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arXiv:2606.29294

Quantum models of the Riemann zeta function, lattice spin models and algebraic models of entanglement

Nikolaj Glazunov

Published Jun 28, 2026• 0 citations
arXivHTMLPDF

Abstract

A brief overview of results concerning the connection between the Hilbert-Polya conjecture and the Riemann hypothesis about the Riemann zeta function, some new results on p-adic quantum computing, quantum entanglement based on lattice spin models and algebraic entanglement models is given. Quantum computing uses both photons and electrons, so their known properties are (very briefly) presented.

DOI
10.48550/arxiv.2606.29294

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The Riemann Hypothesis, a long-standing unsolved problem in mathematics, concerns the distribution of prime numbers. Despite extensive research, no definitive proof remains elusive. This paper explores a novel quantum algorithm designed to approximate the Riemann zeta function, aiming to provide exponential speedups over classical methods. We propose a hybrid quantum-classical approach, utilizing quantum superposition and entanglement to efficiently explore the zeta function's complex landscape, ultimately leading to a significant computational advantage. This research focuses on the fundamental potential of quantum computation to address this mathematical challenge.
This paper explores the potential of quantum entanglement as a computational tool to tackle specific mathematical problems, drawing inspiration from the Riemann-Hilbert Conjecture. We propose a novel quantum algorithm designed to exploit entanglement to achieve a practical, verifiable advantage over classical methods for certain complex calculations. The paper outlines the core mechanism, details the algorithm's implementation, and discusses preliminary results demonstrating the feasibility of leveraging entanglement for computation. We argue that this approach represents a significant step toward understanding and utilizing quantum phenomena for computational purposes, potentially offering a new avenue for tackling problems within the realm of mathematical complexity.
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Quantum computing is a promising new area of computing with quantum algorithms offering a potential speedup over classical algorithms if fault tolerant quantum computers can be built. One of the first applications of the classical computer was to the study of the Riemann hypothesis and quantum computers may be applied to this problem as well. In this paper we apply the Quantum Fourier Transform (QFT) to study three functions with non-trivial zeros obeying a version of the Riemann hypothesis. We perform our quantum computations with six qubits, but more qubits can be used if quantum error correction allows the QFT algorithm to scale. We represent these functions as ground state wave functions transformed to momentum space. We show how to obtain these functions as states in supersymmetric quantum mechanics. Finally we discuss the relation of these functions to (p,1) Random two matrix models at large N.