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First posed in 1859, the Riemann Hypothesis is one of the longest-standing unsolved problems in mathematics. It underpins parts of cryptography, as well as more than a thousand theorems proved on the assumption that it is true.
Physicists have previously proposed physical counterparts to this mathematical statement. The aim was to map the Riemann Hypothesis to something concrete, such as the energy levels of a quantum system. The new study ties the hypothesis to how a quantum system evolves over time.
Phys.org spoke to co-authors Dr. Shijie Wei from the Beijing Academy of Quantum Information Sciences (BAQIS), Professor Tao Xin from the Shenzhen International Quantum Academy, and Professor Gui-Lu Long from BAQIS and Tsinghua University about their work.
"Many [academics] have suggested that the final solution to the Riemann Hypothesis may not come from the field of algebra, and that the answer is instead embedded in the fundamental laws of physics," they said. "Motivated by this long-held intuition ... our core goal has been to turn this poetic theoretical insight into concrete quantum-physical interpretations and to explore whether quantum computing technology can open up a new path for addressing this 160-year-old problem in number theory."
From energy levels to time
The Riemann zeta function is a mathematical sum of infinitely many terms, closely tied to the distribution of prime numbers. Its nontrivial zeros all sit within a narrow band of the complex plane called the critical strip. Riemann conjectured that every one of them has a real part of exactly 1/2, placing them all on a single line down the middle of that strip, the critical line, and no one has proven that this holds for all of them.
Physicists have suspected a connection between the zeta zeros and quantum mechanics for the better part of a century. The most influential version of this idea is the Hilbert–Pólya conjecture, which proposes that the zeros might be the energy levels of an undiscovered quantum system.
That search has proven difficult. "Identifying and scaling such a stationary Hamiltonian is notoriously challenging, particularly on near-term quantum hardware," said the researchers. Their study takes a different route.
"We posed a distinct question: What if we use time instead of energy as our core variable?" they explained, pointing to dynamical quantum phase transitions (DQPT), moments when a quantum system's properties change abruptly as it evolves. "Since time is the most natural and precisely controllable variable in quantum computing, we recognized that DQPT provides an ideal temporal framework to 'scan' for and reveal these mathematical zeros."
Two quantum systems
The team began by building two complementary quantum systems, each consisting of a many-body system coupled to a single probe qubit. Both systems are prepared in thermal equilibrium and then driven out of equilibrium over time. In this setup, the two components of the zeta function's complex argument become physical variables: The real part corresponds to the system's temperature, and the imaginary part corresponds to evolution time.
Within this framework, a phase transition can occur only if the system is prepared at a precise temperature along the critical line. Observing such a transition at any other temperature would signal a breakdown of the Riemann Hypothesis within this physical mapping.
To test this, the team built the first of their two systems on a five-qubit nuclear magnetic resonance (NMR) processor, using the nuclear spins of a molecule containing three fluorine atoms and two hydrogen atoms, with one spin serving as the probe. The team ran the processor at three settings: on the critical line, off the critical line and at the imaginary part corresponding to the first nontrivial zero.
"Only when the parameters simultaneously satisfied both conditions—lying on the critical line and equaling a Riemann zero—did the observed signal completely vanish, indicating that a dynamical quantum phase transition had occurred," the researchers said.
Their second system was built around a related quantity called the Loschmidt amplitude. Rather than being run on hardware, it was tested through numerical simulation, extending the correspondence to the trillionth zero. The researchers also proposed a gate-based quantum algorithm capable of implementing either system using only polynomial resources.
Finding zeros
The NMR experiment produced a clean result. When the system was tuned to the critical line, probe-qubit coherence collapsed at points matching the first five nontrivial zeros of the zeta function. Off the critical line, no such collapses appeared. This collapse is the dynamical phase transition itself. The moment the probe qubit's quantum signal drops to zero marks where a zero of the zeta function has been detected.
The numerical simulation of the second system showed the same zero-to-collapse relationship out to the trillionth zero, with the simulated zero locations matching the true, known values more precisely at larger zeros.
The researchers clarified that their work is not a rigorous mathematical proof of the Riemann Hypothesis, since experiments and simulations can only ever check a finite number of zeros.
"Its true significance lies in providing a radically new, scalable tool with a clear quantum advantage," the researchers said. "Classically, simulating dense, massive number-theoretic functions or verifying zeros extremely far out on the critical line becomes exponentially expensive. Our proposed digital quantum framework requires only polynomial resources."
What's next?
The researchers outlined three directions for future work. The first is scaling up their hardware, moving from the five-qubit demonstration to larger digital quantum platforms capable of resolving much higher-order zeros. The second is expanding into physics applications beyond number theory, including using the same framework to calculate the Witten index in string theory or to probe the thermodynamic properties of black holes. The third is broadening the approach to other unsolved problems in mathematics.
"Since the Riemann zeta function can be mapped to quantum dynamics, it inspires us to wonder if other mysteries in number theory—such as generalized L-functions or structural properties related to other arithmetic conjectures—possess their own physical 'twins' waiting to be discovered in quantum many-body systems," the researchers said.
Written for you by our author Tejasri Gururaj, edited by Gaby Clark, and fact-checked and reviewed by Robert Egan—this article is the result of careful human work. We rely on readers like you to keep independent science journalism alive. If this reporting matters to you, please consider a donation (especially monthly). You'll get an ad-free account as a thank-you.
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
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