The Pauli Probability Spectrum Carries the Pure-State Quantum Fisher Metric

What the math gives to ML

The paper identifies an exact metric-preserving transformation for pure quantum states: the squared, labeled Pauli expectation values form a classical probability distribution whose Fisher matrix is exactly twice the quantum Fisher information matrix. This is valuable because it replaces difficult state-overlap and tangent-vector calculations with derivatives of observable statistics, while preserving the full local geometry only when Pauli labels are retained. The most direct neural-network transfer is a Pauli-spectrum natural-gradient optimizer for neural quantum states or differentiable quantum circuits, using sampled Pauli strings and damping to obtain a practical approximation to the quantum natural-gradient metric.