New review paper on ZX-Calculus – From theory to practice – how diagrammatic calculus aids guides quantum computing in the real world.

“In science…novelty emerges only with difficulty, manifested by resistance, against a background provided by expectation. Initially only the anticipated and usual are experienced even under circumstances where anomaly is later to be observed.”  Thomas Kuhn, The Structure of Scientific Revolutions

Programmers and software developers working in traditional classical computing have long grown used to hearing about quantum computing. However, for the past two decades they have been able to benefit from massive advances in the speed of classical hardware, even as they approach the limits of silicon devices, with resulting spectacular and relentless progress in areas across the spectrum of computational tasks from apps on smartphones to industrial scale machine learning. Over that same period, meanwhile, quantum computers, designed to tackle computational problems that are too difficult for conventional computers to solve, have remained little more than an area of academic study.

The developments in the past year or so suggest that quantum computing is no longer simply an uncertain, far-distant possibility. First, quantum computers are now available to an increasingly large constituency of experimental users. Second, the decisions by organisations such as Microsoft and IBM to offer access to quantum processors on their cloud platforms. To that end, they have published roadmaps that suggest rapid growth in the scale and capacity of quantum computing processors for real-world applications, often in hybrid classical-quantum constructs.

One of the main challenges for computer professionals who have not been exposed to or trained in quantum computing is that any attempt to “get under the hood” leads almost immediately to a realisation that quantum physics is a foreign country. One with symbols, notation and idioms that reflect logical structures that can be incomprehensibly obscure.

Whilst it is likely that quantum computing will eventually evolve to the point where real-world problems will be solved with the use of high-level programming languages that do not require an in-depth knowledge of quantum physics, it is equally the case that to fully benefit from this paradigm shift in computation, one should have at least a basic understanding of the core precepts.

A paper published on the pre-print arXiv by CQC and written by Bob Coecke (CQC), and his associates Quanlong Wang (CQC), Dominic Horsman (Université Grenoble Alpes) and Aleks Kissinger (University of Oxford), provides curious non-specialists, as well as quantum scientists, with a unique and practical tool to come to grips with the core concepts and foundational building blocks of quantum computing. The paper, “Kindergarten Quantum Mechanics Graduates”, is described by Bob Coecke as a “spiritual god child” of papers in 2004 and 2008 that effectively introduced “ZX Calculus” as a diagrammatic form to untangle and make more intuitive the processes that are described by the postulates of quantum physics.

This is not the first time that theoretical physics has seen the benefits of supplementing or replacing complex formulas with diagrams. Feynman’s ingenious diagrammatic notations for describing elementary particles are now as ubiquitous and popular as they were ignored when first introduced. Bob Coecke and Ross Duncan (the original co-founders of ZX-calculus) took matters one step further and entirely replaced the usual formulation of quantum computing. And now, 12 years after ZX was first formalised, it is being adopted and used as a standard tool in quantum computing by an increasing number of developers and organisations.

But then again as Thomas Kuhn reminded us, novelty, true novelty, takes time to seed and grow roots.

The paper can be accessed here in pdf form and here is a link to the arXiv page.

About Cambridge Quantum Computing

Founded in 2014 and backed by some of the world’s leading quantum computing companies, CQC is a global leader in quantum software and quantum algorithms, enabling clients to achieve the most out of rapidly evolving quantum computing hardware. CQC has offices in the UK, USA and Japan. For more information, visit CQC at and on LinkedIn. Access the tket Python module on GitHub.

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