The patterns of crossing threads that make up a knot can be explored mathematically.Credit: Ilie Lupescu/500px via Getty
Are quantum computers worth the billions that are being invested in them? The answer is probably many years away. However, the machines could prove to be particularly suited to solving problems in mathematics — especially in topology, the branch of maths that studies shapes.
In a preprint posted on arXiv in March1, researchers at Quantinuum, a company headquartered in Cambridge, UK, report using their quantum machine H2-2 to distinguish between different types of knot on the basis of topological properties, and show that the method could be faster than those that run on ordinary, or ‘classical’, computers. Quantinuum chief product officer Ilyas Khan says that Helios, a quantum computer that the company expects to release later this year, could get much closer to beating classical supercomputers at analysing fiendishly complicated knots.
Although other groups have already made similar claims of ‘quantum advantage’, typically for ad hoc calculations that have no practical use, classical algorithms tend to catch up eventually. But theoretical results2,3 suggest that for some topology problems, quantum algorithms could be faster than any possible classical counterpart. This is owing to mysterious connections between topology and quantum physics. “That these things are related is mind-blowing, I think,” says Konstantinos Meichanetzidis, a Quantinuum researcher who led the work behind the preprint.
Knotty problems
In that work, Meichanetzidis and his colleagues used a quantum computer to calculate knot ‘invariants’ — numbers that describe particular types of knot. The invariants they looked at were devised by the New Zealand-born mathematician Vaughan Jones.
Knot invariants are typically calculated from patterns of crossings — how the threads in a knot cross over each other when the knot is flattened on a surface — but depend only on the knot’s topological type. In other words, the same knot can be flattened in two different ways, with vastly different crossing patterns, but the knot invariant will still be the same. If two crossing patterns have different knot invariants, it means that they come from knots that are topologically distinct. (However, the converse is not always true: in rare cases, two topologically different knots can give the same invariant.)
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Meichanetzidis’s team implemented a quantum algorithm for calculating the invariants of knots, proposed4 by Jones and computer scientists Dorit Aharonov and Zeph Landau. The algorithm is a series of quantum operations corresponding to the crossings of a flattened knot. The researchers used it to calculate Jones invariants for knots with up to 600 crossings on Quantinuum’s H2-2 quantum computer. This is still well within the scope of classical computing, but the company’s machines should eventually be able to handle 3,000 crossings or so, at which point even the fastest classical supercomputers will run out of steam, says Meichanetzidis.
Mathematically, the theoretical equivalence between knot crossings and quantum algorithms has been known for decades, but only now has the team been able to fully put it into practice, says Aharonov, who is at the Hebrew University of Jerusalem. “I expected that the transition between the languages would be much less efficient,” she adds.
