Can Quantum Computers Solve Math’s Hardest Problem?

The Riemann hypothesis claims that nan locations of prime numbers on nan infinite number statement each adhere to a beautiful and orderly but obscure formula. Yet 167 years aft German mathematician Bernhard Riemann made this guess, and successful spite of a million-dollar bounty, mathematicians still person nary thought really to beryllium it.

Now a squad successful China has managed to encode that look into a beingness strategy and research its workings utilizing a quantum computer. The researchers’ work, an unedited type of which saw early publication past period successful nan diary Nature Communications, makes this absurd mobility astir premier numbers much tangible than ever before.

“It provides a caller position connected nan Riemann hypothesis,” says Shijie Wei of nan Beijing Academy of Quantum Information Sciences, nan study’s co-lead author. He hopes nan activity will beryllium “that quantum computing will service arsenic a powerful avenue for investigating awesome mathematical conjectures.”

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The conception struck Wei a decade ago, erstwhile he saw a speech astir nan exploration of a related mathematical formula, called nan Möbius inversion, pinch a quantum computer. “Inspired by this idea, I thought that possibly nan Riemann presumption tin besides link to quantum systems,” he says.

At nan halfway of nan presumption is nan Riemann zeta function, a gnarly equation involving a sum of infinitely galore pieces. First, you plug successful its input—a number pinch a existent portion and an imaginary portion (the second is “imaginary” because it involves thing seemingly nonsensical: nan quadrate guidelines of –1). Then, erstwhile you cipher that infinite sum, nan consequence is simply a azygous number.

Riemann showed that nan locations of nan zeta function’s “zeros”—the different inputs that origin its infinite sum to precisely adjacent zero—encode nan locations of each nan premier numbers on nan number line. Moreover, he hypothesized that these zeros hap only erstwhile nan input’s existent portion is precisely 1⁄2. If true, this would uncover singular bid hiding beneath nan primes’ evident chaos. But if you find immoderate zero wherever nan input’s existent portion isn’t 1⁄2 (regardless of nan imaginary part’s value), you’ve disproved nan conjecture—and should constitute to nan Clay Mathematics Institute to person your $1 million.

For much than a century, nan problem rested squarely successful nan realm of number theory. But successful 1972 mathematician Hugh Montgomery, past a Ph.D. student astatine nan University of Cambridge, happened to meet nan renowned physicist Freeman Dyson, and nan 2 sewage to talking astir nan Riemann zeta function. They noticed a unusual relationship betwixt its zeros and nan soul workings of nan atomic nucleus.

Ever since, immoderate person wondered if nan secrets to nan primes mightiness dishonesty successful nan microscopic quantum world. Scientists person sought a quantum strategy whose power ever corresponds to an input that makes nan zeta function’s output zero. But nary 1 has recovered specified a hypothetical system.

Wei and his colleagues person projected a caller beingness way to nan mysterious function. It’s a group of interacting atomic nuclei that evolves complete time, sometimes undergoing a sharp, melodramatic translator called a “phase transition,” a beingness alteration astir akin to liquid h2o freezing to crystal aliases boiling to steam.

The researchers showed that this unsocial quantum strategy precisely reflects nan Riemann zeta function. At immoderate fixed moment, nan system’s somesthesia corresponds to nan input’s existent part, and nan magnitude of clip they’ve allowed it to germinate encodes nan input’s imaginary part. The shape modulation occurs only erstwhile some of these inputs, if fed to nan Riemann zeta function, would make its output zero.

This uncovering offered a caller measurement for Wei and his colleagues to hunt for unexpected zeros of nan zeta function. They hole nan strategy astatine a somesthesia truthful that nan input’s existent portion is thing different than 1⁄2. Then they fto nan strategy germinate and hold for a shape transition. If 1 happens, that will mean that nan zeta usability has a zero pinch an input that’s not 1⁄2—disproving nan Riemann hypothesis.

The squad demonstrated that this quantum algorithm “scans” nan zeta usability for zeros faster than immoderate of nan “classical” computers mathematicians typically use. The researchers’ objection utilized a strategy of 5 interacting atoms—the quantum balanced of computational bits, called “qubits.” They haven’t yet travel crossed immoderate zeros that mathematicians person missed, but nan research “offers an analog beingness lens alternatively than conscionable integer calculation,” Wei says. If nan squad tin turn nan strategy to 100 qubits, past it will beryllium capable to cheque much zeros than immoderate existing computer. The study brings Wei and his colleagues a measurement person to Dyson and Montgomery’s dream: Riemann’s timeless mobility mightiness find its reply successful nan beingness world.

“It whitethorn good be,” says Guilu Long of nan Beijing Academy of Quantum Information Sciences, nan study’s elder author, “that nan Riemann Hypothesis and dynamical quantum shape transitions correspond 2 facets of 1 underlying truth.”