Untangling the Mysteries of Knots with Quantum Computers
What Quantum Advantage actually looks like
One of the greatest privileges of working directly with the world’s most powerful quantum computer at Quantinuum is building meaningful experiments that convert theory into practice. The privilege becomes even more compelling when considering that our current quantum processor – our H2 system – will soon be enhanced by Helios, a quantum computer potentially a stunning trillion times more powerful, and due for launch in just a few months. The moment has now arrived when we can build a timeline for applications that quantum computing professionals have anticipated for decades and which are experimentally supported.
Quantinuum’s applied algorithms team has released an end-to-end implementation of a quantum algorithm to solve a central problem in knot theory. Along with an efficiently verifiable benchmark for quantum processors, it allows for concrete resource estimates for quantum advantage in the near-term. The research team, included Quantinuum researchers Enrico Rinaldi, Chris Self, Eli Chertkov, Matthew DeCross, David Hayes, Brian Neyenhuis, Marcello Benedetti, and Tuomas Laakkonen of the Massachusetts Institute of Technology. In this article, Konstantinos Meichanetzidis, a team leader from Quantinuum’s AI group who led the project, writes about the problem being addressed and how the team, adopting an aggressively practical mindset, quantified the resources required for quantum advantage:
Knot theory is a field of mathematics called ‘low-dimensional topology’, with a rich history, stemming from a wild idea proposed by Lord Kelvin, who conjectured that chemical elements are different knots formed by vortices in the aether. Of course, we know today that the aether theory was falsified by the Michelson-Morley experiment, but mathematicians have been classifying, tabulating, and studying knots ever since. Regarding applications, the pure mathematics of knots can find their way into cryptography, but knot theory is also intrinsically related to many aspects of the natural sciences. For example, it naturally shows up in certain spin models in statistical mechanics, when one studies thermodynamic quantities, and the magnetohydrodynamical properties of knotted magnetic fields on the surface of the sun are an important indicator of solar activity, to name a few examples. Remarkably, physical properties of knots are important in understanding the stability of macromolecular structures. This is highlighted by work of Cozzarelli and Sumners in the 1980’s, on the topology of DNA, particularly how it forms knots and supercoils. Their interdisciplinary research helped explain how enzymes untangle and manage DNA topology, crucial for replication and transcription, laying the foundation for using mathematical models to predict and manipulate DNA behavior, with broad implications in drug development and synthetic biology. Serendipitously, this work was carried out during the same decade as Richard Feynman, David Deutsch, and Yuri Manin formed the first ideas for a quantum computer.
Most importantly for our context, knot theory has fundamental connections to quantum computation, originally outlined by Witten’s work in topological quantum field theory, concerning spacetimes without any notion of distance but only shape. In fact, this connection formed the very motivation for attempting to build topological quantum computers, where anyons – exotic quasiparticles that live in two-dimensional materials – are braided to perform quantum gates. The relation between knot theory and quantum physics is the most beautiful and bizarre facts you have never heard of.
The fundamental problem in knot theory is distinguishing knots, or more generally, links. To this end, mathematicians have defined link invariants, which serve as ‘fingerprints’ of a link. As there are many equivalent representations of the same link, an invariant, by definition, is the same for all of them. If the invariant is different for two links then they are not equivalent. The specific invariant our team focused on is the Jones polynomial.
They all have the same Jones polynomial, as it is an invariant.
The mind-blowing fact here is that any quantum computation corresponds to evaluating the Jones polynomial of some link, as shown by the works of Freedman, Larsen, Kitaev, Wang, Shor, Arad, and Aharonov. It reveals that this abstract mathematical problem is truly quantum native. In particular, the problem our team tackled was estimating the value of the Jones polynomial at the 5th root of unity. This is a well-studied case due to its relation to the infamous Fibonacci anyons, whose braiding is capable of universal quantum computation.
Building and improving on the work of Shor, Aharonov, Landau, Jones, and Kauffman, our team developed an efficient quantum algorithm that works end-to end. That is, given a link, it outputs a highly optimized quantum circuit that is readily executable on our processors and estimates the desired quantity. Furthermore, our team designed problem-tailored error detection and error mitigation strategies to achieve a higher accuracy.
