March 3, 2026Open Access

The Semicircle Constraint: A Geometric Framework for Quantum-Classical Correlation

Key Points

Quantum-classical correlation follows a semicircle constraint with radius 0.5, centered at (0.5, 0).
Measurement probability and correlation uniquely achieve maximum coherence at q=0.5, with a gradient variance scaling of q(1-q).
Observational analysis across various quantum states confirms the constraint's validity, achieving a correlation of r=0.943 in testing.
The findings provide insights into barren plateaus in variational quantum algorithms, suggesting more efficient approaches in quantum computing.

Abstract

For any normalized qubit state | = |0 + |1, the measurement probability q = ||² and the quantum-classical correlation Cₐ₂ = |||| = q (1-q) satisfy the constraint (q - 12) ² + Cₐ₂² = 14. This is a semicircle of radius 12 centered at (12, 0) in the (q, Cₐ₂) plane, and it follows from the Born rule and normalization alone. Classical states (q 0 or 1) sit at the endpoints where Cₐ₂ vanishes, while maximum coherence Cₐ₂ = 12 is achieved uniquely at q = 12. The Fisher information metric along this curve is constant, giving a total arc length of. We show that the constraint explains the geometric origin of barren plateaus in variational quantum algorithms: gradient variance scales as q (1-q) = Cₐ₂² and vanishes at the classical endpoints. Hardware validation on IonQ Forte-1 (15 test points, 52 shots each) yields a theory-measurement correlation of r = 0. 943. Code and test files available at https: //github. com/Variably-Constant/QCSemicircleConstraintProof

The Semicircle Constraint: A Geometric Framework for Quantum-Classical Correlation

Key Points

Abstract

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