A Compass on the Quantum State Sphere: The Hopf Ansatz for Arbitrary Pure-State Optimization
Abstract
Optimizing an arbitrary pure state requires more than a circuit that can prepare it: one also needs coordinates, local distance information, and measurable derivative directions. We introduce the Hopf ansatz, a binary-tree circuit for arbitrary normalized real and complex state vectors. Internal angles split probability between subtrees, while leaf phases supply the complex degrees of freedom. The same tree recovers physical angles from known amplitudes, gives an analytic diagonal metric on the state-vector sphere, and prepares each nonzero coordinate tangent as a normalized state. For Hamiltonian objectives and other costs with the same local transition-moment form, each gradient component is a known metric factor times a transition moment between the current state and a tangent state. Signed branch states express these moments through expectation-value measurements. Indexed preparation groups magnitude tangents by tree depth and collects complex phase tangents into one leaf-indexed block. The number of compiled gradient-access circuit families is logarithmic in Hilbert-space dimension; measurement repetitions needed for a prescribed accuracy remain a separate statistical cost. In deterministic real-state benchmarks with exact objectives and gradients and known global optima, metric-aware Hopf updates reach numerical-precision median gaps. Relative to coordinate-Adam baselines, they yield smaller mean final gaps in variational eigensolver tests and concentrate more runs at numerical precision in metrology-inspired tests. The Hopf ansatz thus supplies a compass for arbitrary pure-state optimization, with coordinates, geometry, and measurable directions in one circuit structure.
Related
- Compass in the Mirror: Quantum Backpropagation with the Hopf Ansatz
- The Quantum Sphere: A Physically Realizable Optimization Benchmark with Provable Linear Convergence in White- and Black-Box Settings
- Quantum assisted ghost Gutzwiller ansatz
- Energy Is Not Enough: Leakage-Aware Ansatz Criterion for Variational Quantum Eigensolvers
- Quasi-Normal Modes and thermalization in integrable Quantum Field Theories from the Thermodynamic Bethe Ansatz
BibTeX
@misc{lin2026compass,
title = {{A Compass on the Quantum State Sphere: The Hopf Ansatz for Arbitrary Pure-State Optimization}},
author = {Ruge Lin and Guangxi Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.14231},
eprint = {2607.14231},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.14231},
}
RIS
TY - UNPB
TI - A Compass on the Quantum State Sphere: The Hopf Ansatz for Arbitrary Pure-State Optimization
AU - Ruge Lin
AU - Guangxi Li
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2607.14231
UR - https://arxiv.org/abs/2607.14231
AB - Optimizing an arbitrary pure state requires more than a circuit that can prepare it: one also needs coordinates, local distance information, and measurable derivative directions. We introduce the Hopf ansatz, a binary-tree circuit for arbitrary normalized real and complex state vectors. Internal angles split probability between subtrees, while leaf phases supply the complex degrees of freedom. The same tree recovers physical angles from known amplitudes, gives an analytic diagonal metric on the state-vector sphere, and prepares each nonzero coordinate tangent as a normalized state. For Hamiltonian objectives and other costs with the same local transition-moment form, each gradient component is a known metric factor times a transition moment between the current state and a tangent state. Signed branch states express these moments through expectation-value measurements. Indexed preparation groups magnitude tangents by tree depth and collects complex phase tangents into one leaf-indexed block. The number of compiled gradient-access circuit families is logarithmic in Hilbert-space dimension; measurement repetitions needed for a prescribed accuracy remain a separate statistical cost. In deterministic real-state benchmarks with exact objectives and gradients and known global optima, metric-aware Hopf updates reach numerical-precision median gaps. Relative to coordinate-Adam baselines, they yield smaller mean final gaps in variational eigensolver tests and concentrate more runs at numerical precision in metrology-inspired tests. The Hopf ansatz thus supplies a compass for arbitrary pure-state optimization, with coordinates, geometry, and measurable directions in one circuit structure.
ER -