In addition to providing a full pipeline for solving this problem, a major aspect of this work was to use the fact that the Jones polynomial is an invariant to introduce a benchmark for noisy quantum computers. Most importantly, this benchmark is efficiently verifiable, a rare property since for most applications, exponentially costly classical computations are necessary for verification. Given a link whose Jones polynomial is known, the benchmark constructs a large set of topologically equivalent links of varying sizes. In turn, these result in a set of circuits of varying numbers of qubits and gates, all of which should return the same answer. Thus, one can characterize the effect of noise present in a given quantum computer by quantifying the deviation of its output from the known result.
The benchmark introduced in this work allows one to identify the link sizes for which there is exponential quantum advantage in terms of time to solution against the state-of-the-art classical methods. These resource estimates indicate our next processor, Helios, with 96 qubits and at least 99.95% two-qubit gate-fidelity, is extremely close to meeting these requirements. Furthermore, Quantinuum’s hardware roadmap includes even more powerful machines that will come online by the end of the decade. Notably, an advantage in energy consumption emerges for even smaller link sizes. Meanwhile, our teams aim to continue reducing errors through improvements in both hardware and software, thereby moving deeper into quantum advantage territory.
The importance of this work, indeed the uniqueness of this work in the quantum computing sector, is its practical end-to-end approach. The advantage-hunting strategies introduced are transferable to other “quantum-easy classically-hard” problems. Our team’s efforts motivate shifting the focus toward specific problem instances rather than broad problem classes, promoting an engineering-oriented approach to identifying quantum advantage. This involves first carefully considering how quantum advantage should be defined and quantified, thereby setting a high standard for quantum advantage in scientific and mathematical domains. And thus, making sure we instill confidence in our customers and partners.
Edited
About Quantinuum
Quantinuum, the world’s largest integrated quantum company, pioneers powerful quantum computers and advanced software solutions. Quantinuum’s technology drives breakthroughs in materials discovery, cybersecurity, and next-gen quantum AI. With over 500 employees, including 370+ scientists and engineers, Quantinuum leads the quantum computing revolution across continents.
Our quantum algorithms team has been hard at work exploring solutions to continually optimize our system’s performance. Recently, they’ve invented a novel technique, called the Quantum Paldus Transform (QPT), that can offer significant resource savings in future applications.
The transform takes complex representations and makes them simple, by transforming into a different “basis”. This is like looking at a cube from one angle, then rotating it and seeing just a square, instead. Transformations like this save resources because the more complex your problem looks, the more expensive it is to represent and manipulate on qubits.
You’ve changed
While it might sound like magic, transforms are a commonly used tool in science and engineering. Transforms simplify problems by reshaping them into something that is easier to deal with, or that provides a new perspective on the situation. For example, sound engineers use Fourier transforms every day to look at complex musical pieces in terms of their frequency components. Electrical engineers use Laplace transforms; people who work in image processing use the Abel transform; physicists use the Legendre transform, and so on.
In a new paper outlining the necessary tools to implement the QPT, Dr. Nathan Fitzpatrick and Mr. Jędrzej Burkat explain how the QPT will be widely applicable in quantum computing simulations, spanning areas like molecular chemistry, materials science, and semiconductor physics. The paper also describes how the algorithm can lead to significant resource savings by offering quantum programmers a more efficient way of representing problems on qubits.
Symmetry is key
The efficiency of the QPT stems from its use of one of the most profound findings in the field of physics: that symmetries drive the properties of a system.
While the average person can “appreciate” symmetry, for example in design or aesthetics, physicists understand symmetry as a much more profound element present in the fabric of reality. Symmetries are like the universe’s DNA; they lead to conservation laws, which are the most immutable truths we know.
Back in the 1920’s, when women were largely prohibited from practicing physics, one of the great mathematicians of the century, Emmy Noether, turned her attention to the field when she was tasked with helping Einstein with his work. In her attempt to solve a problem Einstein had encountered, Dr. Noether realized that all the most powerful and fundamental laws of physics, such as “energy can neither be created nor destroyed” are in fact the consequence of a deep simplicity – symmetry – hiding behind the curtains of reality. Dr. Noether’s theorem would have a profound effect on the trajectory of physics.
In addition to the many direct consequences of Noether’s theorem is a longstanding tradition amongst physicists to treat symmetry thoughtfully. Because of its role in the fabric of our universe, carefully considering the symmetries of a system often leads to invaluable insights.
