Untangling the Mysteries of Knots with Quantum Computers
By Konstantinos Meichanetzidis
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.
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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.
Typically, Quantum Error Detection (QED) is viewed as a short-term solution—a non-scalable, stop-gap until full fault tolerance is achieved at scale.
That’s just changed, thanks to a serendipitous discovery made by our team. Now, QED can be used in a much wider context than previously thought. Our team made this discovery while studying the contact process, which describes things like how diseases spread or how water permeates porous materials. In particular, our team was studying the quantum contact process (QCP), a problem they had tackled before, which helps physicists understand things like phase transitions. In the process (pun intended), they came across what senior advanced physicist, Eli Chertkov, described as “a surprising result.”
While examining the problem, the team realized that they could convert detected errors due to noisy hardware into random resets, a key part of the QCP, thus avoiding the exponentially costly overhead of post-selection normally expected in QED.
To understand this better, the team developed a new protocol in which the encoded, or logical, quantum circuit adapts to the noise generated by the quantum computer. They quickly realized that this method could be used to explore other classes of random circuits similar to the ones they were already studying.
The team put it all together on System Model H2 to run a complex simulation, and were surprised to find that they were able to achieve near break-even results, where the logically encoded circuit performed as well as its physical analog, thanks to their clever application of QED. Ultimately, this new protocol will allow QED codes to be used in a scalable way, saving considerable computational resources compared to full quantum error correction (QEC).
Researchers at the crossroads of quantum information, quantum simulation, and many-body physics will take interest in this protocol and use it as a springboard for inventing new use cases for QED.
Stay tuned for more, our team always has new tricks up their sleeves.
Learn mode about System Model H2 with this video:
By Konstantinos Meichanetzidis
When will quantum computers outperform classical ones?
This question has hovered over the field for decades, shaping billion-dollar investments and driving scientific debate.
The question has more meaning in context, as the answer depends on the problem at hand. We already have estimates of the quantum computing resources needed for Shor’s algorithm, which has a superpolynomial advantage for integer factoring over the best-known classical methods, threatening cryptographic protocols. Quantum simulation allows one to glean insights into exotic materials and chemical processes that classical machines struggle to capture, especially when strong correlations are present. But even within these examples, estimates change surprisingly often, carving years off expected timelines. And outside these famous cases, the map to quantum advantage is surprisingly hazy.
Researchers at Quantinuum have taken a fresh step toward drawing this map. In a new theoretical framework, Harry Buhrman, Niklas Galke, and Konstantinos Meichanetzidis introduce the concept of “queasy instances” (quantum easy) – problem instances that are comparatively easy for quantum computers but appear difficult for classical ones.
From Problem Classes to Problem Instances
Traditionally, computer scientists classify problems according to their worst-case difficulty. Consider the problem of Boolean satisfiability, or SAT, where one is given a set of variables (each can be assigned a 0 or a 1) and a set of constraints and must decide whether there exists a variable assignment that satisfies all the constraints. SAT is a canonical NP-complete problem, and so in the worst case, both classical and quantum algorithms are expected to perform badly, which means that the runtime scales exponentially with the number of variables. On the other hand, factoring is believed to be easier for quantum computers than for classical ones. But real-world computing doesn’t deal only in worst cases. Some instances of SAT are trivial; others are nightmares. The same is true for optimization problems in finance, chemistry, or logistics. What if quantum computers have an advantage not across all instances, but only for specific “pockets” of hard instances? This could be very valuable, but worst-case analysis is oblivious to this and declares that there is no quantum advantage.
To make that idea precise, the researchers turned to a tool from theoretical computer science: Kolmogorov complexity. This is a way of measuring how “regular” a string of bits is, based on the length of the shortest program that generates it. A simple string like 0000000000 can be described by a tiny program (“print ten zeros”), while the description of a program that generates a random string exhibiting no pattern is as long as the string itself. From there, the notion of instance complexity was developed: instead of asking “how hard is it to describe this string?”, we ask “how hard is it to solve this particular problem instance (represented by a string)?” For a given SAT formula, for example, its polynomial-time instance complexity is the size of the smallest program that runs in polynomial time and decides whether the formula is satisfiable. This smallest program must be consistently answering all other instances, and it is also allowed to declare “I don’t know”.
In their new work, the team extends this idea into the quantum realm by defining polynomial-time quantum instance complexity as the size of the shortest quantum program that solves a given instance and runs on polynomial time. This makes it possible to directly compare quantum and classical effort, in terms of program description length, on the very same problem instance. If the quantum description is significantly shorter than the classical one, that problem instance is one the researchers call “queasy”: quantum-easy and classically hard. These queasy instances are the precise places where quantum computers offer a provable advantage – and one that may be overlooked under a worst-case analysis.
Why “Queasy”?
