Quantum Natural Gradient Descent

• Quantum Natural Gradient Descent is a quantum optimization method that uses the Fubini–Study metric and quantum Fisher information to guide parameter updates.
• It adapts classical natural gradient descent to the quantum state manifold, enabling efficient optimization for variational algorithms under noisy and mixed-state conditions.
• Empirical studies show QNGD converges faster than Euclidean methods, reducing iterations in tasks such as VQE and QAOA.

Searching arXiv for recent and foundational papers on Quantum Natural Gradient Descent. Quantum Natural Gradient Descent (QNGD), often abbreviated QNG or NatGrad in the variational-quantum literature, is a quantum generalization of natural gradient descent in which the update direction is defined by the information geometry of quantum states rather than by the Euclidean geometry of parameter space. For a parametrized pure state $|\psi(\theta)\rangle = U(\theta)|0\rangle$ and a cost such as $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$, the canonical update is

$$\theta_{t+1}=\theta_t-\eta\,F(\theta_t)^{-1}\nabla C(\theta_t),$$

where $F$ is the quantum Fisher information matrix, equivalently the pull-back of the Fubini–Study metric or the real part of the Quantum Geometric Tensor. In this form, QNGD is steepest gradient descent on the variational manifold of quantum states; later work generalized the same principle to mixed, noisy, and thermal-state models, and also to non-monotone quantum metrics (Stokes et al., 2019, Koczor et al., 2019, Minervini et al., 26 Feb 2025, Sasaki et al., 2024).

1. Geometric formulation and core update rule

The foundational formulation of QNGD begins from the observation that the parameter vector $\theta\in\mathbb{R}^d$ is not itself the physically relevant object. The relevant manifold is the manifold of quantum states prepared by the ansatz. For a normalized pure state $|\psi(\theta)\rangle$, the Quantum Geometric Tensor is

$$G_{ij}(\theta)=\langle\partial_i\psi(\theta)|\partial_j\psi(\theta)\rangle-\langle\partial_i\psi(\theta)|\psi(\theta)\rangle\langle\psi(\theta)|\partial_j\psi(\theta)\rangle,$$

and its real part

$g_{ij}(\theta)\equiv \Re\,G_{ij}(\theta)$

is the Fubini–Study metric tensor. Infinitesimally,

$ds^2=g_{ij}(\theta)\,d\theta^i\,d\theta^j,$

which coincides, to second order, with the squared Fubini–Study angle $$\arccos^2|\langle\psi(\theta)|\psi(\theta+d\theta)\rangle|$$ (Stokes et al., 2019).

Replacing the Euclidean norm by the norm induced by $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$0 yields the manifold-aware steepest-descent problem

$$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$1

whose first-order optimality condition gives

$$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$2

Equivalent constrained formulations use the local Fubini–Study or fidelity-based distance and lead to the same preconditioned direction (Stokes et al., 2019, Wang et al., 2023).

For pure-state ansätze, the matrix elements are commonly written as

$$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$3

or, with a convention differing by a factor of $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$4, as the pure-state quantum Fisher information matrix. When each parametrized gate is $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$5 with Hermitian generator $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$6, the same matrix can be expressed through generator moments and covariances (Wierichs et al., 2020). In the classical limit where a variational quantum state is restricted to real amplitudes $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$7, one finds $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$8, where $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$9 is the classical Fisher matrix, so QNG is a direct generalization of classical natural gradient to fully complex quantum states (Stokes et al., 2019).

Several works also place QNGD in relation to neighboring variational principles. For noise-free unitary circuits it was introduced as mathematically equivalent to stochastic reconfiguration in quantum Monte Carlo settings, and for pure-state ansätze it was shown to be equivalent to McLachlan’s variational imaginary-time evolution (Sasaki et al., 2024, Koczor et al., 2019).

