Untangling the Mysteries of Knots with Quantum Computers
What Quantum Advantage actually looks like
March 25, 2025
By Konstantinos Meichanetzidis
One of the greatest privileges of working directly with the world’s most powerful quantum computer at ĢƵ 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.
ĢƵ’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 ĢƵ 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 ĢƵ’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.
Four equivalent representations of the trefoil knot, the simplest non-trivial knot. They all have the same Jones polynomial, as it is an invariant.
These knots have different Jones polynomials, so they are not equivalent.
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.
Demonstration of the quantum algorithm on the H2 quantum computer for estimating the value of Jones polynomial of a link with ~100 crossings. The raw signal (orange) can be amplified (green) with error detection, and corrected via a problem-tailored error mitigation method (purple), bringing the experimental estimate closer to the actual value (blue).
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 topologicallyequivalent 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, ĢƵ’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.
Rigorous resource estimation of our quantum algorithm pinpoints the exponential quantum advantage quantified in terms of time-to-solution, namely the time necessary for the classical state-of-the-art to reach the same error as the achieved by quantum. The advantagecrossover happens at large link sizes, requiring circuits with ~85 qubits and ~8.5k two-qubit gates, assuming 99.99% two-qubit gate fidelity and 30ms per circuit-layer. The classical algorithms are assumed to run on the Frontier Supercomputer.
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 ĢƵ
ĢƵ, the world’s largest integrated quantum company, pioneers powerful quantum computers and advanced software solutions. ĢƵ’s technology drives breakthroughs in materials discovery, cybersecurity, and next-gen quantum AI. With over 500 employees, including 370+ scientists and engineers, ĢƵ leads the quantum computing revolution across continents.
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March 25, 2026
Celebrating Our First Annual Q-Net Connect!
This month, ĢƵ welcomed its global user community to the first-ever Q-Net Connect, an annual forum designed to spark collaboration, share insights, and accelerate innovation across our full-stack quantum computing platforms. Over two days, users came together not only to learn from one another, but to build the relationships and momentum that we believe will help define the next chapter of quantum computing.
Q-Net Connect 2026 drew over 170 attendees from around the world to Denver, Colorado, including representatives from commercial enterprises and startups, academia and research institutions, and the public sector and non-profits - all users of ĢƵ systems.
The program was packed with inspiring keynotes, technical tracks, and customer presentations. Attendees heard from leaders at ĢƵ, as well as our partners at NVIDIA, JPMorganChase and BlueQubit; professors from the University of New Mexico, the University of Nottingham and Harvard University; national labs, including NIST, Oak Ridge National Laboratory, Sandia National Laboratories and Los Alamos National Laboratory; and other distinguished guests from across the global quantum ecosystem.
Congratulations to Q-Net Connect 2026 Award Recipients!
The mission of the ĢƵ Q-Net user community is to create a space for shared learning, collaboration and connection for those who adopt ĢƵ’s hardware, software and middleware platform. At this year’s Q-Net Connect, we awarded four organizations who made notable efforts to champion this effort.
JPMorganChase received the ‘Guppy Adopter Award’ for their exemplary adoption of our quantum programming language, Guppy, in their research workflows.
Phasecraft, a UK and US-based quantum algorithms startup, received the ‘Rising Star’ award for demonstrating exceptional early impact and advancing science using ĢƵ hardware, which they published in a December 2025 .
Qedma, a quantum software startup, received the ‘Startup Partner Engagement’ award for their sustained engagement with ĢƵ platforms dating back to our first commercially deployed quantum computer, H1.
Anna Dalmasso from the University of Nottingham received our ‘New Student Award’ for her impressive debut project on ĢƵ hardware and for delivering outstanding results as a new Q-Net student user.
Congratulations, again, and thank you to everyone who contributed to the success of the first Q-Net Connect!
Become a Q-Net Member
Q-Net offers year‑round support through user access, developer tools, documentation, trainings, webinars, and events. Members enjoy many exclusive benefits, including being the first to hear about exclusive content, publications and promotional offers.
By joining the community, you will be invited to exclusive gatherings to hear about the latest breakthroughs and connect with industry experts driving quantum innovation. Members also get access to Q‑Net Connect recordings and stay connected for future community updates.
In a follow-up to our recent work with Hiverge using AI to discover algorithms for quantum chemistry, we’ve teamed up with Hiverge, Amazon Web Services (AWS) and NVIDIA to explore using AI to improve algorithms for combinatorial optimization.
With the rapid rise of Large Language Models (LLMs), people started asking “what if AI agents can serve as on-demand algorithm factories?” We have been working with Hiverge, an algorithm discovery company, AWS, and NVIDIA, to explore how LLMs can accelerate quantum computing research.
Hiverge – named for Hive, an AI that can develop algorithms – aims to make quantum algorithm design more accessible to researchers by translating high-level problem descriptions in mostly natural language into executable quantum circuits. The Hive takes the researcher’s initial sketch of an algorithm, as well as special constraints the researcher enumerates, and evolves it to a new algorithm that better meets the researcher’s needs. The output is expressed in terms of a familiar programming language, like Guppy or , making it particularly easy to implement.
