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arXiv:2511.11199

The Riemann Hypothesis Emerges in Dynamical Quantum Phase Transitions

ShiJie Wei, Yue Zhai, Quanfeng Lu, Wentao Yang, Pan Gao, Chao Wei, Junda Song, Franco Nori, Tao Xin, GuiLu Long

quant-phhep-thmath-phSubmitted Nov 14, 2025• Announced Nov 17, 2025
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Abstract

The Riemann Hypothesis (RH), one of the most profound unsolved problems in mathematics, concerns the nontrivial zeros of the Riemann zeta function. Establishing connections between the RH and physical phenomena could offer new perspectives on its physical origin and verification. Here, we establish a direct correspondence between the nontrivial zeros of the zeta function and dynamical quantum phase transitions (DQPTs) in two realizable quantum systems, characterized by the averaged accumulated phase factor and the Loschmidt amplitude, respectively. This precise correspondence reveals that the RH can be viewed as the emergence of DQPTs at a specific temperature. We experimentally demonstrate this correspondence on a five-qubit spin-based system and further propose an universal quantum simulation framework for efficiently realizing both systems with polynomial resources, offering a quantum advantage for numerical verification of the RH. These findings uncover an intrinsic link between nonequilibrium critical dynamics and the RH, positioning quantum computing as a powerful platform for exploring one of mathematics' most enduring conjectures and beyond.

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The Riemann Hypothesis manifested in dynamical quantum phase transitions

Shijie Wei, Yue Zhai, Quanfeng Lu +7 more

Abstract The Riemann Hypothesis (RH), one of the most profound unsolved problems in mathematics, concerns the nontrivial zeros of the Riemann zeta function. Linking these zeros to physical phenomena offers new perspectives on their origin and verification. Here we establish a direct correspondence between these zeros and dynamical quantum phase transitions in two quantum many-body systems, characterized by the average accumulated phase factor and the Loschmidt amplitude respectively. This precise correspondence recasts the RH as the emergence of phase transitions at a unique temperature and identifies it as a previously unknown transition mechanism. We demonstrate this correspondence in a proof-of-principle experiment on a quantum processor. Moreover, we propose a quantum computational framework that implements both systems with polynomial resources, suggesting quantum advantage in probing the hypothesis. Our work bridges nonequilibrium quantum dynamics and number theory, positioning quantum computing as a powerful platform for exploring mathematical conjectures, phase transitions and beyond.

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The Hilbert-Pólya conjecture asserts that the imaginary parts of the nontrivial zeros of the Riemann zeta function (the Riemann zeros) are the eigenvalues of a self-adjoint operator (a quantum mechanical Hamiltonian, in the physical sense), as a promising approach to prove the Riemann hypothesis (cf.\cite{SH2011}). Instead of the eigenvalues, in this paper we consider observable-geometric phases as the realization of the Riemann zeros in a periodically driven quantum system, which were introduced in \cite{Chen2020} for the study of geometric quantum computation. To this end, we further introduce the notion of non-Abelian observable-geometric phases, involving which we give an approach to finding a physical system to study the Riemann zeros. Since the observable-geometric phases are connected with the geometry of the observable space according to the evolution of the Heisenberg equation, this sheds some light on the investigation of the Riemann hypothesis.

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The universal scaling of critical behavior in phase transitions is a cornerstone of physics. Dynamical quantum phase transitions (DQPTs) are their nonequilibrium analogues: abrupt nonanalyticities that emerge as a quantum system evolves in time. Yet the hardness and cost of detecting this phenomenon remain largely unexplored. We prove that estimating DQPT to a certain precision is intractable even for quantum computers, whereas deciding a subsystem variant of DQPT is as hard as simulating generic quantum circuits, implying a provable exponential quantum advantage. Furthermore, to search for critical times of local DQPTs, we show a quadratically faster quantum algorithm that estimates observables of Hamiltonian dynamics at multiple time points with Heisenberg-limited precision and sublinear scaling in the number of time points. Moreover, through encoding classical evolution into quantum dynamics, our framework enables broader quantum speedups for detecting anomalous phenomena in classical systems.

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In this paper, I explore the intriguing connection between hidden variables in quantum mechanics and the nontrivial zeros of the Riemann zeta function.Despite the apparent randomness in quantum phenomena, I demonstrate that deterministic hidden variables can coexist without violating Bell's inequalities, particularly in scenarios involving small angles.Using MATLAB simulations and a simplified space-time model, I illustrate this relationship, shedding light on the fundamental nature of quantum systems.This research not only deepens our understanding of quantum mechanics but also opens new avenues for investigating the Riemann zeta function's implications, potentially transforming our approach to both fields.

