A Five-Qubit Machine Made the Riemann Hypothesis Show Up as a Physical Phase Transition

A team of Chinese physicists has tied the Riemann Hypothesis — the most famous unsolved problem in mathematics — to a measurable event inside a quantum system, and then watched that event happen on a five-qubit processor.

The hypothesis, posed by Bernhard Riemann in 1859, concerns the Riemann zeta function, an infinite sum bound up with how prime numbers are distributed. Its nontrivial zeros all fall inside a narrow band of the complex plane called the critical strip, and Riemann conjectured that every one of them sits exactly on the line down the middle of that strip, with a real part of 1/2. Nobody has proved it. More than a thousand theorems have been proved on the assumption that it is true, and parts of modern cryptography lean on it.

Physicists have suspected for most of a century that the zeros are hiding something physical. The best-known version of that hunch is the Hilbert–Pólya conjecture, which proposes that the zeros are the energy levels of some undiscovered quantum system. That search has gone nowhere useful on real hardware. "Identifying and scaling such a stationary Hamiltonian is notoriously challenging, particularly on near-term quantum hardware," the researchers said. So they changed the variable. "We posed a distinct question: What if we use time instead of energy as our core variable?" said co-authors Shijie Wei of the Beijing Academy of Quantum Information Sciences, Tao Xin of the Shenzhen International Quantum Academy and Gui-Lu Long of BAQIS and Tsinghua University.

Their target was a dynamical quantum phase transition — a moment when a system's properties change abruptly as it evolves. The team built two complementary setups, each pairing a many-body system with a single probe qubit, prepared in thermal equilibrium and then driven out of it. In that construction, the two halves of the zeta function's complex argument become physical knobs: the real part is the system's temperature, and the imaginary part is elapsed time. Within the mapping, a phase transition can only occur if the system is prepared at a temperature that sits precisely on the critical line. Seeing one anywhere else would mean the Riemann Hypothesis fails inside this physical picture.

The first system ran on a five-qubit nuclear magnetic resonance processor, using the nuclear spins of a molecule with three fluorine and two hydrogen atoms, one spin acting as the probe. Tuned to the critical line, the probe qubit's coherence collapsed at points matching the first five nontrivial zeros of the zeta function. Off the critical line, no collapses appeared at all. "Only when the parameters simultaneously satisfied both conditions — lying on the critical line and equaling a Riemann zero — did the observed signal completely vanish," the researchers said. A second system, built around a related quantity called the Loschmidt amplitude and tested in numerical simulation rather than hardware, reproduced the same behavior out to the trillionth zero, with accuracy improving for the larger ones.

The authors are careful about what this is not. An experiment can check only finitely many zeros, so nothing here proves the hypothesis. "Its true significance lies in providing a radically new, scalable tool with a clear quantum advantage," they said, noting that verifying zeros far out on the critical line is exponentially expensive classically while their digital quantum framework needs only polynomial resources. Next steps run from bigger processors to applications well outside number theory — calculating the Witten index in string theory, probing the thermodynamics of black holes, and hunting for physical "twins" of other arithmetic conjectures. The work appears in Nature Communications.