In a new paper, Quantinuum scientists have perfected a way of doing maths with diagrams instead of symbols
Doing mathematical physics with diagrams instead of traditional formalism allows researchers to tackle difficult problems in an intuitive and mathematically strict way that opens the door to new insights and solutions. The new calculus we are developing that we refer to as ZX calculus, also known as Penrose Spin Calculus, has applications in fields as diverse as quantum chemistry, condensed matter physics, and loop quantum gravity.
In a recent paper on the arXiv, Quantinuum researchers Harny Wang, Razin A. Shaikh, and Boldizsár Poór have proven the “completeness” of this ZX calculus in finite dimensions, meaning that one can now use diagrams instead of linear algebra to perform calculations in finite dimensional quantum mechanics. This is a remarkable achievement.
“Now very complicated formulas in quantum chemistry and loop quantum gravity can be derived by diagrams,” said co-author Harny Wang.
Physicists have used graphical calculus for a long time. They are used widely in quantum field theory, in the form of Feynman diagrams, or in gravitational theory, in the form of Penrose diagrams. Graphical calculation strategies are generally very well appreciated as they replace a lot of difficult and tedious ‘formal’ mathematics with a simpler, more intuitive, but no less accurate diagrammatic approach.
Our researcher’s work on ZX and ZXW calculus (a near cousin to ZX) is the latest but most innovative shift from “shut up and calculate” to “depict and rewrite”, a shift that many researchers are sure to welcome.
ZX calculus was initially developed by scientists as a tool for working on problems in quantum mechanics that require intricate calculations. ZX calculus, created by Professor Bob Coecke and Dr. Ross Duncan, both of whom are senior scientists at Quantinuum, has developed over the course of 15 years, leading to a growing global community of researchers. This most recent paper marks the transition of important parts of ZX from ‘a work in progress’ to something that is fully formed.
Both ZX and ZXW calculus are known for efficiently expressing quantum relations such as entanglement. It is hoped these new formalisms may uncover connections between some of the most challenging problems in science and quantum computing.
Distinguished physicist Carlo Rovelli has already expressed interest in using ZX and ZXW graphical calculus for his work, stating “Indeed, there are concrete steps in place to translate quantum gravity problems into quantum computing problems, and I have hope that the powerful conceptual and technical tools developed by Bob [Coecke], Harny [Wang] and their collaborators could play a major role in this.”
In addition to interest from the gravity community, ZX is being adopted in the wider quantum computing community. Dr. Peter Shor recently worked with colleagues to develop an algorithm that maps Clifford encoders to graphical representations in the ZX calculus.
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.
If we are to create ‘next-gen’ AI that takes full advantage of the power of quantum computers, we need to start with quantum native transformers. Today we announce yet again that Quantinuum continues to lead by demonstrating concrete progress — advancing from theoretical models to real quantum deployment.
The future of AI won't be built on yesterday’s tech. If we're serious about creating next-generation AI that unlocks the full promise of quantum computing, then we must build quantum-native models—designed for quantum, from the ground up.
Around this time last year, we introduced Quixer, a state-of-the-art quantum-native transformer. Today, we’re thrilled to announce a major milestone: one year on, Quixer is now running natively on quantum hardware.
Why this matters: Quantum AI, born native
This marks a turning point for the industry: realizing quantum-native AI opens a world of possibilities.
Classical transformers revolutionized AI. They power everything from ChatGPT to real-time translation, computer vision, drug discovery, and algorithmic trading. Now, Quixer sets the stage for a similar leap — but for quantum-native computation. Because quantum computers differ fundamentally from classical computers, we expect a whole new host of valuable applications to emerge.
Achieving that future requires models that are efficient, scalable, and actually run on today’s quantum hardware.
That’s what we’ve built.
What makes Quixer different?
Until Quixer, quantum transformers were the result of a brute force “copy-paste” approach: taking the math from a classical model and putting it onto a quantum circuit. However, this approach does not account for the considerable differences between quantum and classical architectures, leading to substantial resource requirements.
Quixer is different: it’s not a translation – it's an innovation.
With Quixer, our team introduced an explicitly quantum transformer, built from the ground up using quantum algorithmic primitives. Because Quixer is tailored for quantum circuits, it's more resource efficient than most competing approaches.
As quantum computing advances toward fault tolerance, Quixer is built to scale with it.
What’s next for Quixer?
We’ve already deployed Quixer on real-world data: genomic sequence analysis, a high-impact classification task in biotech. We're happy to report that its performance is already approaching that of classical models, even in this first implementation.
This is just the beginning.
Looking ahead, we’ll explore using Quixer anywhere classical transformers have proven to be useful; such as language modeling, image classification, quantum chemistry, and beyond. More excitingly, we expect use cases to emerge that are quantum-specific, impossible on classical hardware.
