Quantum Natural Gradient

• Quantum natural gradient is a geometry-aware optimization approach that employs the Fubini–Study metric to guide parameter updates in variational quantum algorithms for improved convergence.
• Hamiltonian-aware and weighted variants tailor the metric to problem-specific structures, reducing measurement overhead while maintaining robust performance.
• Adaptive, momentum, and stochastic enhancements extend QNG to noisy and mixed-state contexts, ensuring efficient resource scaling in VQE, QNNs, and QAOA.

Quantum natural gradient (QNG) is a geometry-aware optimization paradigm for variational quantum algorithms (VQAs), most prominently variational quantum eigensolvers (VQE), quantum neural networks (QNNs), and quantum approximate optimization algorithms (QAOA). QNG generalizes Amari’s classical natural gradient to the curved quantum state manifold, using the Fubini–Study (quantum Fisher) metric to precondition parameter updates. This approach yields algorithms that are invariant under reparameterization, robust to ill-conditioning, and empirically achieve faster convergence than standard gradient descent. Recent advancements include Hamiltonian-aware quantum natural gradients, weighted and approximate quantum natural gradient metrics, efficient resource-saving schemes, and generalizations beyond monotone quantum Fisher metrics.

1. Theoretical Foundations of Quantum Natural Gradient

Quantum natural gradient emerges from information geometry on the manifold of variational quantum states $|\psi(\theta)\rangle=U(\theta)|0\rangle$. The Fubini–Study metric (the real part of the quantum geometric tensor)

$$g_{ij}(\theta) = \mathrm{Re} \left[ \langle\partial_i\psi|\partial_j\psi\rangle - \langle\partial_i\psi|\psi\rangle\langle\psi|\partial_j\psi\rangle \right]$$

measures the infinitesimal distance between quantum states under parameter changes. The QNG update at iteration $t$ is

$$\theta_{t+1} = \theta_t - \eta\, [g(\theta_t)]^{-1}\, \nabla_\theta E(\theta_t),$$

where $$E(\theta) = \langle\psi(\theta)|H|\psi(\theta)\rangle$$, $\nabla_\theta E$ denotes the Euclidean gradient, and $g(\theta)$ is the Fubini–Study metric evaluated at $\theta$ (Stokes et al., 2019).

This update is derived by constraining steps to remain within a fixed Fubini–Study (fidelity) distance and solving a constrained Lagrangian optimization. In the mixed-state scenario, QNG is generalized using appropriate quantum Fisher information matrices such as the symmetric logarithmic derivative (SLD), Kubo-Mori, and Wigner-Yanase metrics, each associated with different monotone Riemannian geometries (Minervini et al., 26 Feb 2025).

2. Hamiltonian-Aware and Weighted Quantum Natural Gradient Metrics

The standard QNG metric is constructed without explicit reference to the target Hamiltonian, leading to a generic geometry over the full quantum state space. To exploit problem structure and minimize resource overhead, Hamiltonian-aware and weighted QNG variants have been developed.

Hamiltonian-Aware QNG (H-QNG):

The metric is pulled back from the subspace spanned by the Hamiltonian’s Pauli terms: $$G_{ij}(\theta) = \sum_{r=1}^v a_r^2 \, \mathrm{Tr}[\partial_i\rho_\theta P_r]\, \mathrm{Tr}[\partial_j\rho_\theta P_r],$$ where $H = \sum_{r=1}^v a_r P_r$ and $$\rho_\theta = |\psi(\theta)\rangle\langle\psi(\theta)|$$. The rescaled H-QNG metric is

$$g^H_{ij}(\theta) = (2\sqrt{\sum_r a_r^2})^{-1} G_{ij}(\theta).$$

This metric can be assembled with no extra quantum measurements beyond those for the gradient (Shi et al., 18 Nov 2025).

Weighted Approximate QNG (WA-QNG):

For $k$-local Hamiltonians $H = \sum_m h_m H_m$, a weighted, subsystem-aware metric is constructed as

$$W(\theta) = \frac{2}{\sum_m h_m^2} \sum_{m} h_m^2 T_m(\theta),$$

where $T_m$ is the Hilbert-Schmidt metric tensor for the reduced $k$-qubit state $\rho_m$. This produces a Gauss–Newton-like update targeting the physical curvatures most relevant for the cost (Shi et al., 7 Apr 2025).

Both approaches lead to sample/measurement complexities per iteration that match vanilla gradient descent, overcoming the quadratic scaling in standard QNG.

3. Algorithmic Realizations and Resource Scaling

QNG algorithms proceed by constructing the metric and the gradient at each step, followed by solving a linear system for the natural gradient direction.

• Standard QNG: Measure all $O(m^2)$ entries of $g_{ij}(\theta)$ (or suitable approximation), yielding $O(m^2)$ quantum circuit calls per iteration.
• H-QNG and WA-QNG: The metric can be formed from the measurements already acquired for gradient evaluation, with overall $O(mv)$ quantum resources per step, analogous to vanilla gradient descent, even for large Hamiltonians and many qubits (Shi et al., 18 Nov 2025, Shi et al., 7 Apr 2025).