2. Metric tensors, mixed states, and generalized quantum geometries

The pure-state Fubini–Study metric is only the first member of a broader family. For arbitrary quantum states $$\theta_{t+1}=\theta_t-\eta\,F(\theta_t)^{-1}\nabla C(\theta_t),$$0 generated by completely positive trace-preserving maps, the natural replacement is the quantum Fisher information defined through the symmetric logarithmic derivatives $$\theta_{t+1}=\theta_t-\eta\,F(\theta_t)^{-1}\nabla C(\theta_t),$$1, which satisfy

$$\theta_{t+1}=\theta_t-\eta\,F(\theta_t)^{-1}\nabla C(\theta_t),$$2

The corresponding matrix is

$$\theta_{t+1}=\theta_t-\eta\,F(\theta_t)^{-1}\nabla C(\theta_t),$$3

This yields the mixed-state natural-gradient update

$$\theta_{t+1}=\theta_t-\eta\,F(\theta_t)^{-1}\nabla C(\theta_t),$$4

for costs such as $$\theta_{t+1}=\theta_t-\eta\,F(\theta_t)^{-1}\nabla C(\theta_t),$$5 or any smooth objective (Koczor et al., 2019).

Noisy and non-unitary generalizations emphasize that the geometry of density operators need not be restricted to exact QFI estimation. A modification of Error Suppression by Derangements and Virtual Distillation enables an experimentally efficient approximation of the QFI via the Hilbert–Schmidt metric tensor

$$\theta_{t+1}=\theta_t-\eta\,F(\theta_t)^{-1}\nabla C(\theta_t),$$6

The same work proves that the geometry of typical noisy quantum states is approximately identical in either the Hilbert–Schmidt metric or as characterized by the QFI, and numerical simulations on noisy circuits report that this metric-aware optimizer can significantly outperform other variational techniques (Koczor et al., 2019).

Thermal-state extensions replace the pure-state Fubini–Study geometry by metrics native to Gibbs families. For parameterized thermal states

$$\theta_{t+1}=\theta_t-\eta\,F(\theta_t)^{-1}\nabla C(\theta_t),$$7

closed-form expressions were established for the Fisher–Bures and Kubo–Mori information matrices, together with quantum algorithms for estimating their entries via classical sampling, Hamiltonian simulation, and the Hadamard test (Patel et al., 2024). For parameterized quantum circuits initialized with thermal states, exact methods were later given for three quantum generalizations of the Fisher information matrix—Fisher–Bures, Wigner–Yanase, and Kubo–Mori—with the ordering

$$\theta_{t+1}=\theta_t-\eta\,F(\theta_t)^{-1}\nabla C(\theta_t),$$8

and with unbiased estimation procedures based on the Hadamard test, classical random sampling, and Hamiltonian simulation (Minervini et al., 26 Feb 2025).

A distinct generalization removes monotonicity as a design principle. Conventional QNG is built from the symmetric-logarithmic-derivative metric, one of the monotone metrics singled out by CPTP monotonicity. The non-monotone program retains the generic update

$$\theta_{t+1}=\theta_t-\eta\,F(\theta_t)^{-1}\nabla C(\theta_t),$$9

but allows a broader family of Petz-function-induced metrics, including those derived from sandwiched Rényi-$F$0 divergences. That work states that monotonicity is a crucial condition for conventional QNG to be optimal, and provides analytical and numerical evidence that non-monotone QNG can outperform conventional QNG based on the SLD metric in terms of convergence speed (Sasaki et al., 2024).

3. Metric estimation, approximations, and computational structure

In practical VQAs, the central difficulty is not the form of the update but the cost of obtaining a usable metric tensor. A direct VQE loop with QNG typically proceeds by estimating the energy and the ordinary gradient, assembling the Fubini–Study matrix from overlaps of derivative states, regularizing the metric, and solving the linear system for the natural-gradient step. In one explicit workflow, the ordinary gradient is estimated by analytic parameter-shift, symmetric finite-difference, or SPSA; the Fubini–Study matrix is built from estimates of $F$1 and $F$2; and Tikhonov regularization is applied through $F$3 before inversion (Wierichs et al., 2020).

The resulting asymptotic cost is typically quadratic in the number of parameters. In one detailed accounting for VQE, if $F$4 is the number of parameters, $F$5 the number of Pauli terms in the Hamiltonian, $F$6 the shots per measurement basis for energy and gradient, $F$7 the shots for metric elements, and $F$8 the time to run one circuit of depth $F$9 plus readout, then the ordinary gradient by symmetric finite-difference costs $\theta\in\mathbb{R}^d$0, the Fubini matrix costs $\theta\in\mathbb{R}^d$1, and the NatGrad per-epoch cost scales as $\theta\in\mathbb{R}^d$2. The same source notes that in spin-chain models with translation symmetry $\theta\in\mathbb{R}^d$3 is small, so the Fubini cost is dominant for NatGrad, whereas in chemistry applications with $\theta\in\mathbb{R}^d$4 the relative overhead may be smaller (Wierichs et al., 2020).

