Again and again, the quantum signal faded almost to nothing before returning, tracing a rhythmic pattern that matched one of mathematics’ oldest unsolved mysteries. In a carefully engineered quantum system, those fleeting moments of disappearance aligned with the elusive nontrivial zeros of the Riemann zeta function, allowing the researchers to reinterpret the famous Riemann hypothesis as a specific kind of quantum phase transition and to reproduce key signatures on a quantum processor.
For more than a century and a half, the Riemann hypothesis has stood among the deepest unsolved problems in mathematics. At its heart lies the Riemann zeta function, a mathematical object whose nontrivial zeros are believed to all lie along a single vertical line in the complex plane, known as the critical line where the real part equals 1/2. Despite enormous numerical evidence supporting this idea, no proof has been found.
The new research approaches the mystery from an entirely different direction. Instead of searching directly for a mathematical proof, the authors built quantum systems whose measurable physical behavior reproduces the structure of the zeta function itself. In these systems, the locations of the zeta function’s nontrivial zeros correspond directly to dynamical quantum phase transitions, or DQPTs—nonequilibrium transitions that occur during the time evolution of a quantum system rather than under changing temperature or pressure.
The researchers argue that this correspondence works in both directions. The zeros become observable through quantum dynamics, while the Riemann hypothesis itself can be reformulated as the statement that a particular quantum system experiences these dynamical phase transitions only at one unique inverse temperature.
Turning a mathematical function into quantum dynamics
The study begins by constructing a quantum system with an unusual energy spectrum. Instead of evenly spaced energy levels or another familiar arrangement, each level is assigned an energy proportional to the natural logarithm of an integer. This logarithmic spectrum is the essential ingredient because it naturally reproduces the mathematical terms that appear in the Dirichlet representation of the Riemann zeta function.
The system is first prepared in thermal equilibrium. It is then driven out of equilibrium using carefully designed interaction Hamiltonians. Rather than examining the energy levels directly, the researchers monitor specific measurable quantities that evolve over time.
They developed two complementary quantum systems.
In the first, the key observable is an average accumulated phase factor, denoted by L(β, t). In the second, the central quantity is the Loschmidt amplitude, a measure of how closely an evolving quantum state resembles its initial state.
Although the two systems differ in their dynamics and observables, both ultimately encode the same mathematical object.
As the system evolves, these observables oscillate. At particular moments they vanish before reviving again. According to the theoretical framework, those vanishing points coincide exactly with the nontrivial zeros of the Riemann zeta function.
Phase transitions appear at the zeta zeros
The researchers identify these disappearances as dynamical quantum phase transitions.
Unlike conventional phase transitions such as ice melting into water, DQPTs occur during the evolution of quantum systems over time. They are marked by nonanalytic behavior in quantities analogous to free energy.
Within the proposed framework, whenever the zeta function reaches one of its nontrivial zeros, the corresponding quantum observable falls to zero and the dynamical free-energy density becomes singular.
The work further examines the associated entropy behavior. According to the authors, entropy changes asymmetrically as the system crosses these critical points.
For values of the inverse temperature slightly below a zero, entropy increases, corresponding to a loss of quantum coherence and information flowing into internal degrees of freedom. Slightly above the zero, entropy decreases, reflecting information flowing back into the system and a partial recovery of coherence.
The authors interpret each nontrivial zero as a nonequilibrium critical point where the direction of information flow reverses.
A new interpretation of the Riemann hypothesis
One of the most striking aspects of the work is the way it reframes the famous mathematical conjecture.
Rather than expressing the Riemann hypothesis solely as a statement about complex numbers, the authors reinterpret it as a statement about the behavior of a physical quantum system.
If the hypothesis is correct, then these dynamical quantum phase transitions occur only when the dimensionless inverse temperature equals β = 1/2, corresponding to the critical line of the zeta function.
Away from this value, the theory predicts that the characteristic vanishing-and-revival behavior disappears.
In this picture, the hypothesis becomes equivalent to saying that a specially constructed quantum many-body system possesses a unique nonequilibrium critical phase transition occurring exclusively along that critical line.
The authors also argue for the converse interpretation: the Riemann hypothesis itself identifies what they describe as a previously unknown mechanism capable of generating dynamical quantum phase transitions.
Watching the phenomenon unfold on a quantum processor
To test the theory experimentally, the team implemented the first quantum system using a five-qubit nuclear magnetic resonance quantum processor.
