The Inverse Born Rule Fallacy: On the Informational Limits of Phase-Locked Amplitude Encoding

TL;DR

This paper critically examines the limitations of phase-locked amplitude encoding in quantum machine learning, highlighting its inability to capture non-commutative quantum structures essential for true quantum advantage.

Contribution

It demonstrates that phase-locked amplitude encoding fails to represent non-commutative structures and proposes dynamical Hamiltonian encoding as a more effective alternative.

Findings

Square-root mapping is phase-deaf and abelianizes the Hilbert space.

Basis changes do not replicate quantum computational power.

Dynamical Hamiltonian encoding introduces non-commutative evolution.

Abstract

In Quantum Machine Learning (QML) and Quantum Finance, amplitude encoding is often motivated by its logarithmic storage capacity arXiv:1307.0411. This paradigm typically relies on the mapping $\psi = \sqrt{P}$, treating the quantum state as a derivative of a classical probability distribution $P$. By restricting the data manifold to the positive real orthant $\mathcal{S}^+$, the accessible Hilbert space is effectively abelianized, rendering the representation ``phase-deaf''. We rigorously establish that while $P$ is a projection of $|\psi|^2$, the simple square-root mapping fails to recover the non-commutative structure necessary for genuine quantum advantage in classification tasks. Furthermore, we clarify why applying basis changes (like Hadamard gates) to these states fails to replicate the computational power of active phase-kickback mechanisms. Finally, we advocate for Dynamical Hamiltonian Encoding (based on QIFT), where data generates non-commutative evolution rather than serving as a static, phase-locked vector.