Can quantum computers solve math’s hardest problem?

Can quantum computers solve math’s hardest problem? A $1 million math mystery may soon be resolved in a physics lab. The Riemann hypothesis claims that the locations of prime numbers along an infinite number line follow a complex and hidden formula. Despite $1 million offered and 167 years of attempts, mathematicians still lack a proof.

Now, a team in China has encoded this formula into a physical system and used a quantum computer to explore its workings. This study, published in Nature Communications, brings this abstract question about prime numbers closer to reality. Shijie Wei, co-lead author of the study, believes quantum computing could serve as a powerful tool for investigating major mathematical conjectures.

The Riemann hypothesis centers on the Riemann zeta function, an equation involving an infinite sum of numbers. By plugging in an input (a number with real and imaginary parts), the result is a single number. The Riemann hypothesis posits that the "zeros" of this function—where the sum equals zero—correspond to prime number locations.

If a zero is found where the real part of the input isn't 1/2, the hypothesis is disproved, and the mathematician who discovers it will receive $1 million. For over a century, the problem remained in number theory. However, in 1972, scientists noticed a connection between the function's zeros and atomic nuclei, sparking interest in the quantum world.

Wei and his colleagues propose encoding the Riemann zeta function into a set of interacting atomic nuclei. This quantum system reflects the zeta function, with temperature representing the real part of the input and the evolution time encoding the imaginary part. A phase transition occurring in the system indicates a zero input not equal to 1/2, potentially disproving the Riemann hypothesis.

Using a system of five interacting atoms (quantum bits or "qubits"), the researchers found that their quantum algorithm scans the zeta function for zeros faster than classical computers. While they haven't yet disproved the hypothesis, their experiment offers a physical lens to explore prime numbers. As the system grows to 100 qubits, it will be able to check more zeros than any existing computer.

This study brings mathematicians closer to solving Dyson and Montgomery's dream, suggesting that the Riemann Hypothesis and quantum phase transitions might be connected.