To reduce this overhead, the foundational QNG work introduced a block-diagonal approximation for layered ansätze. If

$\theta\in\mathbb{R}^d$5

with each $\theta\in\mathbb{R}^d$6 a product of commuting Pauli rotations, then one ignores inter-layer couplings and computes a local block

$\theta\in\mathbb{R}^d$7

Under this approximation, each optimization step requires $\theta\in\mathbb{R}^d$8 circuit evaluations for the parameter-shift gradient and $\theta\in\mathbb{R}^d$9 further runs to measure the commuting observables needed for the blocks, for a total quantum cost per step of approximately $|\psi(\theta)\rangle$0 circuit evaluations (Stokes et al., 2019).

There are also specialized estimators tailored to hardware. In a silicon-photonic experiment, a Simultaneous Perturbation Stochastic Approximation estimator of the metric used four state overlaps $|\psi(\theta)\rangle$1 and the matrix estimate

$|\psi(\theta)\rangle$2

followed by exponential smoothing and the stabilized inverse power $|\psi(\theta)\rangle$3 with $|\psi(\theta)\rangle$4. For a $|\psi(\theta)\rangle$5 ansatz, the per-iteration overhead was reported as $|\psi(\theta)\rangle$6 runs for the gradient, $|\psi(\theta)\rangle$7 runs for the SPSA overlap metric, and additional runs for the cost-function projectors, yet the total circuit-execution time was reduced by $|\psi(\theta)\rangle$8 on average because the iteration count decreased substantially (Wang et al., 2023).

On the classical-simulation side, QNG introduces its own bottleneck. For unitary ansätze, one analysis showed that state-vector simulation does not dominate the runtime; rather, the Fisher information matrix does. A recurrence-based strategy computes the metric exactly in $|\psi(\theta)\rangle$9 gates and $$G_{ij}(\theta)=\langle\partial_i\psi(\theta)|\partial_j\psi(\theta)\rangle-\langle\partial_i\psi(\theta)|\psi(\theta)\rangle\langle\psi(\theta)|\partial_j\psi(\theta)\rangle,$$0 state-vectors, improving over a naïve $$G_{ij}(\theta)=\langle\partial_i\psi(\theta)|\partial_j\psi(\theta)\rangle-\langle\partial_i\psi(\theta)|\psi(\theta)\rangle\langle\psi(\theta)|\partial_j\psi(\theta)\rangle,$$1 gate count while relying only on “apply gate,” “clone state,” and “inner product” primitives (Jones, 2020).

A more radical complexity reduction appears in Quantum Natural Stochastic Pairwise Coordinate Descent. There, an ensemble-based quantum Fisher information metric is estimated through a random $$G_{ij}(\theta)=\langle\partial_i\psi(\theta)|\partial_j\psi(\theta)\rangle-\langle\partial_i\psi(\theta)|\psi(\theta)\rangle\langle\psi(\theta)|\partial_j\psi(\theta)\rangle,$$2 coordinate block using single-shot measurements, and the update only touches a randomly selected parameter pair. The method uses $$G_{ij}(\theta)=\langle\partial_i\psi(\theta)|\partial_j\psi(\theta)\rangle-\langle\partial_i\psi(\theta)|\psi(\theta)\rangle\langle\psi(\theta)|\partial_j\psi(\theta)\rangle,$$3 samples for the gradient and $$G_{ij}(\theta)=\langle\partial_i\psi(\theta)|\partial_j\psi(\theta)\rangle-\langle\partial_i\psi(\theta)|\psi(\theta)\rangle\langle\psi(\theta)|\partial_j\psi(\theta)\rangle,$$4 for the metric per iteration, giving $$G_{ij}(\theta)=\langle\partial_i\psi(\theta)|\partial_j\psi(\theta)\rangle-\langle\partial_i\psi(\theta)|\psi(\theta)\rangle\langle\psi(\theta)|\partial_j\psi(\theta)\rangle,$$5 quantum measurements per step, in contrast with the $$G_{ij}(\theta)=\langle\partial_i\psi(\theta)|\partial_j\psi(\theta)\rangle-\langle\partial_i\psi(\theta)|\psi(\theta)\rangle\langle\psi(\theta)|\partial_j\psi(\theta)\rangle,$$6 copy complexity attributed to conventional full-metric QNGD (Sohail et al., 2024).

