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.
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 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.
This paper indicates the problem of the famous Riemann hypothesis (RH), which has been well-verified by a definite answering method using a Bose-Einstein Condensate (BEC) phase. We adopt mathematical induction, mappings, and laser photons governed by electromagnetically induced transparency (EIT) to examine the existence of the RH. In considering the well-developed as Riemann zeta function, we find that the existence of RH has a corrected and self-consistent solution. Specifically, there is the only one pole at s = 1 on the complex plane for Riemann’s functions, which generalizes to all non-trivial zeros while s > 1. The essential solution is based on the BEC phases and on the nature of the laser photon(s). This work also incorporates Heisenberg commutators [ x ^ , p ^ ]=1/2 in the field of quantum mechanics. We found that a satisfactory solution for the RH would be incomplete without the formalism of Heisenberg commutators, BEC phases, and EIT effects. Ultimately, we propose the application of qubits in connection with the RH.
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.
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.
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.