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The Riemann Hypothesis (RH), one of the most profound unsolved problems in mathematics, concerns the nontrivial zeros of the Riemann zeta function. Establishing connections between the RH and physical phenomena could offer new perspectives on its physical origin and verification. Here, we establish a direct correspondence between the nontrivial zeros of the zeta function and dynamical quantum phase transitions (DQPTs) in two realizable quantum systems, characterized by the averaged accumulated phase factor and the Loschmidt amplitude, respectively. This precise correspondence reveals that the RH can be viewed as the emergence of DQPTs at a specific temperature. We experimentally demonstrate this correspondence on a five-qubit spin-based system and further propose an universal quantum simulation framework for efficiently realizing both systems with polynomial resources, offering a quantum advantage for numerical verification of the RH. These findings uncover an intrinsic link between nonequilibrium critical dynamics and the RH, positioning quantum computing as a powerful platform for exploring one of mathematics' most enduring conjectures and beyond.
Abstract The Riemann Hypothesis (RH), one of the most profound unsolved problems in mathematics, concerns the nontrivial zeros of the Riemann zeta function. Linking these zeros to physical phenomena offers new perspectives on their origin and verification. Here we establish a direct correspondence between these zeros and dynamical quantum phase transitions in two quantum many-body systems, characterized by the average accumulated phase factor and the Loschmidt amplitude respectively. This precise correspondence recasts the RH as the emergence of phase transitions at a unique temperature and identifies it as a previously unknown transition mechanism. We demonstrate this correspondence in a proof-of-principle experiment on a quantum processor. Moreover, we propose a quantum computational framework that implements both systems with polynomial resources, suggesting quantum advantage in probing the hypothesis. Our work bridges nonequilibrium quantum dynamics and number theory, positioning quantum computing as a powerful platform for exploring mathematical conjectures, phase transitions and beyond.
The Hilbert-Pólya conjecture asserts that the imaginary parts of the nontrivial zeros of the Riemann zeta function (the Riemann zeros) are the eigenvalues of a self-adjoint operator (a quantum mechanical Hamiltonian, in the physical sense), as a promising approach to prove the Riemann hypothesis (cf.\cite{SH2011}). Instead of the eigenvalues, in this paper we consider observable-geometric phases as the realization of the Riemann zeros in a periodically driven quantum system, which were introduced in \cite{Chen2020} for the study of geometric quantum computation. To this end, we further introduce the notion of non-Abelian observable-geometric phases, involving which we give an approach to finding a physical system to study the Riemann zeros. Since the observable-geometric phases are connected with the geometry of the observable space according to the evolution of the Heisenberg equation, this sheds some light on the investigation of the Riemann hypothesis.
The universal scaling of critical behavior in phase transitions is a cornerstone of physics. Dynamical quantum phase transitions (DQPTs) are their nonequilibrium analogues: abrupt nonanalyticities that emerge as a quantum system evolves in time. Yet the hardness and cost of detecting this phenomenon remain largely unexplored. We prove that estimating DQPT to a certain precision is intractable even for quantum computers, whereas deciding a subsystem variant of DQPT is as hard as simulating generic quantum circuits, implying a provable exponential quantum advantage. Furthermore, to search for critical times of local DQPTs, we show a quadratically faster quantum algorithm that estimates observables of Hamiltonian dynamics at multiple time points with Heisenberg-limited precision and sublinear scaling in the number of time points. Moreover, through encoding classical evolution into quantum dynamics, our framework enables broader quantum speedups for detecting anomalous phenomena in classical systems.
Dynamical quantum phase transitions (DQPT) provide a non-equilibrium framework for probing quantum criticality, where the non-analytic kinks in the Loschmidt echo rate function λ(t) signal the crossing of Fisher zeros across the real-time axis. In disordered many-body localized (MBL) systems, spatial inhomogeneity typically induces multi-frequency dephasing that melts these non-analytic signatures into diffuse crossovers. Here, we demonstrate that deterministic on-site potential landscapes derived from the unfolded non-trivial zeros of the Riemann zeta function prevent the thermalized melting of DQPT signatures through Gaussian Unitary Ensemble (GUE) level repulsion. While strong unengineered Riemann potentials (W = 2.0J) induce multi-frequency spectral beating resulting in dense multi-kink profiles, we formulate an augmented Lagrangian variational framework that directly targets the trade-off between isochronous periodicity and singular non-analyticity. By minimizing a multi-objective functional encompassing Fisher zero proximity, high-frequency beating penalties, and GUE level-repulsion kernels, the optimization converges to a dual-champion configuration h*. Under quantum quench dynamics of a one-dimensional transverse-field Ising chain (L=10, 1024-dimensional Hilbert space), this engineered landscape drives the Loschmidt echo down to the numerical machine-precision floor (L_min = 2.03e-13), generating an ultra-sharp super-kink with peak amplitude λ_max = 2.9219, nearly three times the baseline transition height. These results reveal a direct route for engineering singular non-equilibrium criticality in digital quantum processors.
This paper investigates the Riemann Hypothesis through the lens of quantum entanglement, proposing a novel theoretical model that posits the emergence of 'quantum branches' within the zeta function distribution. We explore a quantum-mechanical framework to simulate the evolution of the zeta function, incorporating entanglement as a fundamental process driving the formation of these branches. The goal is to provide a potential pathway for understanding the hypothesis and its connection to the nascent field of quantum computation. This work aims to initiate a discussion about the potential for a fundamentally new approach to the Riemann Hypothesis.
In this paper, I explore the intriguing connection between hidden variables in quantum mechanics and the nontrivial zeros of the Riemann zeta function.Despite the apparent randomness in quantum phenomena, I demonstrate that deterministic hidden variables can coexist without violating Bell's inequalities, particularly in scenarios involving small angles.Using MATLAB simulations and a simplified space-time model, I illustrate this relationship, shedding light on the fundamental nature of quantum systems.This research not only deepens our understanding of quantum mechanics but also opens new avenues for investigating the Riemann zeta function's implications, potentially transforming our approach to both fields.