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

• URL: http://arxiv.org/abs/2602.21350v1
• Date: Tue, 24 Feb 2026 20:37:13 GMT
• Title: The Inverse Born Rule Fallacy: On the Informational Limits of Phase-Locked Amplitude Encoding
• Authors: Sebastian Zając, Jacob L. Cybulski, Bartosz Dziewit, Tomasz Kulpa,
• Abstract summary: In Quantum Machine Learning (QML) and Quantum Finance, amplitude encoding is often motivated by its logarithmic storage capacity arXiv:1307.0411.<n>This paradigm typically relies on the mapping $= sqrtP$, treating the quantum state as a derivative of a classical probability distribution $P$.<n>We rigorously establish that while $P$ is a projection of $||2$, the simple square-root mapping fails to recover the non-commutative structure necessary for genuine quantum advantage in classification tasks.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• A Sublinear-Time Quantum Algorithm for High-Dimensional Reaction Rates [0.06524460254566902]
  We introduce an algorithm that overcomes the exponential decay in success probability of quantum algorithms for non-dimensional dynamics.<n>We also use this technique to directly estimate matrix elements without exponential decay.<n>While specialized classical dissipative algorithms may outperform these bounds in practice, this demonstrates a rigorous route toward quantum advantage.
  arXiv  Detail & Related papers  (2026-01-21T23:20:46Z)
• Hamiltonian Decoded Quantum Interferometry [69.7049555871155]
  We introduce Hamiltonian Decoded Quantum Interferometry (HDQI)<n>HDQI utilizes coherent measurements and the symplectic representation of the Pauli group to reduce Gibbs sampling and Hamiltonian Bellians.<n>We show that HDQI efficiently prepares Gibbs states at arbitrary temperatures for a class of physically motivated commuting Hamiltonians.
  arXiv  Detail & Related papers  (2025-10-09T08:06:15Z)
• Quantum approximate optimization of bosonic finite-state systems [0.0]
  Formulating problems in nature inherently described by finite $D$-dimensional states requires mapping the qudit Hilbert space to that of multiqubit.<n>Here we propose to employ the Hamiltonian-based quantum approximate optimization algorithm (QAOA) through devising appropriate mixing Hamiltonians.<n>We apply this framework to quantum approximate thermalization and find the ground state of the repulsive Bose-Hubbard model in the strong and weak interaction regimes.
  arXiv  Detail & Related papers  (2025-10-07T04:44:13Z)
• Explicit Quantum Circuits for Simulating Linear Differential Equations via Dilation [0.0]
  We present a concrete pipeline that connects the dilation formalism with explicit quantum circuit constructions.<n>On the analytical side, we introduce a discretization of the continuous dilation operator that is tailored for quantum implementation.<n>We prove that the resulting scheme achieves a global error bound of order $O(M-3/2)$, up to exponentially small boundary effects.
  arXiv  Detail & Related papers  (2025-09-20T18:54:49Z)
• Quantum Homogenization as a Quantum Steady State Protocol on NISQ Hardware [42.52549987351643]
  Quantum homogenization is a reservoir-based quantum state approximation protocol.<n>We extend the standard quantum homogenization protocol to the dynamically-equivalent ($mathttSWAP$)$alpha$ formulation.<n>We show that our proposed protocol yields a completely positive, trace preserving (CPTP) map under which the code subspace is correctable.
  arXiv  Detail & Related papers  (2024-12-19T05:50:54Z)
• Partition function estimation with a quantum coin toss [0.0]
  Estimating quantum partition functions is a critical task in a variety of fields.<n>This paper introduces a quantum algorithm for estimating the partition function $Z_beta$ of a generic Hamiltonian $H$ up to multiplicative error.
  arXiv  Detail & Related papers  (2024-11-26T19:01:19Z)
• The Power of Unentangled Quantum Proofs with Non-negative Amplitudes [55.90795112399611]
  We study the power of unentangled quantum proofs with non-negative amplitudes, a class which we denote $textQMA+(2)$. In particular, we design global protocols for small set expansion, unique games, and PCP verification. We show that QMA(2) is equal to $textQMA+(2)$ provided the gap of the latter is a sufficiently large constant.
  arXiv  Detail & Related papers  (2024-02-29T01:35:46Z)
• Variational method for learning Quantum Channels via Stinespring Dilation on neutral atom systems [0.0]
  We propose a method to approximate an arbitrary target quantum channel by variationally constructing equivalent unitary operations on an extended system.<n>We also present an experimentally feasible approach to extrapolate the quantum channel in discrete time steps beyond the period covered by the training data.
  arXiv  Detail & Related papers  (2023-09-19T13:06:44Z)
• A lower bound on the space overhead of fault-tolerant quantum computation [51.723084600243716]
  The threshold theorem is a fundamental result in the theory of fault-tolerant quantum computation. We prove an exponential upper bound on the maximal length of fault-tolerant quantum computation with amplitude noise.
  arXiv  Detail & Related papers  (2022-01-31T22:19:49Z)
• Realization of arbitrary doubly-controlled quantum phase gates [62.997667081978825]
  We introduce a high-fidelity gate set inspired by a proposal for near-term quantum advantage in optimization problems. By orchestrating coherent, multi-level control over three transmon qutrits, we synthesize a family of deterministic, continuous-angle quantum phase gates acting in the natural three-qubit computational basis.
  arXiv  Detail & Related papers  (2021-08-03T17:49:09Z)
• Gaussian conversion protocols for cubic phase state generation [104.23865519192793]
  Universal quantum computing with continuous variables requires non-Gaussian resources. The cubic phase state is a non-Gaussian state whose experimental implementation has so far remained elusive. We introduce two protocols that allow for the conversion of a non-Gaussian state to a cubic phase state.
  arXiv  Detail & Related papers  (2020-07-07T09:19:49Z)
• Quantum Gram-Schmidt Processes and Their Application to Efficient State Read-out for Quantum Algorithms [87.04438831673063]
  We present an efficient read-out protocol that yields the classical vector form of the generated state. Our protocol suits the case that the output state lies in the row space of the input matrix. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure.
  arXiv  Detail & Related papers  (2020-04-14T11:05:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.