The Riemann Hypothesis Emerges in Dynamical Quantum Phase Transitions

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.

• The paper demonstrates that dynamical quantum phase transitions (DQPTs) occur exclusively at the RH critical value (β=1/2), directly mapping the nontrivial zeros of the zeta function.
• It employs a logarithmic energy spectrum and probe qubit measurements to reveal coherence zeros that align with known Riemann zeros.
• Quantum simulations and NMR experiments validate the approach, offering a quadratic speedup over classical methods for verifying number-theoretic conjectures.

Emergence of the Riemann Hypothesis in Quantum Critical Dynamics

Introduction and Theoretical Foundation

This work establishes a rigorous correspondence between the nontrivial zeros of the Riemann zeta function, central to the Riemann Hypothesis (RH), and dynamical quantum phase transitions (DQPTs) in quantum many-body systems. By constructing two classes of engineered quantum systems—one measured via the probe qubit's accumulated phase and the other via the Loschmidt amplitude—the authors demonstrate that DQPTs occur precisely when, and only when, the system parameters map to the critical line $\beta = 1/2$ of the zeta function; this criticality thus provides a physical manifestation of the RH.

The construction begins with a logarithmic energy spectrum $E_n = \log n$, enabling the preparation of thermal states whose quantum dynamics encode the analytic features of the zeta function. The probe spin's coherence, $\mathcal{L}(\beta, t)$, and the generalized Loschmidt amplitude, $\mathcal{G}(\beta, t)$, each provide experimentally accessible observables that vanish uniquely at the nontrivial zeros of $\zeta(s)$. The mapping between these non-analyticities (DQPTs) and the Riemann zeros is formally exact in the thermodynamic and infinite-time limits.

Figure 1: One-to-one correspondence between the Riemann zeta function and the dynamics of engineered quantum many-body systems, illustrating the mapping between Riemann zeros and DQPTs.

Experimental Realization with Nuclear Spins

The theoretical predictions are validated on a five-qubit NMR setup using the 1-bromo-2,4,5-trifluorobenzene molecule, with controllable initialization to the thermal states required by the construction. The Hamiltonian is realized via a combination of scalar and dipolar couplings, and probe qubit measurements extract both real and imaginary components of $\mathcal{L}(\beta, t)$.

Figure 2: Physical system and quantum circuit emulating the Riemann zeta function using five nuclear spins, with F$_1$ as the probe qubit for phase and coherence measurements.

The experimental results for $\beta = 0.5$ display repeated coherence zeros in $\mathcal{L}(\beta, t)$, whose positions match the known nontrivial Riemann zeros to within $\Delta t < 0.01$—well within expected finite-size effects. For off-critical $\beta$, such coherence zeros disappear, providing direct experimental evidence that DQPTs occur exclusively at the RH critical line.

Figure 3: Probe spin coherence dynamics $\mathcal{L}(\beta, t)$ aligning with the Riemann zeros at $\beta = 0.5$, and absent for $\beta\neq 0.5$; locations of the zeros extracted from the simultaneously vanishing real and imaginary parts.

Moreover, the singularities in the rescaled free energy density $\mathcal{F}_1(\beta,t)$ precisely correspond to zero crossings, exhibiting non-analytic behavior only at the RH line. These features guarantee that the RH is recast as a statement about the exclusive emergence of nonequilibrium quantum criticality at $\beta=1/2$.

Figure 4: Non-analytic behavior of free energy density $\mathcal{F}_1(\beta,t)$ as $\beta$ crosses the RH critical value; the singularity is a unique marker of the quantum DQPT corresponding to a Riemann zero.

Quantum Simulation and Algorithmic Framework

The authors extend their results to large imaginary parts and high-order Riemann zeros by mapping the simulation to digital quantum circuits deployable on noiseless gate-based quantum computers. Efficient oracles are constructed for (1) logarithmic Hamiltonians, (2) state preparation with an inverse power-law envelope (to any prescribed $\varepsilon$), and (3) precise control over time evolution. The formal resource analysis shows that the quantum verification of Riemann zeros through such algorithms outperforms classical methods by at least a quadratic speedup in $|t|$, especially near the critical strip $\Re(s)=1/2$.

Figure 5: Decomposition of the initial state and time-evolution operator on quantum circuits using polynomial and logarithmic oracles; preparation to arbitrary accuracy with polynomial resources.

Numerical simulations on multi-spin Hamiltonians—up to 18 qubits—demonstrate preservation of DQPT-zero correspondence for the $10^{12}$-th Riemann zero, with the deviation between quantum estimates and analytical locations decreasing with increasing $t$ (see also Supplementary Figure 1).

Implications, Numerical Claims, and Theoretical Impact

Key Numerical Results and Claims:

• Direct mapping of DQPTs to Riemann zeros, with experimentally observed coherence zeros agreeing (within experimental uncertainty) to the actual nontrivial zeros at $\beta=1/2$.
• Absence of DQPTs at off-critical $\beta$, in strict alignment with the RH; no nontrivial zeros are detected outside the putative critical line.
• Polynomial resource quantum algorithms for thermal state preparation and Hamiltonian evolution, achieving sample and gate complexities $$\mathcal{O}(\delta^{-1}|t|^{1/4}\mathrm{polylog}(|t|,1/\delta))$$ for $\beta=1/2$, representing a quadratic quantum speedup over the optimal classical Riemann-Siegel evaluation.
• Robust scaling: the approach generalizes to exponentially large $|t|$ (i.e., arbitrarily high Riemann zeros), as demonstrated by circuit-based simulations.

Contradictory/Strong Statements:

• Physical criticality and RH equivalence: The discovery that DQPTs occur solely at $\beta=1/2$ in this quantum framework is posited as a physical encoding of the RH itself—if any DQPT (zero) occurred at $\beta\neq 1/2$, the RH would directly fail.
• Quantum advantage for number theoretic verification: The resource scalings and explicit constructions position quantum algorithms as the most efficient available probe for the Riemann zeros, particularly in regions where classical zero-free verifications are computationally intractable.

Prospects for Further Research and Quantum-Physical Implications

This work paves the way for leveraging quantum simulators not just for abstract mathematical discovery, but for concrete verification tasks in analytic number theory, opening the RH (and related problems) to scrutiny by future, large-scale quantum computation platforms. Furthermore, the mapping between quantum phase structure and the fine properties of zeta-function zeros suggests deeper connections between quantum chaos diagnostics (e.g., OTOCs, spectral statistics) and arithmetic properties—potentially leading to a broader class of quantum algorithms for special functions or number theoretic benchmarks.

On a practical level, algorithmic improvements for state preparation and controlled time evolution in these settings could also benefit other applications in many-body physics where engineered non-integrable Hamiltonians are desired. The demonstration of direct quantum evaluation of DQPTs linked to zeta zeros could serve as a new experimental probe for quantum-critical behavior and cross-validation of quantum hardware for scientific discovery in mathematics.

Conclusion

The direct correspondence established here between nontrivial Riemann zeros and DQPTs represents a significant theoretical convergence between analytic number theory and quantum nonequilibrium physics. The results provide experimental evidence, algorithmic pathways, and quantum resource estimates for using quantum computing as a verification tool for deep mathematical conjectures. Extensions to broader classes of number-theoretic functions and pursuit of a deeper operator-theoretic understanding of these correspondences will be natural future directions, with wide-ranging implications for mathematics, physics, and quantum information science.