580196947
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Online Inference of Constrained Optimization: Primal-Dual Optimality and Sequential Quadratic Programming [55.848340925419286]
  We study online statistical inference for the solutions of quadratic optimization problems with equality and inequality constraints.<n>We develop a sequential programming (SSQP) method to solve these problems, where the step direction is computed by sequentially performing an approximation of the objective and a linear approximation of the constraints.<n>We show that our method global almost moving-average convergence and exhibits local normality with an optimal primal-dual limiting matrix in the sense of Hjek and Le Cam.
  arXiv  Detail & Related papers  (2025-11-27T06:16:17Z)
• An Introduction to the Quantum Approximate Optimization Algorithm [51.56484100374058]
  The tutorial begins by outlining variational quantum circuits and QUBO problems.<n>Next, it explores the QAOA in detail, covering its Hamiltonian formulation, gate decomposition, and example applications.<n>The tutorial extends these concepts to higher-order Hamiltonians and discussing the associated symmetries and circuit construction.
  arXiv  Detail & Related papers  (2025-11-23T09:54:20Z)
• Variational quantum algorithms with exact geodesic transport [0.0]
  Variational quantum algorithms (VQAs) are promising candidates for near-term applications of quantum computers.<n>We introduce exact-geodesic VQAs, a curvature-aware framework that enables analytic Riemannian optimization of variational quantum circuits.
  arXiv  Detail & Related papers  (2025-06-20T18:00:10Z)
• Decentralized Optimization on Compact Submanifolds by Quantized Riemannian Gradient Tracking [45.147301546565316]
  This paper considers the problem of decentralized optimization on compact submanifolds.<n>We propose an algorithm where agents update variables using quantized variables.<n>To the best of our knowledge, this is the first algorithm to achieve an $mathcalO (1/K)$ convergence rate in the presence of quantization.
  arXiv  Detail & Related papers  (2025-06-09T01:57:25Z)
• Application of Langevin Dynamics to Advance the Quantum Natural Gradient Optimization Algorithm [43.21662368414268]
  Quantum Natural Gradient (QNG) algorithm for optimization of variational quantum circuits has been proposed recently.<n>Momentum-QNG is more effective to escape local minima and plateaus in the variational parameter space.<n>Best result is obtained for the Sherrington-Kirkpatrick model in the strong spin glass regime.
  arXiv  Detail & Related papers  (2024-09-03T15:21:16Z)
• Trainability maximization using estimation of distribution algorithms assisted by surrogate modelling for quantum architecture search [8.226785409557598]
  Quantum architecture search (QAS) involves optimizing both the quantum parametric circuit configuration but also its parameters for a variational quantum algorithm. In this paper, we aim to achieve two improvements in QAS: (1) to reduce the number of measurements by an online surrogate model of the evaluation process that aggressively discards architectures of poor performance; (2) to avoid training the circuits when BPs are present. We experimentally validate our proposal for the variational quantum eigensolver and showcase that our algorithm is able to find solutions that have been previously proposed in the literature for the Hamiltonians; but also to outperform the state of the
  arXiv  Detail & Related papers  (2024-07-29T15:22:39Z)
• Fast Quantum Process Tomography via Riemannian Gradient Descent [3.1406146587437904]
  Constrained optimization plays a crucial role in the fields of quantum physics and quantum information science. One specific issue is that of quantum process tomography, in which the goal is to retrieve the underlying quantum process based on a given set of measurement data.
  arXiv  Detail & Related papers  (2024-04-29T16:28:14Z)
• Random coordinate descent: a simple alternative for optimizing parameterized quantum circuits [4.112419132722306]
  This paper introduces a random coordinate descent algorithm as a practical and easy-to-implement alternative to the full gradient descent algorithm. Motivated by the behavior of measurement noise in the practical optimization of parameterized quantum circuits, this paper presents an optimization problem setting amenable to analysis.
  arXiv  Detail & Related papers  (2023-10-31T18:55:45Z)
• Quadratic Unconstrained Binary Optimisation via Quantum-Inspired Annealing [58.720142291102135]
  We present a classical algorithm to find approximate solutions to instances of quadratic unconstrained binary optimisation. We benchmark our approach for large scale problem instances with tuneable hardness and planted solutions.
  arXiv  Detail & Related papers  (2021-08-18T09:26:17Z)
• Avoiding local minima in Variational Quantum Algorithms with Neural Networks [0.0]
  Variational Quantum Algorithms have emerged as a leading paradigm for near-term computation. In this paper we present two algorithms within benchmark them on instances of the gradient landscape problem. We suggest that our approach suggests that the cost landscape is a fruitful path to improving near-term quantum computing algorithms.
  arXiv  Detail & Related papers  (2021-04-07T07:07:28Z)
• A Dynamical Systems Approach for Convergence of the Bayesian EM Algorithm [59.99439951055238]
  We show how (discrete-time) Lyapunov stability theory can serve as a powerful tool to aid, or even lead, in the analysis (and potential design) of optimization algorithms that are not necessarily gradient-based. The particular ML problem that this paper focuses on is that of parameter estimation in an incomplete-data Bayesian framework via the popular optimization algorithm known as maximum a posteriori expectation-maximization (MAP-EM) We show that fast convergence (linear or quadratic) is achieved, which could have been difficult to unveil without our adopted S&C approach.
  arXiv  Detail & Related papers  (2020-06-23T01:34:18Z)
• IDEAL: Inexact DEcentralized Accelerated Augmented Lagrangian Method [64.15649345392822]
  We introduce a framework for designing primal methods under the decentralized optimization setting where local functions are smooth and strongly convex. Our approach consists of approximately solving a sequence of sub-problems induced by the accelerated augmented Lagrangian method. When coupled with accelerated gradient descent, our framework yields a novel primal algorithm whose convergence rate is optimal and matched by recently derived lower bounds.
  arXiv  Detail & Related papers  (2020-06-11T18:49:06Z)
• Optimal Randomized First-Order Methods for Least-Squares Problems [56.05635751529922]
  This class of algorithms encompasses several randomized methods among the fastest solvers for least-squares problems. We focus on two classical embeddings, namely, Gaussian projections and subsampled Hadamard transforms. Our resulting algorithm yields the best complexity known for solving least-squares problems with no condition number dependence.
  arXiv  Detail & Related papers  (2020-02-21T17:45:32Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.