4. Empirical behavior in variational eigensolvers and related VQAs

Empirical work on QNGD has concentrated on the optimization pathologies of VQEs and QAOA. In early numerical demonstrations, minimizing $$G_{ij}(\theta)=\langle\partial_i\psi(\theta)|\partial_j\psi(\theta)\rangle-\langle\partial_i\psi(\theta)|\psi(\theta)\rangle\langle\psi(\theta)|\partial_j\psi(\theta)\rangle,$$7 on $$G_{ij}(\theta)=\langle\partial_i\psi(\theta)|\partial_j\psi(\theta)\rangle-\langle\partial_i\psi(\theta)|\psi(\theta)\rangle\langle\psi(\theta)|\partial_j\psi(\theta)\rangle,$$8 qubits with $$G_{ij}(\theta)=\langle\partial_i\psi(\theta)|\partial_j\psi(\theta)\rangle-\langle\partial_i\psi(\theta)|\psi(\theta)\rangle\langle\psi(\theta)|\partial_j\psi(\theta)\rangle,$$9 layers of alternating $g_{ij}(\theta)\equiv \Re\,G_{ij}(\theta)$0 rotations and CZ entanglers, using $g_{ij}(\theta)\equiv \Re\,G_{ij}(\theta)$1 shots per expectation, showed that vanilla gradient descent fails to find the global minimum, Adam eventually converges but requires many iterations, and QNG—both with block-diagonal and diagonal metric approximations—converges in far fewer steps; the advantage persists as the circuit is made deeper (Stokes et al., 2019).

A systematic VQE study on the transverse-field Ising model sharpened this picture. For a QAOA-type ansatz with depth $g_{ij}(\theta)\equiv \Re\,G_{ij}(\theta)$2, $g_{ij}(\theta)\equiv \Re\,G_{ij}(\theta)$3 parameters, and $g_{ij}(\theta)\equiv \Re\,G_{ij}(\theta)$4 random starts, BFGS was reported to fail beyond $g_{ij}(\theta)\equiv \Re\,G_{ij}(\theta)$5, ADAM with $g_{ij}(\theta)\equiv \Re\,G_{ij}(\theta)$6 to converge up to $g_{ij}(\theta)\equiv \Re\,G_{ij}(\theta)$7 but with epoch count scaling $g_{ij}(\theta)\equiv \Re\,G_{ij}(\theta)$8 and rapidly exceeding $g_{ij}(\theta)\equiv \Re\,G_{ij}(\theta)$9 epochs above a threshold $ds^2=g_{ij}(\theta)\,d\theta^i\,d\theta^j,$0, and NatGrad with regularization and $ds^2=g_{ij}(\theta)\,d\theta^i\,d\theta^j,$1 to always find the global minimum up to $ds^2=g_{ij}(\theta)\,d\theta^i\,d\theta^j,$2, with epoch scaling $ds^2=g_{ij}(\theta)\,d\theta^i\,d\theta^j,$3 and low variance. The same study also found that seemingly benign overparametrization degrades the performance of all optimizers, with BFGS and ADAM failing more often and more severely than NatGrad (Wierichs et al., 2020).

The overparametrized-QAOA experiments are especially notable because they contradict the common expectation that more parameters help first-order training. When one or two Pauli-$ds^2=g_{ij}(\theta)\,d\theta^i\,d\theta^j,$4 rotation layers were added, the BFGS success fraction fell below $ds^2=g_{ij}(\theta)\,d\theta^i\,d\theta^j,$5 for two extra layers at $ds^2=g_{ij}(\theta)\,d\theta^i\,d\theta^j,$6, ADAM with $ds^2=g_{ij}(\theta)\,d\theta^i\,d\theta^j,$7 had success rate zero by $ds^2=g_{ij}(\theta)\,d\theta^i\,d\theta^j,$8, and NatGrad with $ds^2=g_{ij}(\theta)\,d\theta^i\,d\theta^j,$9 maintained success rates above $$\arccos^2|\langle\psi(\theta)|\psi(\theta+d\theta)\rangle|$$0 even with two extra layers. On the Heisenberg XXZ model with Trotter ansatz, BFGS failed for $$\arccos^2|\langle\psi(\theta)|\psi(\theta+d\theta)\rangle|$$1, ADAM with $$\arccos^2|\langle\psi(\theta)|\psi(\theta+d\theta)\rangle|$$2 showed abrupt jumps in epoch count at $$\arccos^2|\langle\psi(\theta)|\psi(\theta+d\theta)\rangle|$$3 and $$\arccos^2|\langle\psi(\theta)|\psi(\theta+d\theta)\rangle|$$4, and NatGrad with $$\arccos^2|\langle\psi(\theta)|\psi(\theta+d\theta)\rangle|$$5 occasionally stalled because of tiny parameter-step interrupts in flat regions but still attained better final precision than BFGS (Wierichs et al., 2020).

