arXiv subjects

Hantao Nie

Publications and source records attributed to Hantao Nie.

5 recordsLinked to original sources

arXiv2026

Pauli-structured preconditioning for quantum linear system solvers

Preconditioning is a fundamental technique for accelerating classical linear system solvers, and understanding when its benefits persist in quantum linear system (QLS) solvers is important for assessing the practical resource requirements of quantum linear algebra. In QLS algorithms, however, the potential advantage of preconditioning may be offset by the normalization overhead incurred by composing separate block-encodings of the system matrix and the preconditioner, as observed in recent work. This limitation leaves open whether additional algebraic structure can make preconditioning effective in quantum access models. Motivated by this question, we show that Pauli-structured representations of both the system matrix and the preconditioner allow the preconditioned operator to be accessed through regrouped Pauli expansions. In this setting, algebraic regrouping of Pauli products can reduce the Pauli coefficient weight of the preconditioned operator, thereby altering the normalization parameters relevant to quantum algorithms. We derive explicit size and coefficient-weight bounds for the regrouped Pauli representations, and we trace their consequences for both direct block-encoding constructions and randomized Pauli linear system solvers. These results identify when Pauli-structured preconditioning can reduce the effective complexity parameters of QLS algorithms, rather than merely improving the classical condition number. Numerical experiments on a finite-dimensional synthetic benchmark show reductions in norm-aware direct block-encoding diagnostics and in the randomized QLS per-sample depth proxy.

quant-ph↗

arXiv2026

Nystr\"om Approximation on Manifolds

Computations on a manifold often involve constructing an operator on the tangent space and computing its inverse, which can be time-consuming in many applications. In order to reduce the computational costs and preserve the benign properties of tangent operators, we develop the Riemannian Nystr\"om approximation on manifolds, a low-rank approximation of tangent operators through subspace projections onto the tangent space. The developed approximation is intrinsically constructed and inherits desirable properties from the classical Nystr\"om approximation, e.g., positive semidefiniteness and approximation errors. Instead of the Gaussian sketching, we introduce the Haar--Grassmann sketching condition with a coordinate-free representation, which remains compatible under isometric vector transport across tangent spaces. Moreover, we propose a randomized Newton-type method for optimization on manifolds in which the linear system is constructed via the Riemannian Nystr\"om approximation. Numerical experiments on the SPD and Grassmann manifolds, together with principal geodesic analysis on real data, illustrate that the proposed approximation reduces the computational cost of operators while maintaining comparable accuracy.

math.NA↗

arXiv2026

Quantum circuit design from a retraction-based Riemannian optimization framework

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.

quant-ph↗

arXiv2025

Quantum Alternating Direction Method of Multipliers for Semidefinite Programming

Semidefinite programming (SDP) is a fundamental convex optimization problem with wide-ranging applications. However, solving large-scale instances remains computationally challenging due to the high cost of solving linear systems and performing eigenvalue decompositions. In this paper, we present a quantum alternating direction method of multipliers (QADMM) for SDPs, building on recent advances in quantum computing. An inexact ADMM framework is developed, which tolerates errors in the iterates arising from block-encoding approximation and quantum measurement. Within this robust scheme, we design a polynomial proximal operator to address the semidefinite conic constraints and apply the quantum singular value transformation to accelerate the most costly projection updates. We prove that the scheme converges to an $\epsilon$-optimal solution of the SDP problem under the strong duality assumption. A detailed complexity analysis shows that the QADMM algorithm achieves favorable scaling with respect to dimension compared to the classical ADMM algorithm and quantum interior point methods, highlighting its potential for solving large-scale SDPs.

math.OC↗

arXiv2025

An Augmented Lagrangian Value Function Method for Lower-level Constrained Stochastic Bilevel Optimization

Recently, lower-level constrained bilevel optimization has attracted increasing attention. However, existing methods mostly focus on either deterministic cases or problems with linear constraints. The main challenge in stochastic cases with general constraints is the bias and variance of the hyper-gradient, arising from the inexact solution of the lower-level problem. In this paper, we propose a novel stochastic augmented Lagrangian value function method for solving stochastic bilevel optimization problems with nonlinear lower-level constraints. Our approach reformulates the original bilevel problem using an augmented Lagrangian-based value function and then applies a penalized stochastic gradient method that carefully manages the noise from stochastic oracles. We establish an equivalence between the stochastic single-level reformulation and the original constrained bilevel problem and provide a non-asymptotic rate of convergence for the proposed method. The rate is further enhanced by employing variance reduction techniques. Extensive experiments on synthetic problems and real-world applications demonstrate the effectiveness of our approach.

math.OC↗