Untying the knot

History tells us that Alexander the Great entered Phrygia, a province of the Persian Empire, in the year 333 BC. The events preceding Alexander’s dramatic arrival, however, are shrouded in a bit more mystery and mythology.

Several hundred years earlier—accounts vary but either in the eighth century or the second millennium BC— Phrygia, then an independent nation, found itself without a king. Naturally the gods intervened to rectify this situation, declaring via an oracle that the next person to enter Phrygia in an ox cart would take the throne. Sure enough, shortly after, a peasant farmer named Gordias along with his adopted son Midas, trundled up in a cart drawn by an ox. Again, the mythology diverges. The people chose Midas, a handsome, strapping young man, as king; or else they handed the crown to Gordias, and it was eventually inherited by Midas^[1].

At any rate, the capital of Phrygia was renamed Gordion. Midas, out of gratitude, dedicated his father’s ox cart to the gods and tied it to a post in the city square with a knot so intricate and complicated that it was virtually impossible to untie. And so the cart stood for centuries, governed by an oracular declaration that anyone who could loosen the Gordian knot would be destined to conquer all of Asia.

Enter Alexander, who, with the youthful arrogance of a budding emperor (he was only in his early 20s), decided that the oracle had never specified exactly how the knot should be unraveled. He drew his sword and with a single stroke simply cut through the tangle of rope and then proceeded to establish an empire of 5.2 million square kilometers, reaching as far east and north as the Indus and Oxus rivers. Had he not died prematurely at the age of 32, he may well have completely fulfilled the oracle’s prophecy—or maybe the gods just limited his imperial expansion because of his unorthodox solution to the Gordian knot.

Suffice to say, knots have continued to fascinate ever since. There’s a lot of intricate mathematics underlying knots, and many of these problems grow exponentially in complexity. Recent developments in quantum algorithms are providing novel solutions in knot theory and demonstrating another area where near-term quantum advantage can be realized. Let’s take a closer look at knots and their quantum solutions.

Why knot?
As any boy scout or sailor knows, learning knots is a critical skill. From climbing harnesses to a boat’s rigging and moorings, your life may depend on a properly tied knot. The ancient Inca peoples of South America used systems of knotted strings known as quipu to store numerical information in a decimal system. More recently, mathematicians have also been interested in knots as a curious subset of the field of topology,^[2] with several interesting practical and scientific applications. An MIT paper from 2019 describes two uses of knot theory, one in genetics and the other in statistical mechanics.

DNA, of course, famously takes the shape of a double helix, where both strands are connected by pairs of bases called adenine, cytosine, guanine and thymine. With millions of base pairs in each DNA molecule, the result looks extremely tangled. The MIT authors, Joy Lim and Eve Martin, note that as a cell manipulates its DNA, the enzymes performing the manipulations execute crossings and twists of the DNA strands very similar to those found in common knots. The mathematics of knot theory helps biologists analyze, predict and model how DNA will behave under the influence of different types of enzymes.

In physics, statistical mechanics is the study of the overall behaviours of systems of particles and their state changes—think of ice melting into water, or water evaporating into steam, which is not applicable to an individual molecule but to a large collection of molecules. The functions and graphs that physicists use to model state changes, especially at critical points, turn out to be identical to the mathematics of knots, the moves one can make to transform knots, and especially values known as knot invariants, which I will return to later in this post. So, knot theory and statistical physics can reinforce each other, yielding deeper insights in both fields.

Knot as easy as it looks
Unlike Alexander the Great, mathematicians study knots a bit more rigorously. To make a mathematical knot, take a one-dimensional line segment—a piece of string—and bend it, twist it or pass parts of it over or under each other, then join the ends together so that it forms a closed loop. Geometrically speaking, this is called projecting a circle into three-dimensional space—which is important, because the line or string is not allowed to intersect with itself. It goes over or under at crossing points, but not through itself. A circle with no crossings is the simplest type of knot, actually called the unknot. Cross both ends of a string over each other as if you’re beginning to tie your shoelaces, then join the ends together and you have a knot with three crossing points—known as the trefoil. The more crossing points, the higher the complexity of the knot. Finally, multiple knots can be combined to form links. The Olympic rings are a simple example of a link of five unknots.

