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Journal of Physics A Mathematical and Theoretical

Geometric analysis of variational quantum eigensolver

Zhen Qin

Published Jul 15, 2026• 0 citations
Publisher

Abstract

Abstract The variational quantum eigensolver (VQE) is a fundamental algorithm in quantum computing, yet its theoretical understanding remains fragmented across fixed-ansatz and adaptive-circuit formulations, which are typically analyzed using different frameworks. As a result, a coherent geometric characterization of the VQE remains missing. In this paper, we establish a geometric analysis of VQE in terms of optimization landscape, initialization guarantee, and noise robustness. First, from a Riemannian geometric perspective, we study the optimization landscape via an ansatz-free product-unitary formulation over the unitary group, where each layer unitary is treated as an independent optimization variable, thereby capturing both fixed-ansatz and adaptive-circuit paradigms. For the single-unitary case, we establish explicit linear convergence guarantees for Riemannian gradient descent (RGD) under proper initialization and characterize the global landscape by proving that every critical point is either a global minimum or a strict saddle point. For the product-unitary case, we extend the convergence analysis of RGD and show that the convergence rate deteriorates polynomially with circuit depth, providing a convergence-theoretic explanation of the barren plateau phenomenon through the interplay between Hilbert-space geometry, target-unitary complexity, per-layer expressivity, and circuit depth. Second, we analyze initialization guarantees for the above convergence results. We consider a practical variational circuit composed of small-angle random Pauli rotations and prove that this initialization scheme satisfies the required geometric conditions with high probability. We further characterize how the resulting initialization error depends on the reference state, circuit depth, and random angles. Third, we study robustness under finite-shot measurements for VQE with Pauli-decomposed Hamiltonians. We show that RGD retains linear convergence up to a noise-dominated neighborhood of the global minimizer. Moreover, we extend the analysis to non-uniform measurement allocation under a fixed measurement budget and show that coefficient-adaptive allocation achieves strictly lower statistical error than uniform sampling. Together, these results provide a systematic geometric characterization of the optimization, initialization, and measurement behavior of VQE, offering new insights into its trainability and robustness.