These results anchor a recurring empirical claim in the literature: QNGD often improves reliability of convergence in nonconvex quantum landscapes, particularly when local minima, plateaus, or poor conditioning dominate the behavior of Euclidean first-order methods. At the same time, the same benchmark record shows that QNG is not uniformly immune to pathologies such as flat-region stalling or metric singularity, and that the per-iteration or per-epoch gain must be weighed against metric-estimation overhead (Wierichs et al., 2020).

5. Variants, domain-specific adaptations, and experimental realizations

The QNG framework has been adapted to multiple hardware platforms, cost structures, and distributed settings. In optical quantum circuits, the method was generalized to a complex-valued parameter space containing both real parameters and complex parameters treated through Wirtinger calculus. On two state-preparation tasks—a single-photon source and a Gottesman–Kitaev–Preskill state source—the natural-gradient approach was compared with vanilla gradient descent and Adam. For the single-photon source, an $$\arccos^2|\langle\psi(\theta)|\psi(\theta+d\theta)\rangle|$$6-layer single-mode circuit with $$\arccos^2|\langle\psi(\theta)|\psi(\theta+d\theta)\rangle|$$7 parameters reached fidelity $$\arccos^2|\langle\psi(\theta)|\psi(\theta+d\theta)\rangle|$$8 in approximately $$\arccos^2|\langle\psi(\theta)|\psi(\theta+d\theta)\rangle|$$9 steps for QNGD, compared with approximately $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$00 for Adam and approximately $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$01 for SGD; for the Hex-GKP source, a $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$02-layer circuit with $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$03 parameters achieved cost $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$04 in approximately $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$05 steps for QNGD, while Adam required approximately $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$06 steps and SGD stalled around cost $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$07 (Yao et al., 2021).

A hardware-level realization was reported on a fully programmable silicon photonic chip for the dissociation curve of the He-H$$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$08 cation. There, rigorous full QNG reached $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$09 in approximately $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$10 iterations at $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$11 Å, SPSA-QNG with $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$12 took approximately $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$13 iterations averaged over ten runs, and vanilla gradient descent needed approximately $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$14 iterations. The final state fidelity with the exact ground state was $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$15, and the dissociation-curve energies, after a constant shift of $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$16, lay within $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$17 Hartree of theory across all $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$18 (Wang et al., 2023).

In distributed learning, federated quantum natural gradient descent embeds QNG preconditioning into a federated QNN architecture. Each client computes a local block-diagonal QFIM approximation, applies the pseudoinverse locally, and sends only the preconditioned gradient vector to the server. On MNIST tasks with six local VQC clients and equal-sized training splits, FQNGD was reported to converge in roughly half the rounds of Adam and Adagrad, with binary classification accuracy $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$19 versus $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$20 for Adam and ternary accuracy $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$21 versus $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$22 for Adam. Because each client still transmits only one vector of length $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$23 per round, the reduction in rounds directly reduces total communication cost (Qi et al., 2023).

For structured Hamiltonians, several task-aware preconditioners have been proposed. Weighted Approximate QNG defines

$$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$24

where each $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$25 is the Hilbert–Schmidt metric tensor of a $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$26-qubit reduced state and the weights are $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$27. Under weakly entangling assumptions this method is exactly Gauss–Newton for an approximate weighted nonlinear least-squares reformulation, and numerical experiments on $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$28, $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$29, and $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$30 qubits reported fewer iterations than standard QNG, with lower final energy for the $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$31- and $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$32-qubit Ising cases (Shi et al., 7 Apr 2025). A closely related Hamiltonian-aware QNG constructs a pullback metric directly from the lower-dimensional subspace spanned by the Hamiltonian terms, yielding $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$33 measurement overhead rather than the $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$34 of standard QNG. In noiseless molecular benchmarks, it was reported to reach the same accuracy as QNG with approximately $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$35 fewer circuits on $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$36-qubit H$$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$37 and approximately $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$38 fewer circuits on $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$39-qubit LiH and H$$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$40 (Shi et al., 18 Nov 2025).

