In a new paper, Quantinuum scientists have perfected a way of doing maths with diagrams instead of symbols

June 14, 2024

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.

Kaniah Konkoly-Thege

Kaniah is Chief Legal Counsel and SVP of Government Relations for Quantinuum. In her previous role, she served as General Counsel, Honeywell Quantum Solutions. Prior to Honeywell, she was General Counsel, Honeywell Federal Manufacturing and Technologies, LLC, and Senior Attorney, U.S. Department of Energy. She was Lead Counsel before the Civilian Board of Contract Appeals, the Merit Systems Protection Board, and the Equal Employment Opportunity Commission. Kaniah holds a J.D. from American University, Washington College of Law and B.A., International Relations and Spanish from the College of William and Mary.

Jeff Miller

Jeff Miller is Chief Information Officer for Quantinuum. In his previous role, he served as CIO for Honeywell Quantum Solutions and led a cross-functional team responsible for Information Technology, Cybersecurity, and Physical Security. For Honeywell, Jeff has held numerous management and executive roles in Information Technology, Security, Integrated Supply Chain and Program Management. Jeff holds a B.S., Computer Science, University of Arizona. He is a veteran of the U.S. Navy, attaining the rank of Commander.

Matthew Bohne

Matthew Bohne is the Vice President & Chief Product Security Officer for Honeywell Corporation. He is a passionate cybersecurity leader and executive with a proven track record of building and leading cybersecurity organizations securing energy, industrial, buildings, nuclear, pharmaceutical, and consumer sectors. He is a sought-after expert with deep experience in DevSecOps, critical infrastructure, software engineering, secure SDLC, supply chain security, privacy, and risk management.

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Todd Moore is the Global Vice President of Data Encryption Products at Thales. He is responsible for setting the business line and go to market strategies for an industry leading cybersecurity business. He routinely helps enterprises build solutions for a wide range of complex data security problems and use cases. Todd holds several management and technical degrees from the University of Virginia, Rochester Institute of Technology, Cornell University and Ithaca College. He is active in his community, loves to travel and spends much of his free time supporting his family in pursuing their various passions.

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Retired U.S. Army Major General John Davis is the Vice President, Public Sector for Palo Alto Networks, where he is responsible for expanding cybersecurity initiatives and global policy for the international public sector and assisting governments around the world to prevent successful cyber breaches. Prior to joining Palo Alto Networks, John served as the Senior Military Advisor for Cyber to the Under Secretary of Defense for Policy and served as the Acting Deputy Assistant Secretary of Defense for Cyber Policy.  Prior to this assignment, he served in multiple leadership positions in special operations, cyber, and information operations.