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

Sebastian Zając, Jacob L. Cybulski, Bartosz Dziewit, Tomasz Kulpa

TL;DR

This work rigorously establishes 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, and advocates for Dynamical Hamiltonian Encoding (based on QIFT), where data generates 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 $ψ= \sqrt{P}$, 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 $|ψ|^2$, 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.

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

TL;DR

This work rigorously establishes that while is a projection of , the simple square-root mapping fails to recover the non-commutative structure necessary for genuine quantum advantage in classification tasks, and advocates for Dynamical Hamiltonian Encoding (based on QIFT), where data generates 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 , treating the quantum state as a derivative of a classical probability distribution . By restricting the data manifold to the positive real orthant , the accessible Hilbert space is effectively abelianized, rendering the representation ``phase-deaf''. We rigorously establish that while is a projection of , 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.
Paper Structure (12 sections, 2 theorems, 6 equations)

This paper contains 12 sections, 2 theorems, 6 equations.

Table of Contents

  1. The Ontological Priority of Amplitude over Distribution
  2. The Confusion Between Data and Statistics
  3. The Abelian Trap: Commutativity of the Positive Orthant
  4. The Phase-Deafness and The Passive Data Problem
  5. Linear Mixing vs. Phase Injection
  6. The Diagonal Trap
  7. Restoring Dynamics: The Quantum Information Field Approach
  8. Data as Physical Excitations of the Vacuum
  9. The Suzuki-Trotter "Sandwich" and Information Curvature
  10. From Geometry to Resonance
  11. Spectral Resonance vs. Geometric Overlap
  12. Conclusion

Key Result

theorem 1

Let $\mathcal{A}_{\mathcal{S}^+}$ denote the set of linear operators on $\mathcal{H} \cong \mathbb{C}^N$ that map the real positive orthant $\mathcal{S}^+$ into itself and are self-adjoint. Then $\mathcal{A}_{\mathcal{S}^+}$ is a commutative algebra.

Theorems & Definitions (3)

  • theorem 1: Abelianization of Observables
  • proof
  • corollary 1: Absence of Destructive Interference