When mathematicians study anything, they generally look for two concepts—identities and transformations. Identity is found when two objects—figures, geometric shapes, equations, etc.—have a one-to-one mapping even though they may look very different. Transformations define the series of steps you might have to take to change an object of one type into another. In knot theory, the terminology is a bit different, and we talk about invariants and moves.

Knot invariants are of particular interest, as they help mathematicians distinguish different types, or groups, of knots. One simple invariant is the crossing number—simply put, the minimum number of times a string crosses over itself in a knot. The unknot has a crossing number of zero, and the next simplest knot, the trefoil, has a crossing number of three. It turns out that there is only one group of knots with a crossing number of three, meaning that no matter how you twist and turn your string, if you create a knot with exactly three crossings, it will always be identical to the trefoil. However, the number of possible distinct knots increases quickly with the number of crossings—there are two possible distinct knots with five crossings, seven with seven crossings, 200 with 10 crossings, and the complexity goes up from there.

There are many more complicated knot invariants, and one in particular is the Jones Polynomial, discovered in 1984 by the New Zealand mathematician and Fields medalist Vaughan Jones. Without going into too much detail, the Jones Polynomial provides a measure of the complexity of a knot or link in terms of the number of moves it might take to untie it—to, in other words, reduce it to the unknot. Given the nature of DNA and proteins, the Jones Polynomial helps biochemists analyze and predict the properties of molecules and molecular chains, with direct applicability in health science. In statistical mechanics, it’s again the Jones Polynomial that allows physicists to model the phase transitions of complex systems.

The Jones Polynomial is known to be NP-hard^[3], which you might expect given its applicability to molecular modelling. An article published in Nature in April 2025 suggests that beyond about 3,000 crossings, the best classical supercomputers will fail at the Jones Polynomial.

Quantum Alexander
Enter quantum computing, which takes an equally non-classical but less destructive approach to solving the knot problem than Alexander the Great did. The same Nature article describes work done by quantum hardware vendor Quantinuum to implement an algorithm proposed by Jones himself, along with renowned quantum computer scientist Dorit Aharonov, almost two decades ago, in 2009. The algorithm uses quantum operations to model knot crossings and has been proven to solve the Jones Polynomial for knots of up to 104 crossings. It is expected to scale beyond 3,000 crossings on Quantinuum’s next-generation Helios computer, which would deliver quantum advantage within a year—a considerable achievement.

Even better, solutions to the Jones Polynomial have direct applicability to quantum computing itself. Quantinuum’s researchers, in a March 2025 post on the company’s website, announced the astonishing finding that any quantum computation corresponds directly to a particular solution of the Jones Polynomial. I’m not sure if the reverse is also true—that every possible solution to the Jones Polynomial maps to a quantum computation—but it does open up intriguing possibilities. Studying knot theory just might help inform and guide research into new quantum algorithms, which will help prove the utility of quantum computing.

Because the Jones Polynomial is a knot (and link) invariant, it turns out to also be applicable to managing quantum noise and mitigating the resulting errors. Quantinuum’s researchers devised a benchmarking technique whereby, once the solution to the Jones Polynomial is known for one link, they could create multiple topologically equivalent links of different sizes. Because of the aforementioned mapping of quantum computations to Jones Polynomial solutions, they could then generate new quantum circuits of varying depth and complexity, which should produce the same solution. Comparing the actual results to the expected known result allowed them to measure and map the effects of noise in their system.

The Quantinuum team is continuing its research, exploring broader parallels between the field of topology and quantum mechanics. In particular, the study of how surfaces remain invariant under distortions appears to be directly applicable to entangled systems of quantum particles.

Almost two and a half millennia ago, Alexander the Great took a brute force approach to solving a particularly knotty problem. That may have worked for him in his era, but now in the 21^st century, it’s good to know that quantum computing can provide equally radical but much more constructive contributions.

^[1] In case you’re wondering, yes, this is the same Midas who was granted the fateful golden touch by the gods.

^[2] Topology is the mathematical study of surfaces and shapes. Any topologist will be able to tell you that the only correct answer to the question making the rounds of the internet recently—how many holes does a straw have?—is one.

^[3] I’ve discussed NP-hardness before. It’s an indication that the complexity of a problem grows exponentially or worse, and therefore quickly becomes intractable for classical computing to solve exactly.