Efficient classical simulation strategies for QNG have also been developed, leveraging recursive calculation of the quantum Fisher matrix using $O(P^2)$ gates and constant memory, supporting the simulation of circuits with hundreds of parameters (Jones, 2020).

4. Enhanced Optimizers and Convergence

Adaptive and Momentum Variants:

QNG can be combined with adaptive learning rates (e.g., via Armijo backtracking line search), conjugate-gradient update directions (CQNG), or momentum terms (Momentum-QNG), leading to optimizers that improve convergence robustness and wall-clock efficiency. Empirical studies show that variants such as QNG with geodesic correction (QNGGC) or momentum outperform both basic QNG and classic optimizers (Adam, vanilla SGD) in VQE and QAOA tasks (Halla, 5 Sep 2024, Borysenko et al., 3 Sep 2024, Halla, 10 Jan 2025).

Reparameterization Invariance:

Gradient flows induced by QNG and H-QNG are invariant under smooth invertible reparameterizations of the circuit parameters, ensuring that the optimization trajectory reflects the physical quantum landscape rather than the arbitrary choice of parameterization (Shi et al., 18 Nov 2025, Stokes et al., 2019).

Benchmarks:

On molecules such as H₂, LiH, and H₆, QNG-type updates attain chemical accuracy in 10–15 iterations compared to much slower convergence for vanilla gradient descent, with H-QNG and WA-QNG further reducing quantum resource usage and number of iterations, especially for large or highly local Hamiltonians (Shi et al., 18 Nov 2025, Shi et al., 7 Apr 2025).

5. Practical Extensions: Noise, Mixed States, and Resource-Efficient Approximations

QNG and its generalizations have been extended to:

• Noisy and Non-Unitary Circuits: Mixed-state QNG is defined using monotone quantum Fisher metrics (e.g., SLD, Kubo-Mori, Wigner-Yanase) and can be efficiently approximated for thermal-state initialization or general CPTP parametrizations, leveraging derangement or virtual-distillation circuits and classical-shadow-based estimates (Minervini et al., 26 Feb 2025, Koczor et al., 2019).
• Random and Stochastic-Coordinate Approaches: Random Natural Gradient (RNG) replaces the QNG metric with a randomly sampled classical Fisher information, reducing quantum resources from $O(m^2)$ to $O(m)$ per step. Stochastic-coordinate QNG updates only a subset of parameters with a principal submatrix of the QNG metric, maintaining convergence while reducing quantum overhead (Kolotouros et al., 2023).
• Approximate and Block-Diagonal Metrics: Block-diagonal and diagonal approximations of the QNG metric have been developed for layered ansätze, enabling scalable applications to large circuits and yielding convergence improvements over first-order methods (Stokes et al., 2019).

These developments enable application of geometry-aware optimization on current and near-term (NISQ) quantum hardware, even in the presence of noise and hardware imperfections.

6. Beyond Standard Geometry: Monotonicity, Nonmonotonicity, and Metric Design

Standard QNG is defined with respect to monotone operator metrics (notably the SLD, associated with the maximal monotone Petz function), which ensure invariance under CPTP maps and a well-behaved information geometry. Recent work establishes:

• Optimality of SLD-QNG: The SLD metric yields locally optimal convergence among all monotone-geometry-based QNGs (Sasaki et al., 24 Jan 2024, Miyahara, 21 Oct 2025).
• Nonmonotone Quantum Fisher Geometries: By relaxing monotonicity, QNG can be defined with respect to generalized Petz functions (e.g., from sandwiched Rényi divergences with $\alpha<1/2$), leading to strictly faster convergence per iteration than SLD-QNG, though at the expense of losing monotonicity guarantees (Sasaki et al., 24 Jan 2024, Miyahara, 21 Oct 2025).
• Metric Design via Problem Structure: Selection among monotone and nonmonotone quantum Fisher metrics enables tailoring the geometry to the target optimization problem, including for non-full-rank states or special divergences (Miyahara, 21 Oct 2025).

7. Outlook and Domains of Applicability

Quantum natural gradient constitutes a universal framework for optimization in variational quantum algorithms, embedding the cost function in the correct geometric structure of quantum state space. It proves particularly advantageous for molecular VQE, quantum chemistry, QAOA, and variational state preparation, especially in regimes where local observables or Hamiltonian structure enable efficient metric approximation.

Further areas of application and future research include:

• QNG for Bayesian quantum statistical learning and quantum machine learning (Lopatnikova et al., 2021);
• Natural gradient flows in quantum optimal transport and statistical learning under Wasserstein geometry (Becker et al., 2020);
• Problem-specific metric design and regularization to address ill-conditioning or singularity in the metric tensor.

Novel variants balancing geometric fidelity, measurement cost, noise resilience, and implementation overhead continue to broaden the practical impact of quantum natural gradient methods across quantum algorithmic domains.