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arXiv:2604.05627 · Journal of Physics A Mathematical and Theoretical

Loss-aware state space geometry for quantum variational algorithms

Ankit Gill, Kunal Pal

quant-phSubmitted Apr 7, 2026• Announced Apr 8, 2026• 0 citations
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Abstract

The natural gradient descent optimisation technique is an efficient optimising protocol for broad classes of classical and quantum systems that takes the underlying geometry of the parameter manifold into account by means of using either the Fisher information metric of the classical probability distribution function or the Fubini-Study tensor of the associated parametrised quantum states in the consequent update rules. Even though the natural gradient descent procedure utilises the geometry of the space of probability or states, it is, however, insensitive to the measure of parametrised distance on the space of possible outcomes when the corresponding optimising problem is considered for the expectation value of a classical or quantum observable with respect to the probability distribution or the quantum state. In this work, we introduce a generic optimising principle, where the intrinsic geometry of the space of outcomes has been taken into account suitably, either by using an ambient space construction with a base statistical manifold with the usual Fisher information metric (or the Fubini-Study tensor), where the loss hypersurface is embedded to, or by means of a first-principle construction from the overlap of nearby quantum states on the projective Hilbert space. This construction as well as a family of conformal variants yields a form of loss-aware natural gradient updates that rescale the effective step size while preserving the descent direction. We benchmark the resulting optimisers on variational quantum circuit examples and on a classical neural network task, finding that, while the standard natural gradient remains the most robust on average, the proposed conformal schemes can improve best-case convergence in favourable regimes.