The processor used a molecule of 1-bromo-2,4,5-trifluorobenzene dissolved in a liquid-crystal solvent. Three fluorine nuclei and two hydrogen nuclei served as the five qubits. One qubit acted as a probe while the remaining four formed a sixteen-dimensional working system.
Preparing the required quantum state presented two major challenges.
First, the researchers had to create thermal-equivalent states whose level populations matched the mathematical weighting required by the zeta function.
Second, they had to implement controlled quantum evolutions that generated precisely the desired logarithmic phase factors.
Using shaped control pulses optimized through a gradient ascent pulse technique, they achieved simulated control fidelities exceeding 99.5% before performing the experiments.
The quantum signal repeatedly vanished where mathematics predicted
The experimental program consisted of three main tests.
In one experiment, the inverse temperature was fixed at β = 0.5, corresponding to the critical line predicted by the Riemann hypothesis, while the evolution time varied.
The measured probe-spin coherence repeatedly vanished and revived. Polynomial fitting identified coherence zeros at
14.12, 20.96, 25.09, 30.44, and 32.93.
These closely matched the theoretical locations of the first several nontrivial zeros, whose expected imaginary parts are
14.13, 21.02, 25.01, 30.43, and 32.94.
In another experiment, the researchers instead prepared the system with β = 0.3, away from the critical line.
Under those conditions, the characteristic vanishing-and-revival behavior disappeared, and no significant coherence zeros were observed.
A third experiment fixed the evolution time at the first nontrivial zero while varying β between 0.1 and 0.9.
The measured coherence displayed a clear minimum at β = 0.5, supporting the predicted special role of the critical line within the constructed system.
A second quantum model reaches vastly larger zeros
The researchers also introduced a second quantum construction designed to probe much larger nontrivial zeros.
Instead of relying on the average accumulated phase factor, this system measures the Loschmidt amplitude.
Numerical simulations demonstrated that relatively modest quantum systems could already access extremely high-order zeros.
A three-spin simulation reproduced zeros whose imaginary parts lay between 420 and 450.
Expanding to ten spins allowed access to zeros around 6.595 × 10⁶, corresponding to approximately the 13,502,344th through 13,502,366th nontrivial zeros.
An eighteen-spin system successfully reproduced behavior associated with the 10¹²th nontrivial zero and neighboring ones.
The authors report that agreement between estimated and exact zero locations improves as the evolution time increases, making the correspondence increasingly accurate for larger zeros.
Building a scalable quantum computing framework
Beyond the laboratory demonstration, the paper develops a gate-based quantum computing framework intended to reproduce both quantum systems efficiently.
Rather than requiring exponentially large resources, the authors describe algorithms for preparing the necessary quantum states and implementing the required logarithmic Hamiltonians using polynomial resources.
The framework includes methods for preparing the required thermal-like quantum states, implementing logarithmic time evolution, and constructing the controlled operations needed by both physical models.
Using these components, the researchers derive computational complexity estimates for evaluating the zeta function and probing its zeros.
According to their analysis, the quantum approach can reduce the dependence on the imaginary part of the zeta argument from the square-root scaling associated with direct Riemann–Siegel evaluation to at most a one-quarter power, yielding at least a quadratic speedup for the verification task they analyze.
The paper emphasizes that this represents a framework for probing and numerically verifying zeros rather than a proof of the Riemann hypothesis itself.
What the work does—and does not claim
The authors present their results as establishing a direct correspondence between engineered quantum dynamics and the mathematical structure of the Riemann zeta function.
Within their framework, the hypothesis acquires a physical interpretation: if it is true, then the specially constructed quantum system undergoes dynamical quantum phase transitions only at β = 1/2.
At the same time, the study does not claim to prove the Riemann hypothesis.
Instead, it introduces an experimentally realizable setting in which the hypothesis can be investigated through measurable quantum phenomena.
The researchers also note that the framework naturally extends beyond the Riemann zeta function. They suggest it could be adapted to related mathematical objects such as Dirichlet L-functions, potentially creating broader number-theoretic benchmarks for quantum computing.
Finally, they point to several open directions, including exploring connections between these zeta-driven quantum dynamics and other indicators of quantum information flow and quantum chaos, while continuing to investigate whether quantum computing can accelerate the computationally demanding tasks involved in studying one of mathematics’ most enduring unsolved problems.
Publication details
Shijie Wei et al, The Riemann Hypothesis manifested in dynamical quantum phase transitions, Nature Communications (2026). DOI: 10.1038/s41467-026-74935-8. On arXiv: arxiv.org/abs/2511.11199
