We give noise-robust, Probably Approximately Correct (PAC) guarantees of global $\varepsilon$-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 $r = O(\log p)$ for $p$ 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 $\varepsilon$-optimal region with confidence $1-δ$ scales with the curvature dimension $r$ rather than the ambient dimension $p$. With probability at least $1-δ$, the algorithm outputs a region in which all points are $\varepsilon$-optimal, and at least one lies within a bounded neighborhood of the global minimum. The resulting complexity is quasi-polynomial in $p$ and $\varepsilon^{-1}$ and logarithmic in $δ^{-1}$. This identifies a geometric regime in which high-probability global optimization remains feasible despite shot noise.
We give noise-robust, Probably Approximately Correct (PAC) guarantees of global ε-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(logp) for p 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-optimal region with confidence 1−δ scales with the curvature dimension r rather than the ambient dimension p. With probability at least 1−δ, the algorithm outputs a region in which all points are ε-optimal, and at least one lies within a bounded neighborhood of the global minimum. The resulting complexity is quasi-polynomial in p and ε−1 and logarithmic in δ−1. This identifies a geometric regime in which high-probability global optimization remains feasible despite shot noise.
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.
We formalize the exact optimization boundaries, physical time evolution minimums, and finite-resource measurement constraints that govern the performance of the Quantum Natural Gradient (QNG) update rule on parameterized variational manifolds. First, we state and prove Theorem 4.1 to characterize the exact convergence properties of QNG updates under bounded Riemannian metrics. Crucially, we isolate the causal mechanism linking ansatz expressivity to geometric degradation, proving that the pulled-back Fubini–Study metric elements inherit the exponential O(2−N ) variance collapse dictated by Haar measure integration over random unitary circuit layouts. This scaling is shown to drive an exponential inflation of the metric condition number, establishing a clear geometric origin for the numerical tracking instabilities that stall optimization vectors in large-scale parameter spaces. Second, we formalize the Information-Geometric Quantum Brachistochrone (Theorem 4.2), proving that under instantaneous velocity limitations derived from the system’s energy variance, the absolute minimum time T required to map a state vector between two configurations is bounded strictly below by the integrated Riemannian geodesic distance Lgeodesic. This result demonstrates that flat Euclidean metrics severely underestimate physical time requirements near highly entangled boundaries where local manifold curvature dominates. Finally, we derive a sharp Shot-Noise Metric Concentration Bound (Theorem 4.3), resolving an exact cyclic trace simplification to prove that finite-shot perturbation errors concentrate according to a structural noise alignment factor κ(θ) = Tr(Σg−1)/∥∆θ∥2g . Multi-trial multinomial simulations validate this relation across all shot budgets (102 to 105), verifying that the empirical breach probability tracks safely below the analytical trace constraint line. This bound exposes a fundamental engineering trade-off: as an optimization path approaches ill conditioned regions where the minimum eigenvalue collapses (λmin → 0), the tracking framework requires a quadratic expansion of measurement resources to suppress shot noise and stabilize control loops.
Variational quantum algorithms, such as the Variational Quantum Eigensolver (VQE), are severely limited on superconducting quantum processing units (QPUs) by physical shot noise, vanishing gradients, and metered credit limits. Standard gradient-descent algorithms (e.g., SGD, Adam) are unstable under the large variance of noisy estimators, often diverging or getting trapped in local minima. In this work, we present a resource-efficient optimization framework that combines low-rank projection (GaLore [1]) and second-order diagonal Hessian curvature scaling (Sophia [2]) to bypass these noise barriers in non-Abelian variational spaces (where the sequential Givens rotations do not commute and thus do not form an Abelian subgroup of U(N)). By offloading gradient and Hessian evaluations to a classical processor using a polynomial fermionic shadow tomography state reconstruction [3, 4], we eliminate parameter-shift circuit execution overhead and bypass the QPU noise floor. We execute this closed-loop GaLore-Sophia VQE on the 156-qubit IBM Heron r2 architecture (ibm_marrakesh), achieving a convergence energy of −1.378 Hartrees at step 3—reaching 97.4% of the topologically restricted 3-parameter Givens chain geometric limit of -\sqrt{2} Hartrees in under 70 seconds of QPU time. We clarify that this benchmark is measured exclusively against this sub-optimal parameter subspace rather than the exact molecular ground state of -\sqrt{5} Hartrees, which requires richer ansatz topologies. Furthermore, we analyze the parameter-count stability of the ansatz, proving that the VQE ansatz limit systematically approaches the ground state energy as parameter count scales. Finally, to rigorously validate the stability of quantum information preservation, we execute a physical temporal parity sweep on ibm_marrakesh (Job ID: d8ishf1e8nrc73bint50, 1000 shots). Fitting the physical decay yields a Lyapunov decay time τL ≈ 6.13 steps and prediction horizon τ ≈ 4.68 steps (at 86.5% accuracy threshold), confirming the universal DHP ratio τ / τL ≈ 0.763 on generational Heron hardware, matching the stable cross-device trend observed in P25 [5] and P27 [6].
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.
Variational Quantum Eigensolver (VQE) faces significant challenges due to hardware noise and the presence of barren plateaus and local traps in the optimization landscape. To mitigate the detrimental effects of these issues, we introduce a general formalism that optimizes hardware resource utilization and accuracy by projecting VQE optimizations onto a reduced-dimensional subspace, followed by a set of posteriori corrections. Our method partitions the ansatz into a lower dimensional principal subspace and a higher-dimensional auxiliary subspace based on a conjecture of temporal hierarchy present among the parameters during optimization. The adiabatic approximation exploits this hierarchy, restricting optimization to the lower dimensional principal subspace only. This is followed by an efficient higher dimensional auxiliary space reconstruction without the need to perform variational optimization. These reconstructed auxiliary parameters are subsequently included in the cost-function via a set of auxiliary subspace corrections (ASC), leading to a "plummeting effect" in the energy landscape toward a more optimal minimum without utilizing any additional quantum hardware resources. Numerical simulations show that, when integrated with any chemistry-inspired ansatz, our method can provide one to two orders of magnitude better estimation of the minima. In addition, based on the adiabatic approximation, we introduce a novel initialization strategy driven by unitary rotation generators for accelerated convergence of gradient-informed dynamic quantum algorithms. Our method shows heuristic evidence of alleviating the effects of local traps, facilitating convergence toward a more optimal minimum.