DOI
10.1088/1751-8121/ae8b60

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Abstract The variational quantum eigensolver (VQE) is a fundamental algorithm in quantum computing, yet its theoretical understanding remains fragmented across fixed-ansatz and adaptive-circuit formulations, which are typically analyzed using different frameworks. As a result, a coherent geometric characterization of the VQE remains missing. In this paper, we establish a geometric analysis of VQE in terms of optimization landscape, initialization guarantee, and noise robustness. First, from a Riemannian geometric perspective, we study the optimization landscape via an ansatz-free product-unitary formulation over the unitary group, where each layer unitary is treated as an independent optimization variable, thereby capturing both fixed-ansatz and adaptive-circuit paradigms. For the single-unitary case, we establish explicit linear convergence guarantees for Riemannian gradient descent (RGD) under proper initialization and characterize the global landscape by proving that every critical point is either a global minimum or a strict saddle point. For the product-unitary case, we extend the convergence analysis of RGD and show that the convergence rate deteriorates polynomially with circuit depth, providing a convergence-theoretic explanation of the barren plateau phenomenon through the interplay between Hilbert-space geometry, target-unitary complexity, per-layer expressivity, and circuit depth. Second, we analyze initialization guarantees for the above convergence results. We consider a practical variational circuit composed of small-angle random Pauli rotations and prove that this initialization scheme satisfies the required geometric conditions with high probability. We further characterize how the resulting initialization error depends on the reference state, circuit depth, and random angles. Third, we study robustness under finite-shot measurements for VQE with Pauli-decomposed Hamiltonians. We show that RGD retains linear convergence up to a noise-dominated neighborhood of the global minimizer. Moreover, we extend the analysis to non-uniform measurement allocation under a fixed measurement budget and show that coefficient-adaptive allocation achieves strictly lower statistical error than uniform sampling. Together, these results provide a systematic geometric characterization of the optimization, initialization, and measurement behavior of VQE, offering new insights into its trainability and robustness.
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.
Variational Quantum Algorithms (VQAs) are a class of hybrid quantum-classical algorithms that leverage on classical optimization tools to find the optimal parameters for a parameterized quantum circuit. One relevant application of VQAs is the Variational Quantum Eigensolver (VQE), which aims at steering the output of the quantum circuit to the ground state of a certain Hamiltonian. Recent works have provided global convergence guarantees for VQEs under suitable local surjectivity and smoothness hypotheses, but little has been done in characterizing convergence of these algorithms when the underlying quantum circuit is affected by noise. In this work, we derive an upper bound on the error on the optimal parameters of a VQE under the effect of different coherent and incoherent noise processes. We then procced to show robust convergence guarantees of the algorithm to the perturbed optimal parameters. Our work provides novel theoretical insight into the behavior of VQAs subject to noise. Furthermore, we accompany our results with numerical simulations implemented via Pennylane.
The Variational Quantum Eigensolver (VQE) is a promising candidate for quantum applications on near-term Noisy Intermediate-Scale Quantum (NISQ) computers. Despite a lot of empirical studies and recent progress in theoretical understanding of VQE's optimization landscape, the convergence for optimizing VQE is far less understood. We provide the first rigorous analysis of the convergence of VQEs in the over-parameterization regime. By connecting the training dynamics with the Riemannian Gradient Flow on the unit-sphere, we establish a threshold on the sufficient number of parameters for efficient convergence, which depends polynomially on the system dimension and the spectral ratio, a property of the problem Hamiltonian, and could be resilient to gradient noise to some extent. We further illustrate that this overparameterization threshold could be vastly reduced for specific VQE instances by establishing an ansatz-dependent threshold paralleling our main result. We showcase that our ansatz-dependent threshold could serve as a proxy of the trainability of different VQE ansatzes without performing empirical experiments, which hence leads to a principled way of evaluating ansatz design. Finally, we conclude with a comprehensive empirical study that supports our theoretical findings.
We give noise-robust, Probably Approximately Correct (PAC) guarantees of global ε\varepsilonε-optimality for the Variational Quantum Eigensolver under explicit geometric conditions. For periodic ansatzes with bounded generators -- yielding a globally Lipschitz cost landscape on a toroidal parameter space -- we assume that the low-energy region containing the global minimum is a Morse--Bott submanifold whose normal Hessian has rank r=O(log⁡p)r = O(\log p)r=O(logp) for ppp parameters, and which satisfies polynomial fiber regularity with respect to coordinate-aligned, embedded flats. This low-curvature-dimensional structure serves as a model for regimes in which only a small number of directions control energy variation, and is consistent with mechanisms such as strong parameter tying together with locality in specific multiscale and tied shallow architectures. Under this assumption, the sample complexity required to find an ε\varepsilonε-optimal region with confidence 1−δ1-δ1−δ scales with the curvature dimension rrr rather than the ambient dimension ppp. With probability at least 1−δ1-δ1−δ, the algorithm outputs a region in which all points are ε\varepsilonε-optimal, and at least one lies within a bounded neighborhood of the global minimum. The resulting complexity is quasi-polynomial in ppp and ε−1\varepsilon^{-1}ε−1 and logarithmic in δ−1δ^{-1}δ−1. This identifies a geometric regime in which high-probability global optimization remains feasible despite shot noise.
Variational Quantum Eigensolver (VQE) algorithms suffer from barren plateaus, where gradients vanish with system size and circuit depth. Although many mitigation strategies exist, their connection to convergence performance under different iteration budgets remains unclear. Moreover, a systematic analysis identifying which state-of-the-art mitigation techniques perform best under specific scenarios is also lacking. We benchmark four approaches, Local-Global, Adiabatic, State Efficient Ansatz (SEA), and Pretrained VQE, against standard VQE on molecular systems from 4 to 14 qubits, analyzing gradient variance up to 50 layers and convergence over 1000 iterations. Our results show that the impact of gradient preservation is iteration-dependent. In the 14-qubit BeH2 system, Pretrained VQE outperforms SEA at 100 iterations despite lower gradient variance, but SEA becomes 2.2x more accurate at 1000 iterations. For smaller systems, SEA achieves near-exact energies (H2: 10^-5 Ha, LiH: 2x10^-4 Ha) with fidelities 0.999, while standard methods plateau early. The results demonstrate that robust barren plateau mitigation depends on aligning the chosen strategy with both system size and available computational budget, rather than treating gradient variance as the sole predictor of performance.
Standard Zero-Noise Extrapolation (ZNE) in Variational Quantum Eigensolvers (VQE) assumes a single global noise scaling parameter, failing to capture heterogeneous and non-linear noise propagation across layered quantum circuits. We simulate VQE parameter optimisation at a challenging L=6 hardware-efficient depth, extracting the exact expectation values at each iteration. When the optimal parameters theta_star are found, they are fixed, and the multi-parameter noise surface is mapped. We propose a multi-parameter noise framework, modeling expectation values as a function of layer-wise noise scaling factors E(lambda_1, lambda_2, ..., lambda_L). Using structured unitary folding U_l = U_l (U_l^dagger U_l)^((lambda_l - 1) / 2), we independently scale the noise contribution of each ansatz layer and empirically map the resulting multi-dimensional noise surface. We analyze the separability of inter-layer noise interactions, identify layers with disproportionate error contributions via the gradient dE/dlambda_l, and compare scalar ZNE with multivariate polynomial regression and Gaussian Process (GP) extrapolation across H_2, LiH, and BeH_2 under depolarizing, amplitude damping, and coherent over-rotation noise. Our results demonstrate that noise is non-uniform and partially coupled across layers, and that layer-resolved ZNE with higher-order extrapolation can improve accuracy and variance under realistic NISQ noise models. These findings provide geometric insight into noise propagation in variational circuits and motivate targeted, layer-specific error mitigation strategies.