Optimizer-level modifications further diversify the family. Momentum-QNG derives from a discretized Langevin equation and updates

$$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$41

with benchmark evidence on portfolio VQE and QAOA minimum-vertex-cover instances indicating lower mean $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$42 and improved convergence behavior relative to basic QNG in regimes with barriers or plateaus (Borysenko et al., 2024). Modified Conjugate Quantum Natural Gradient combines QNG with nonlinear conjugate-gradient search directions and a two-dimensional line-search over $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$43; in simulations on H$$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$44 and Heisenberg chains up to $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$45 qubits, it was reported to converge $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$46–$$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$47 faster than QNG on H$$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$48 and to outperform QNG across tested qubit counts and depths (Halla, 10 Jan 2025).

6. Limitations, controversies, and current directions

The principal limitation of QNGD is computational. Multiple sources state that full-metric estimation requires $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$49 quantum measurements and $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$50 classical inversion, or equivalent $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$51 and $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$52 scaling depending on notation. This motivates block-diagonal, diagonal, SPSA, low-rank, moving-average, and coordinate-sampled approximations, but these approximations trade geometric fidelity against cost and noise sensitivity (Stokes et al., 2019, Wierichs et al., 2020, Lisart-Liebermann et al., 23 Apr 2025).

Metric singularity and ill-conditioning are a second recurrent issue. In simple VQE case studies, the pure-state Fisher matrix becomes singular at poles or on separable submanifolds, and natural-gradient trajectories can overshoot and oscillate indefinitely unless the step size is reduced adaptively. Regularization by $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$53, pseudoinversion, step-size decay, or switching to neighboring methods such as imaginary-time evolution are explicit remedies proposed in the literature (Yamamoto, 2019).

A common misconception is that QNG always dominates Euclidean methods. The published record is more conditional. Adaptive QNGD with Armijo backtracking was shown to consistently outperform the original fixed-step QNGD, whose competitiveness depends on near-optimal step-size tuning; however, the same benchmarking also found that a simple Euclidean SGD algorithm equipped with the same adaptive scheme can yield performances similar to the QNG scheme with optimal step size (Atif et al., 2022). Likewise, in shallow-QAOA MaxCut benchmarks under shot noise, block-QNGD reduced iteration counts on very small circuits, but a secant-penalized quasi-Newton method, SP-BFGS, was reported as the recommended optimizer when the shot budget or circuit depth precluded high-precision metric estimation, because QNGD suffered extra shot noise in every entry of the QFIM and the inversion amplified statistical errors (Lisart-Liebermann et al., 23 Apr 2025).

Another ongoing controversy concerns the choice of metric itself. Conventional QNG is built from the symmetric-logarithmic-derivative metric and inherits monotonicity under CPTP maps. The generalized non-monotone program argues that this criterion is not necessary for optimization, proves that conventional QNG is optimal only under the monotonicity requirement, and reports $$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$54–$$C(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$55 faster collapse of the cost in a single-qubit mixed-state experiment for certain Rényi-induced non-monotone metrics (Sasaki et al., 2024). This does not invalidate the standard Fubini–Study or SLD formulation; it instead reframes QNG as a design space of quantum metrics rather than as a single fixed optimizer.

Current directions extend that design space in several dimensions already present in the literature: exact mixed-state metrics for thermal-state initialization, subsystem-weighted and Hamiltonian-aware pullback metrics, communication-aware federated natural gradients, and stochastic estimators with dimension-independent per-step sample complexity (Minervini et al., 26 Feb 2025, Shi et al., 7 Apr 2025, Shi et al., 18 Nov 2025, Sohail et al., 2024). A plausible implication is that “quantum natural gradient descent” now names a family of geometry-aware preconditioners whose practical performance depends as much on metric choice and estimation strategy as on the basic update formula itself.