Finite-Dimensional Transfer Operators and the Hilbert-Pólya Conjecture: An Exploratory Negative

Abstract

This essay examines whether the quantum dimension reduction pipeline for hidden Markov models, which employs finite-dimensional transfer operators with spectral gaps, offers a structural pathway to the Hilbert-Pólya interpretation of the Riemann Hypothesis.

Overview

This paper analyzes a recent quantum information preprint on compressing Hidden Markov Models (HMMs) using infinite Matrix Product States (iMPS) and deterministic dilation. The source work introduces a transfer operator E that governs the quantum memory of a stochastic process, and uses its spectral properties to enable compression.

The Attempted Bridge

The essay proposes a speculative correspondence between the transfer operator E of a quantum HMM and the hypothetical Hilbert-Pólya Hamiltonian H whose eigenvalues are the imaginary parts of the Riemann zeros. The superficial appeal lies in the shared vocabulary of "spectral gaps," "deterministic" underlying dynamics, and the compression of information via spectral truncation (Schmidt decomposition).

The Negative Verdict

The analogy is rated a failed structural correspondence. The source operator is a finite-dimensional completely positive trace-preserving map with at most d² eigenvalues, while the Riemann zeros form an infinite set on the critical line. The source operator is not self-adjoint, and its spectrum lies in the unit disk rather than the real line. The "deterministic" property in the source refers to unifilarity of the HMM graph, not to the integrability of a quantum system. Consequently, no formal dictionary can be established without violating the axioms of either domain.

What Would Be Needed

A viable bridge would require an infinite-dimensional transfer operator (such as those appearing in the Selberg zeta function or Mayer's operator for the Gauss map) where the determinant yields the zeta function and the spectrum is infinite. The source paper's finite-dimensional compression framework is inherently unsuited to the analytic number theory of the Riemann Hypothesis.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.