The Hopf Ansatz

How can a universal state-preparation circuit also provide coordinates, local geometry, and measurable directions for optimization? The Hopf ansatz uses one binary tree to prepare arbitrary normalized real and complex state vectors, recover angles from known amplitudes, and expose the metric and coordinate tangents.

Ruge Lin and Guangxi Li — A Compass on the Quantum State Sphere: The Hopf Ansatz for Arbitrary Pure-State Optimization (2026).

Result	Scientific meaning
Real and complex Hopf charts	A dimension-matched representation of the normalized state-vector sphere, with an explicit amplitude-to-angle inverse
Diagonal pullback metric	Subtree probability masses and leaf occupations supply local geometric weights
Preparable coordinate tangents	Each nonzero tangent becomes a normalized state on the same circuit skeleton
Structured gradient access	Transition moments and signed branches organize derivative access by tree depth and phase family
Geometry-aware optimization	The chart supports sphere-based updates and the paper's matched numerical studies

For $N=2^n$, the real chart has $N-1$ parameters and the complex chart has $2N-1$, retaining the global phase of the state vector. The complex diagonal metric is the ambient round-sphere metric; the real metric agrees with the pullback Fubini–Study metric. Coordinate singularities and the distinction between compiled circuit families and measurement repetitions are part of the scientific account.

Read the science

The complete scientific account contains the paper's main text and every appendix, with original equation, statement, algorithm, figure, table, and reference identifiers. Follow the topic route below, or use the reviewer crosswalk to find a specific item from the paper.

To understand	Read
The question and relation to earlier work	Motivation
State preparation, inverse coordinates, and the metric	Chart and geometry and algorithms
Tangent states, transition moments, and indexed access	Gradient access
A complete concrete construction	Four-qubit example
Jacobians, sphere updates, and optimizer mechanisms	Geometry-native optimizers
QFI and fixed-readout CFI objectives	Metrological objectives
Full analytical arguments	Proofs
Numerical comparisons and their interpretation	Numerical studies and experimental details
Implementation conventions and API	Engineering guide

Each result has one scientific home. The engineering guide supplies bit order, angle conventions, boundary handling, and implementation details; the bibliography preserves the paper's attribution.

Learn the project

The single educational anchor is Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Edition. The educational bridge selects relevant sections, translates conventions, and develops the geometry needed to connect that foundation to this work.

The 30-milestone map contains 12 background/preliminaries milestones and 18 narrative/innovation milestones. Both phases progress from local objects and mechanisms to the complete scientific story. Every milestone has an understanding target, dependencies, a conceptual connection, and source links.

Verify and reproduce

Start with the reviewer route: paper item → full statement and proof → implementation → executable check. The reproduction guide gives environment setup, commands, data generation, figure reproduction, and expected results.

Use Python 3.10 or newer. From the repository root:

python -m venv .venv
source .venv/bin/activate
python -m pip install -r requirements.txt
python hopf_utils.py

Install requirements-optional.txt for the explicit Qibo circuit checks. The small checks exercise coordinate maps, metrics, tangents, and logical gate schedules; the full experiments regenerate the numerical studies. General analytical statements are established by their written proofs.

These supplementary implementation figures are described in Experiments and evidence; the original paper figures appear with their captions in the scientific account.

Companion paper

Hopf-QBP develops shared finite-shot records for the complete Hopf-coordinate gradient. Its paper is Compass in the Mirror: Quantum Backpropagation with the Hopf Ansatz. The repositories use the same educational anchor and reading structure and are independently runnable.

Citation

Ruge Lin and Guangxi Li, “A Compass on the Quantum State Sphere: The Hopf Ansatz for Arbitrary Pure-State Optimization,” arXiv:2607.14231 (2026).

@misc{lin2026hopf,
  title = {A Compass on the Quantum State Sphere: The Hopf Ansatz for Arbitrary Pure-State Optimization},
  author = {Lin, Ruge and Li, Guangxi},
  year = {2026},
  eprint = {2607.14231},
  archivePrefix = {arXiv},
  primaryClass = {quant-ph},
  url = {https://arxiv.org/abs/2607.14231}
}

Machine-readable citation metadata is available in CITATION.cff.

For questions about the article, feel free to contact Ruge Lin at [email protected].

License

The software is released under the MIT License. The scientific account identifies the authors and the arXiv source of the paper.