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.