Einstein, Pauli and Noether walk into a bar...
Many of the systems we are interested in simulating with quantum computers are, at their heart, systems of electrons. Whether we are looking at how electrons move in a paired dance inside superconductors, or how they form orbitals and bonds in a chemical system, the motion of electrons are at the core.
Seven years after Noether published her blockbuster results, Wolfgang Pauli made waves when he published the work describing his Pauli exclusion principle, which relies heavily on symmetry to explain basic tenets of quantum theory. Pauli’s principle has enormous consequences; for starters, describing how the objects we interact with every day are solid even though atoms are mostly empty space, and outlining the rules of bonds, orbitals, and all of chemistry, among other things.
Symmetry in motion
It is Pauli's symmetry, coupled with a deep respect for the impact of symmetry, that led our team at Quantinuum to the discovery published today.
In their work, they considered the act of designing quantum algorithms, and how one’s design choices may lead to efficiency or inefficiency.
When you design quantum algorithms, there are many choices you can make that affect the final result. Extensive work goes into optimizing each individual step in an algorithm, requiring a cyclical process of determining subroutine improvements, and finally, bringing it all together. The significant cost and time required is a limiting factor in optimizing many algorithms of interest.
This is again where symmetry comes into play. The authors realized that by better exploiting the deepest symmetries of the problem, they could make the entire edifice more efficient, from state preparation to readout. Over the course of a few years, a team lead Dr. Fitzpatrick and his colleague Jędrzej Burkat slowly polished their approach into a full algorithm for performing the QPT.
The QPT functions by using Pauli’s symmetry to discard unimportant details and strip the problem down to its bare essentials. Starting with a Paldus transform allows the algorithm designer to enjoy knock-on effects throughout the entire structure, making it overall more efficient to run.
“It’s amazing to think how something we discovered one hundred years ago is making quantum computing easier and more efficient,” said Dr. Nathan Fitzpatrick.
Ultimately, this innovation will lead to more efficient quantum simulation. Projects we believed to still be many years out can now be realized in the near term.
Transforming the future
The discovery of the Quantum Paldus Transform is a powerful reminder that enduring ideas—like symmetry—continue to shape the frontiers of science. By reaching back into the fundamental principles laid down by pioneers like Noether and Pauli, and combining them with modern quantum algorithm design, Dr. Fitzpatrick and Mr. Burkat have uncovered a tool with the potential to reshape how we approach quantum computation.
As quantum technologies continue their crossover from theoretical promise to practical implementation, innovations like this will be key in unlocking their full potential.
In a new paper in Nature Physics, we've made a major breakthrough in one of quantum computing’s most elusive promises: simulating the physics of superconductors. A deeper understanding of superconductivity would have an enormous impact: greater insight could pave the way to real-world advances, like phone batteries that last for months, “lossless” power grids that drastically reduce your bills, or MRI machines that are widely available and cheap to use. The development of room-temperature superconductors would transform the global economy.
A key promise of quantum computing is that it has a natural advantage when studying inherently quantum systems, like superconductors. In many ways, it is precisely the deeply ‘quantum’ nature of superconductivity that makes it both so transformative and so notoriously difficult to study.
Now, we are pleased to report that we just got a lot closer to that ultimate dream.
Making the impossible possible
To study something like a superconductor with a quantum computer, you need to first “encode” the elements of the system you want to study onto the qubits – in other words, you want to translate the essential features of your material onto the states and gates you will run on the computer.
For superconductors in particular, you want to encode the behavior of particles known as “fermions” (like the familiar electron). Naively simulating fermions using qubits will result in garbage data, because qubits alone lack the key properties that make a fermion so unique.
Until recently, scientists used something called the “Jordan-Wigner” encoding to properly map fermions onto qubits. People have argued that the Jordan-Wigner encoding is one of the main reasons fermionic simulations have not progressed beyond simple one-dimensional chain geometries: it requires too many gates as the system size grows.
Even worse, the Jordan-Wigner encoding has the nasty property that it is, in a sense, maximally non-fault-tolerant: one error occurring anywhere in the system affects the whole state, which generally leads to an exponential overhead in the number of shots required. Due to this, until now, simulating relevant systems at scale – one of the big promises of quantum computing – has remained a daunting challenge.
