Why Random-Matching Markov Chains Resist a Spectral Bridge to the Riemann Hypothesis: An Exploratory Assessment

Abstract

This essay examines the reversible Markov chain governing Pauli-weight diffusion in the shallow random Clifford circuits of Anand et al. (2026).

The Source Structure

Anand, Gorokhovsky, Hritz, and Sun study random Clifford circuits built from random perfect matchings on n qubits. Their key technical tool is a reduction of the second-moment dynamics of these circuits to a reversible Markov chain on binary support strings (the “Pauli weight chain”). The chain has a stationary distribution π(w) ∝ 3^w (n choose w) and exhibits logarithmic hitting-time bounds. The transition operator is self-adjoint with respect to the inner product weighted by π.

The Proposed Analogy

The Hilbert–Pólya conjecture posits that the non-trivial zeros of the Riemann zeta function correspond to the eigenvalues of a self-adjoint operator on an infinite-dimensional Hilbert space. Because the Pauli-weight chain is defined by a self-adjoint (reversible) transition matrix, it superficially invites comparison: both involve spectra of self-adjoint operators, and both are studied via their spectral gaps and level statistics.

Assessment of the Bridge

We rate this correspondence a suggestive metaphor. While the property of self-adjointness provides a formal similarity, the disanalogies are severe: the Markov chain acts on a finite state space, lacks any functional equation or Euler product, and possesses a spectral gap incompatible with the unbounded accumulation of zeta zeros. We propose testing whether the finite-chain eigenvalue spacing nevertheless mimics the GUE statistics observed in the zeta zeros; a negative result would definitively refute the bridge.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.