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The natural gradient descent optimisation technique is an efficient optimising protocol for broad classes of classical and quantum systems that takes the underlying geometry of the parameter manifold into account by means of using either the Fisher information metric of the classical probability distribution function or the Fubini-Study tensor of the associated parametrised quantum states in the consequent update rules. Even though the natural gradient descent procedure utilises the geometry of the space of probability or states, it is, however, insensitive to the measure of parametrised distance on the space of possible outcomes when the corresponding optimising problem is considered for the expectation value of a classical or quantum observable with respect to the probability distribution or the quantum state. In this work, we introduce a generic optimising principle, where the intrinsic geometry of the space of outcomes has been taken into account suitably, either by using an ambient space construction with a base statistical manifold with the usual Fisher information metric (or the Fubini-Study tensor), where the loss hypersurface is embedded to, or by means of a first-principle construction from the overlap of nearby quantum states on the projective Hilbert space. This construction as well as a family of conformal variants yields a form of loss-aware natural gradient updates that rescale the effective step size while preserving the descent direction. We benchmark the resulting optimisers on variational quantum circuit examples and on a classical neural network task, finding that, while the standard natural gradient remains the most robust on average, the proposed conformal schemes can improve best-case convergence in favourable regimes.
The Variational Quantum Eigensolver (VQE) is one of the most promising algorithms for current quantum devices. It employs a classical optimizer to iteratively update the parameters of a variational quantum circuit in order to search for the ground state of a given Hamiltonian. The efficacy of VQEs largely depends on the optimizer employed. Recent studies suggest that Quantum Natural Gradient Descent (QNG) can achieve faster convergence than vanilla gradient descent (VG), but at the cost of additional quantum resources to estimate Fubini-Study metric tensor in each optimization step. The Fubini-Study metric tensor used in QNG is related to the entire quantum state space and does not incorporate information about the target Hamiltonian. To take advantage of the structure of the Hamiltonian and address the limitation of additional computational cost in QNG, we propose Hamiltonian-aware Quantum Natural Gradient Descent (H-QNG). In H-QNG, we propose to use the Riemannian pullback metric induced from the lower-dimensional subspace spanned by the Hamiltonian terms onto the parameter space. We show that H-QNG inherits the desirable features of both approaches: the low quantum computational cost of VG and the reparameterization-invariance of QNG. We also validate its performance through numerical experiments on molecular Hamiltonians, showing that H-QNG achieves faster convergence to chemical accuracy while requiring fewer quantum computational resources.
A quantum generalization of Natural Gradient Descent is presented as part of a general-purpose optimization framework for variational quantum circuits. The optimization dynamics is interpreted as moving in the steepest descent direction with respect to the Quantum Information Geometry, corresponding to the real part of the Quantum Geometric Tensor (QGT), also known as the Fubini-Study metric tensor. An efficient algorithm is presented for computing a block-diagonal approximation to the Fubini-Study metric tensor for parametrized quantum circuits, which may be of independent interest.
Variational quantum circuits are promising tools whose efficacy depends on their optimisation method. For noise-free unitary circuits, the quantum generalisation of natural gradient descent was recently introduced. The method can be shown to be equivalent to imaginary time evolution, and is highly effective due to a metric tensor reconciling the classical parameter space to the device's Hilbert space. Here we generalise quantum natural gradient to consider arbitrary quantum states (both mixed and pure) via completely positive maps; thus our circuits can incorporate both imperfect unitary gates and fundamentally non-unitary operations such as measurements. Whereas the unitary variant relates to classical Fisher information, here we find that quantum Fisher information defines the core metric in the space of density operators. Numerical simulations indicate that our approach can outperform other variational techniques when circuit noise is present. We finally assess the practical feasibility of our implementation and argue that its scalability is only limited by the number and quality of imperfect gates and not by the number of qubits.
ABSTRACT The lack of both a consistently effective and broadly reliable classical optimizer for variational quantum algorithms and a clear physical understanding of how the Fubini‐Study (FS) metric evolves during optimization remains a central challenge for practical variational quantum eigensolver (VQE) applications. In this work, we systematically investigate the quantum natural gradient (QNG) as a geometry‐aware optimization strategy and benchmark its performance against commonly used classical optimizers, including gradient‐based, derivative‐free, and adaptive methods. Using representative molecular systems, we analyze energy convergence, parameter‐space trajectories, and the dynamical evolution of the FS metric tensor. QNG consistently achieves faster and more stable convergence in the studied molecules by aligning parameter updates with the intrinsic curvature of the quantum state manifold. The evolution of the metric reveals a rapid suppression of irrelevant parameter directions and a reduced state distinguishability near the ground state. Obtained under ideal and simplified noise conditions, these results clarify the geometric origin of QNG advantage and highlight the potential suitability of geometry‐aware optimization for robust VQE implementation on near‐term quantum devices.
Variational quantum algorithms (VQAs) are commonly developed under the assumption that a parameterized circuit corresponds to a single, well-defined physical realization. In practice, this assumption is not satisfied. Differences in compilation, gate ordering, and hardware-level perturbations lead to multiple admissible circuit realizations, each associated with a distinct objective over the same parameter space. The resulting optimization problem is therefore not defined by a single loss function, but by a family of configuration-dependent objectives whose aggregate behaviour must be considered. This study formulates optimization explicitly at the level of such configuration families. A consistent extension of natural-gradient methods is developed in which both the descent direction and the local geometric structure are obtained through averaging across configurations. The resulting update preserves the structure of quantum natural gradient methods while ensuring alignment with the family-defined objective. The approach is evaluated using exact statevector simulations on transverse-field Ising and Heisenberg Hamiltonians under controlled variability. Across all tested regimes, the method produces lower average loss, substantially reduced dispersion across configurations, and improved worst-case performance relative to both configuration-specific optimization and gradient-averaged baselines. These results show that when the objective depends on a set of admissible circuit realizations, optimization must be constructed at the same level to obtain stable and reliable solutions.
Variational quantum algorithms are promising tools whose efficacy depends on their optimization method. For noise-free unitary circuits, the quantum generalization of natural gradient descent has been introduced and shown to be equivalent to imaginary time evolution: the approach is effective due to a metric tensor reconciling the classical parameter space to the device's Hilbert space. Here we generalize quantum natural gradient to consider arbitrary quantum states (both mixed and pure) via completely positive maps; thus our circuits can incorporate both imperfect unitary gates and fundamentally nonunitary operations such as measurements. We employ the quantum Fisher information (QFI) as the core metric in the space of density operators. A modification of the error suppression by derangements (ESD) and virtual distillation (VD) techniques enables an accurate and experimentally efficient approximation of the QFI via the Hilbert-Schmidt metric tensor using prior results on the dominant eigenvector of noisy quantum states. Our rigorous proof also establishes the fundamental observation that the geometry of typical noisy quantum states is (approximately) identical in either the Hilbert-Schmidt metric or as characterized by the QFI. In numerical simulations of noisy quantum circuits we demonstrate the practicality of our approach and confirm it can significantly outperform other variational techniques.