This milestone isn’t just about one model. It’s a signal that the quantum AI era has begun, and that Quantinuum is leading the charge with real results, not empty hype.
Stay tuned. The revolution is only getting started.
Our team is participating in ISC High Performance 2025 (ISC 2025) from June 10-13 in Hamburg, Germany!
As quantum computing accelerates, so does the urgency to integrate its capabilities into today’s high-performance computing (HPC) and AI environments. At ISC 2025, meet the Quantinuum team to learn how the highest performing quantum systems on the market, combined with advanced software and powerful collaborations, are helping organizations take the next step in their compute strategy.
Quantinuum is leading the industry across every major vector: performance, hybrid integration, scientific innovation, global collaboration and ease of access.
- Our industry-leading quantum computer holds the record for performance with a Quantum Volume of 2²³ = 8,388,608 and the highest fidelity on a commercially available QPU available to our users every time they access our systems.
- Our systems have been validated by a #1 ranking against competitors in a recent benchmarking study by Jülich Research Centre.
- We’ve laid out a clear roadmap to reach universal, fully fault-tolerant quantum computing by the end of the decade and will launch our next-generation system, Helios, later this year.
- We are advancing real-world hybrid compute with partners such as RIKEN, NVIDIA, SoftBank, STFC Hartree Center and are pioneering applications such as our own GenQAI framework.
Exhibit Hall
From June 10–13, in Hamburg, Germany, visit us at Booth B40 in the Exhibition Hall or attend one of our technical talks to explore how our quantum technologies are pushing the boundaries of what’s possible across HPC.
Presentations & Demos
Throughout ISC, our team will present on the most important topics in HPC and quantum computing integration—from near-term hybrid use cases to hardware innovations and future roadmaps.
Multicore World Networking Event
- Monday, June 9 | 7:00pm – 9:00 PM at Hofbräu Wirtshaus Esplanade
In partnership with Multicore World, join us for a Quantinuum-sponsored Happy Hour to explore the present and future of quantum computing with Quantinuum CCO, Dr. Nash Palaniswamy, and network with our team.
Register here
H1 x CUDA-Q Demonstration
- All Week at Booth B40
We’re showcasing a live demonstration of NVIDIA’s CUDA-Q platform running on Quantinuum’s industry-leading quantum hardware. This new integration paves the way for hybrid compute solutions in optimization, AI, and chemistry.
Register for a demo
HPC Solutions Forum
- Wednesday, June 11 | 2:20 – 2:40 PM
“Enabling Scientific Discovery with Generative Quantum AI” – Presented by Maud Einhorn, Technical Account Executive at Quantinuum, discover how hybrid quantum-classical workflows are powering novel use cases in scientific discovery.
See You There!
Whether you're exploring hybrid solutions today or planning for large-scale quantum deployment tomorrow, ISC 2025 is the place to begin the conversation.
We look forward to seeing you in Hamburg!
Quantinuum has once again raised the bar—setting a record in teleportation, and advancing our leadership in the race toward universal fault-tolerant quantum computing.
Last year, we published a paper in Science demonstrating the first-ever fault-tolerant teleportation of a logical qubit. At the time, we outlined how crucial teleportation is to realize large-scale fault tolerant quantum computers. Given the high degree of system performance and capabilities required to run the protocol (e.g., multiple qubits, high-fidelity state-preparation, entangling operations, mid-circuit measurement, etc.), teleportation is recognized as an excellent measure of system maturity.
Today we’re building on last year’s breakthrough, having recently achieved a record logical teleportation fidelity of 99.82% – up from 97.5% in last year’s result. What’s more, our logical qubit teleportation fidelity now exceeds our physical qubit teleportation fidelity, passing the break-even point that establishes our H2 system as the gold standard for complex quantum operations.
This progress reflects the strength and flexibility of our Quantum Charge Coupled Device (QCCD) architecture. The native high fidelity of our QCCD architecture enables us to perform highly complex demonstrations like this that nobody else has yet to match. Further, our ability to perform conditional logic and real-time decoding was crucial for implementing the Steane error correction code used in this work, and our all-to-all connectivity was essential for performing the high-fidelity transversal gates that drove the protocol.
Teleportation schemes like this allow us to “trade space for time,” meaning that we can do quantum error correction more quickly, reducing our time to solution. Additionally, teleportation enables long-range communication during logical computation, which translates to higher connectivity in logical algorithms, improving computational power.
This demonstration underscores our ongoing commitment to reducing logical error rates, which is critical for realizing the promise of quantum computing. Quantinuum continues to lead in quantum hardware performance, algorithms, and error correction—and we’ll extend our leadership come the launch of our next generation system, Helios, in just a matter of months.