The playful name captures the imbalance between classical and quantum effort. A queasy instance is one that makes classical algorithms struggle, i.e. their shortest descriptions of efficient programs that decide them are long and unwieldy, while a quantum computer can handle the same instance with a much simpler, faster, and shorter program. In other words, these instances make classical computers “queasy,” while quantum ones solve them efficiently and finding them quantum-easy. The key point of these definitions lies in demonstrating that they yield reasonable results for well-known optimisation problems.
By carefully analysing a mapping from the problem of integer factoring to SAT (which is possible because factoring is inside NP and SAT is NP-complete) the researchers prove that there exist infinitely many queasy SAT instances. SAT is one of the most central and well-studied problems in computer science that finds numerous applications in the real-world. The significant realisation that this theoretical framework highlights is that SAT is not expected to yield a blanket quantum advantage, but within it lie islands of queasiness – special cases where quantum algorithms decisively win.
Algorithmic Utility
Finding a queasy instance is exciting in itself, but there is more to this story. Surprisingly, within the new framework it is demonstrated that when a quantum algorithm solves a queasy instance, it does much more than solve that single case. Because the program that solves it is so compact, the same program can provably solve an exponentially large set of other instances, as well. Interestingly, the size of this set depends exponentially on the queasiness of the instance!
Think of it like discovering a special shortcut through a maze. Once you’ve found the trick, it doesn’t just solve that one path, but reveals a pattern that helps you solve many other similarly built mazes, too (even if not optimally). This property is called algorithmic utility, and it means that queasy instances are not isolated curiosities. Each one can open a doorway to a whole corridor with other doors, behind which quantum advantage might lie.
A North Star for the Field
Queasy instances are more than a mathematical curiosity; this is a new framework that provides a language for quantum advantage. Even though the quantities defined in the paper are theoretical, involving Turing machines and viewing programs as abstract bitstrings, they can be approximated in practice by taking an experimental and engineering approach. This work serves as a foundation for pursuing quantum advantage by targeting problem instances and proving that in principle this can be a fruitful endeavour.
The researchers see a parallel with the rise of machine learning. The idea of neural networks existed for decades along with small scale analogue and digital implementations, but only when GPUs enabled large-scale trial and error did they explode into practical use. Quantum computing, they suggest, is on the cusp of its own heuristic era. “Quristics” will be prominent in finding queasy instances, which have the right structure so that classical methods struggle but quantum algorithms can exploit, to eventually arrive at solutions to typical real-world problems. After all, quantum computing is well-suited for small-data big-compute problems, and our framework employs the concepts to quantify that; instance complexity captures both their size and the amount of compute required to solve them.
Most importantly, queasy instances shift the conversation. Instead of asking the broad question of when quantum computers will surpass classical ones, we can now rigorously ask where they do. The queasy framework provides a language and a compass for navigating the rugged and jagged computational landscape, pointing researchers, engineers, and industries toward quantum advantage.
From September 16th – 18th, Quantum World Congress (QWC) brought together visionaries, policymakers, researchers, investors, and students from across the globe to discuss the future of quantum computing in Tysons, Virginia.
Quantinuum is forging the path to universal, fully fault-tolerant quantum computing with our integrated full-stack. With our quantum experts were on site, we showcased the latest on Quantinuum Systems, the world’s highest-performing, commercially available quantum computers, our new software stack featuring the key additions of Guppy and Selene, our path to error correction, and more.
Highlights from QWC
Dr. Patty Lee Named the Industry Pioneer in Quantum
The Quantum Leadership Awards celebrate visionaries transforming quantum science into global impact. This year at QWC, Dr. Patty Lee, our Chief Scientist for Hardware Technology Development, was named the Industry Pioneer in Quantum! This honor celebrates her more than two decades of leadership in quantum computing and her pivotal role advancing the world’s leading trapped-ion systems. Watch the Award Ceremony here.
Keynote with Quantinuum's CEO, Dr. Rajeeb Hazra
At QWC 2024, Quantinuum’s President & CEO, Dr. Rajeeb “Raj” Hazra, took the stage to showcase our commitment to advancing quantum technologies through the unveiling of our roadmap to universal, fully fault-tolerant quantum computing by the end of this decade. This year at QWC 2025, Raj shared the progress we’ve made over the last year in advancing quantum computing on both commercial and technical fronts and exciting insights on what’s to come from Quantinuum. Access the full session here.
Panel Session: Policy Priorities for Responsible Quantum and AI
As part of the Track Sessions on Government & Security, Quantinuum’s Director of Government Relations, Ryan McKenney, discussed “Policy Priorities for Responsible Quantum and AI” with Jim Cook from Actions to Impact Strategies and Paul Stimers from Quantum Industry Coalition.
Fireside Chat: Establishing a Pro-Innovation Regulatory Framework
During the Track Session on Industry Advancement, Quantinuum’s Chief Legal Officer, Kaniah Konkoly-Thege, and Director of Government Relations, Ryan McKenney, discussed the importance of “Establishing a Pro-Innovation Regulatory Framework”.