The AI is called a “Hive” because it is a collective of LLM agents, all of whom are editing the same codebase. In this work, the Hive was made up of LLM powerhouses such as Gemini, ChatGPT, Claude, Llama, as well as which was accessed through AWS’ Amazon Bedrock service. Many models are included because researchers know that diversity is a strength – just like a team of human researchers working in a group, a variety of perspectives often leads to the strongest result.
Once the LLMs are assembled, the Hive calls on them to do the work writing the desired algorithm; no new training is required. The algorithms are then executed and their ‘fitness’ (how well they solve the problem) is measured. Unfit programs do not survive, while the fittest ones evolve to the next generation. This process repeats, much like the evolutionary process of nature itself.
After evolution, the fittest algorithm is selected by the researchers and tested on other instances of the problem. This is a crucial step as the researchers want to understand how well it can generalize.
In this most recent work, the joint team explored how AI can assist in the discovery of heuristic quantum optimization algorithms, a class of algorithms aimed at improving efficiency across critical workstreams. These span challenges like optimal power grid dispatch and storage placement, arranging fuel inside nuclear reactors, and molecular design and reaction pathway optimization in drug, material, and chemical discovery—where solutions could translate into maximizing operational efficiency, dramatic reduction in costs, and rapid acceleration in innovation.
In other AI approaches, such as reinforcement learning, models are trained to solve a problem, but the resulting "algorithm" is effectively ‘hidden’ within a neural network. Here, the algorithm is written in Guppy or CUDA-Q (or Python), making it human-interpretable and easier to deploy on new problem instances.
This work leveraged the NVIDIA CUDA-Q platform, running on powerful NVIDIA GPUs made accessible by AWS. It’s state-of-the art accelerated computing was crucial; the research explored highly complex problems, challenges that lie at the edge of classical computing capacity. Before running anything on ĢƵ’s quantum computer, the researchers first used NVIDIA accelerated computing to simulate the quantum algorithms and assess their fitness. Once a promising algorithm is discovered, it could then be deployed on quantum hardware, creating an exciting new approach for scaling quantum algorithm design.
More broadly, this work points to one of many ways in which classical compute, AI, and quantum computing are most powerful in symbiosis. AI can be used to improve quantum, as demonstrated here, just as quantum can be used to extend AI. Looking ahead, we envision AI evolving programs that express a combination of algorithmic primitives, much like human mathematicians, such as Peter Shor and Lov Grover, have done. After all, both humans and AI can learn from each other.
As quantum computing power grows, so does the difficulty of error correction. Meeting that demand requires tight integration with high-performance classical computing, which is why we’ve partnered with NVIDIA to push the boundaries of real-time decoding performance.
Realizing the full power of quantum computing requires more than just qubits, it requires error rates low enough to run meaningful algorithms at scale. Physical qubits are sensitive to noise, which limits their capacity to handle calculations beyond a certain scale. To move beyond these limits, physical qubits must be combined into logical qubits, with errors continuously detected and corrected in real time before they can propagate and corrupt the calculation. This approach, known as fault tolerance, is a foundational requirement for any quantum computer intended to solve problems of real-world significance.
Part of the challenge of fault tolerance is the computational complexity of correcting errors in real time. Doing so involves sending the error syndrome data to a classical co-processor, solving a complex mathematical problem on that processor, then sending the resulting correction back to the quantum processor - all fast enough that it doesn’t slow down the quantum computation. For this reason, Quantum Error Correction (QEC) is currently one of the most demanding use-cases for tight coupling between classical and quantum computing.
Given the difficulty of the task, we have partnered with NVIDIA, leaders in accelerated computing. With the help of NVIDIA’s ultra-fast GPUs (and the GPU-accelerated BP-OSD decoder developed by NVIDIA as part of library), we were able to demonstrate real-time decoding of Helios’ qubits, all in a system that can be connected directly to our quantum processors using .
While real-time decoding has been demonstrated before (notably, by our own scientists in this study), previous demonstrations were limited in their scalability and complexity.
In this demonstration, we used Brings’ code, a high-rate code that is possible with our all-to-all connectivity, to encode our physical qubits into noise-resilient logical qubits. Once we had them encoded, we ran gates as well as let them idle to see if we could catch and correct errors quickly and efficiently. We submitted the circuits via both as well as our own Guppy language, underlining our commitment to accessible, ecosystem-friendly quantum computing.
The results were excellent: we were able to perform low-latency decoding that returned results in the time we needed, even for the faster clock cycles that we expect in future generation machines.
A key part of the achievement here is that we performed something called “correlated” decoding. In correlated decoding, you offload work that would normally be performed on the QPU onto the classical decoder. This is because, in ‘standard’ decoding, as you improve your error correction capabilities, it takes more and more time on the QPU. Correlated decoding elides this cost, saving QPU time for the tasks that only the quantum computer can do.
Stay tuned for our forthcoming paper with all the details.