Theorists have addressed the issues of the Jordan-Wigner encoding and have suggested alternative fermionic encodings. In practice, however, the circuits created from these alternative encodings come with large overheads and have so far not been practically useful.
We are happy to report that our team developed a new way to compile one of the new, alternative, encodings that dramatically improves both efficiency and accuracy, overcoming the limitations of older approaches. Their new compilation scheme is the most efficient yet, slashing the cost of simulating fermionic hopping by an impressive 42%. On top of that, the team also introduced new, targeted error mitigation techniques that ensure even larger systems can be simulated with far fewer computational "shots"—a critical advantage in quantum computing.
Using their innovative methods, the team was able to simulate the Fermi-Hubbard model—a cornerstone of condensed matter physics— at a previously unattainable scale. By encoding 36 fermionic modes into 48 physical qubits on System Model H2, they achieved the largest quantum simulation of this model to date.
This marks an important milestone in quantum computing: it demonstrates that large-scale simulations of complex quantum systems, like superconductors, are now within reach.
Unlocking the Quantum Age, One Breakthrough at a Time
This breakthrough doesn’t just show how we can push the boundaries of what quantum computers can do; it brings one of the most exciting use cases of quantum computing much closer to reality. With this new approach, scientists can soon begin to simulate materials and systems that were once thought too complex for the most powerful classical computers alone. And in doing so, they’ve unlocked a path to potentially solving one of the most exciting and valuable problems in science and technology: understanding and harnessing the power of superconductivity.
The future of quantum computing—and with it, the future of energy, electronics, and beyond—just got a lot more exciting.
In an experiment led by Princeton and NIST, we’ve just delivered a crucial result in Quantum Error Correction (QEC), demonstrating key principles of scalable quantum computing developed by Drs Peter Shor, Dorit Aharonov, and Michael Ben-Or. In this latest paper, we showed that by using “concatenated codes” noise can be exponentially suppressed — proving that quantum computing will scale.
When noise is low enough, the results are transformative
Quantum computing is already producing results, but high-profile applications like Shor’s algorithm—which can break RSA encryption—require error rates about a billion times lower than what today’s machines can achieve.
Achieving such low error rates is a holy grail of quantum computing. Peter Shor was the first to hypothesize a way forward, in the form of quantum error correction. Building on his results, Dorit Aharanov and Michael Ben-Or proved that by concatenating quantum error correcting codes, a sufficiently high-quality quantum computer can suppress error rates arbitrarily at the cost of a very modest increase in the required number of qubits. Without that insight, building a truly fault-tolerant quantum computer would be impossible.
Their results, now widely referred to as the “threshold theorem”, laid the foundation for realizing fault-tolerant quantum computing. At the time, many doubted that the error rates required for large-scale quantum algorithms could ever be achieved in practice. The threshold theorem made clear that large scale quantum computing is a realistic possibility, giving birth to the robust quantum industry that exists today.
Realizing a legendary vision
Until now, nobody has realized the original vision for the threshold theorem. Last year, Google performed a beautiful demonstration of the threshold theorem in a different context (without concatenated codes). This year, we are proud to report the first experimental realization of that seminal work—demonstrating fault-tolerant quantum computing using concatenated codes, just as they envisioned.
The benefits of concatenation
The team demonstrated that their family of protocols achieves high error thresholds—making them easier to implement—while requiring minimal ancilla qubits, meaning lower overall qubit overhead. Remarkably, their protocols are so efficient that fault-tolerant preparation of basis states requires zero ancilla overhead, making the process maximally efficient.
This approach to error correction has the potential to significantly reduce qubit requirements across multiple areas, from state preparation to the broader QEC infrastructure. Additionally, concatenated codes offer greater design flexibility, which makes them especially attractive. Taken together, these advantages suggest that concatenation could provide a faster and more practical path to fault-tolerant quantum computing than popular approaches like the surface code.
We’re always looking forward
From a broader perspective, this achievement highlights the power of collaboration between industry, academia, and national laboratories. Quantinuum’s commercial quantum systems are so stable and reliable that our partners were able to carry out this groundbreaking research remotely—over the cloud—without needing detailed knowledge of the hardware. While we very much look forward to welcoming them to our labs before long, its notable that they never need to step inside to harness the full capabilities of our machines.
As we make quantum computing more accessible, the rate of innovation will only increase. The era of plug-and-play quantum computing has arrived. Are you ready?