A Controlled Null Result for Per-Gate Riemannian Optimization in VQE
Abstract
Parametrized gates in a variational quantum eigensolver (VQE) live on a product of unitary groups, which makes Riemannian optimization a natural candidate for improving training. We test this candidate in the clean per-gate setting: each single-qubit gate is optimized directly on U (2) and compared with Adam on an unconstrained Euler-angle chart representing the same physical gate, initialized from the same unitaries and tuned with a separate learning-rate sweep. On a validated hydrogen-chain benchmark, we find no measurable practical advantage from the Riemannian machinery. The direct manifold optimizer and the Euler-coordinate baseline are interleaved in final error and success rate; the classical coordinate baseline is not slower in iterations on the successful cells; and the choices of Riemannian metric, retraction, and momentum vector transport do not change the conclusion. We also identify four failure modes that can manufacture an apparent positive result: a trivial product-state benchmark, a projected-matrix Adam baseline that discards most of its own step, a non-unitary inherited Cayley retraction that lets the reported energy pass below the variational minimum, and a learning-rate grid whose optimum sits at its boundary. Each failure mode produced a plausible multi-seed result in our own workflow before being caught. The practical output of this paper is therefore a controlled null result and a validation checklist for VQE optimizer comparisons.
Related
- A connectedness result for convex polynomial multi-objective optimization without boundedness
- Centered Weak Discrete Riemannian Gradients: A Unified Framework for Riemannian Optimization
- Shape Optimization and Bernoulli Free Boundary Problems for the Riemannian p-Laplacian
- MPFA: A Pareto Front Approximation Method for Riemannian Bi-objective Optimization
- A Stochastic Riemannian Alternating Descent Ascent Method for Nonsmooth Composite Expectation Optimization on Riemannian Manifolds
BibTeX
@misc{belozerova2026controlled,
title = {{A Controlled Null Result for Per-Gate Riemannian Optimization in VQE}},
author = {Polina Belozerova and Pavel Osinenko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17395},
eprint = {2610.17395},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17395},
}
RIS
TY - UNPB
TI - A Controlled Null Result for Per-Gate Riemannian Optimization in VQE
AU - Polina Belozerova
AU - Pavel Osinenko
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17395
UR - https://arxiv.org/abs/2610.17395
AB - Parametrized gates in a variational quantum eigensolver (VQE) live on a product of unitary groups, which makes Riemannian optimization a natural candidate for improving training. We test this candidate in the clean per-gate setting: each single-qubit gate is optimized directly on U (2) and compared with Adam on an unconstrained Euler-angle chart representing the same physical gate, initialized from the same unitaries and tuned with a separate learning-rate sweep. On a validated hydrogen-chain benchmark, we find no measurable practical advantage from the Riemannian machinery. The direct manifold optimizer and the Euler-coordinate baseline are interleaved in final error and success rate; the classical coordinate baseline is not slower in iterations on the successful cells; and the choices of Riemannian metric, retraction, and momentum vector transport do not change the conclusion. We also identify four failure modes that can manufacture an apparent positive result: a trivial product-state benchmark, a projected-matrix Adam baseline that discards most of its own step, a non-unitary inherited Cayley retraction that lets the reported energy pass below the variational minimum, and a learning-rate grid whose optimum sits at its boundary. Each failure mode produced a plausible multi-seed result in our own workflow before being caught. The practical output of this paper is therefore a controlled null result and a validation checklist for VQE optimizer comparisons.
ER -