Quantinuum researchers are developing a unique approach to mathematics, promising new solutions to old problems

Quantinuum researchers have been working on a different kind of calculus, called graphical calculus, where one does math with pictures instead of symbols. Doing math with pictures allows researchers to tackle old, “unsolvable”, problems in an intuitive and mathematically strict way — allowing for new insights and solutions. The new calculus we are developing, called ZXW calculus, has already proven remarkably useful for reasoning with Hamiltonians, understanding light-matter interactions, and quantum machine learning.

In a recent paper on the arXiv [[2309.13014] Completeness of qufinite ZXW calculus, a graphical language for mixed-dimensional quantum computing (arxiv.org)], Quantinuum researchers have proven “completeness” of their ZXW calculus in “qfinite” dimensions, making it useful in even more domains. This is a lot to unpack (don’t worry — we will tell you what “qfinite” means!) so I (TR) spoke with lead researcher Quanlong Wang (QW) about the work.

TR: What does it mean to prove completeness? What are some specific consequences?

QW: By proving completeness, we prove that anything you can do with finite dimensional linear algebra, you can now do with diagrams. Put another way, we show that the rules of qufinite ZXW calculus are sufficient to derive all equalities of matrices. It means we can perform all reasoning with diagrams without ever going to the underlying matrices. Doing so, we show that this calculus is equivalent to finite dimensional Hilbert spaces. In particular, Qufinite ZXW calculus can be used to study properties of tensor networks, as each tensor can be represented as a qufinite ZXW diagram and equalities can be derived via rewriting of research including quantum many-body systems, quantum computing, and machine learning, the qufinite ZXW calculus may also provide new insights into these fields.

TR: Can you tell me what “qfinite” means?

QW: Qfinite describes the dimension. In standard “qubit”-based quantum computing, every quantum state or linear map is based on dimension 2, i.e., every state space is of dimension a power of 2. Similarly, for “qudit” quantum computing, every state space is of dimension a power of d which is a fixed integer. Now, we have put together all the qudit graphical calculi in a single framework, so the dimension of a state space could be any integer, not necessarily to be a power of a fixed integer d. We use “qufinite” to refer to this situation rather than qubit or qudit.

TR: Are we working on extending this to the infinite dimensional case? What will that mean for the applicability of ZXW calculus?

QW: We are working on the infinite dimensional case, and it will mean that we can solve things like the Schrödinger equation using diagrams.

TR: How does your work on ZXW calculus fit into Quantinuum’s larger goal to “build the world’s best quantum computers”?

QW: Our work on ZXW calculus is aimed at creating a versatile framework for the design and validation of quantum algorithms. The best computers come with the best software and algorithms.

TR: Graphical calculus already existed before this work, for example many readers are probably familiar with Feynman diagrams. Can you explain what is new in your work?

QW: Now, for the first time, we have a graphical rewriting framework which is equivalent to the category of all finite dimensional Hilbert spaces, in other words, for finite dimensional quantum theory (including quantum computing). Now, thanks to this work, we can just do rewriting of diagrams without any matrix calculations. In contrast, all the previous graphical calculi for quantum computing (including ZX calculus) can only do rewriting for a fragment of quantum computing (qubit quantum computing or qudit quantum computing for a fixed dimension d).

TR: Do you think ZXW calculus provides a more accessible entrypoint to quantum theory than linear algebra? Do you think that this new approach could lead to new insights into quantum mechanics? Do you think it makes quantum technology more accessible to the “workforce of the future”?

QW: Yes, ZXW provides a more accessible entry point to quantum theory than linear algebra, because diagrams are intuitive and the only calculation on diagrams is diagrammatic rewriting (replacing a part of a diagram with another), rather than low level matrix calculation. The preliminary results of an ongoing experiment with high school students show that it is possible to teach teenagers quantum theory using these diagrams. The qufinite ZXW calculus is equivalent to the finite dimensional Hilbert space formalism for quantum theory, but with everything represented and reasoned by diagrams, so it could lead to new insights into quantum theory, especially identifying structures in quantum mechanics.

TR: Do you see your work as paradigm-shifting? If yes, how?

QW: Yes, we think of this work as a part of the paradigm-shift which takes us from “shut up and calculate” to “depict and rewrite”. In that sense, we are looking at all of finite dimensional quantum theory in a diagrammatic way rather than matrix calculation.

TR: What’s next for you in terms of research? What are you the most excited about?

QW: We would like to apply the qufinite ZXW calculus to study mixed-dimensional quantum algorithms, efficient tensor network contraction, and symmetries of certain systems in quantum chemistry. Furthermore, we would like to use this mixed-dimensional framework to develop a quantum programming language due to its richness on types and its power for reasoning about quantum computing.

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Historical Note:

ZXW calculus is an extension of ZX calculus, which was developed by scientists as a new tool for tackling quantum mechanics — a challenge that was first set by John von Neumann in 1936, just three years after he competed the Mathematical Foundations for Quantum Mechanics. ZX calculus, whose progenitors include Professor Bob Coecke and Dr Ross Duncan, both senior scientists at Quantinuum, was developed over the course of 15 years by a growing community of researchers spanning Europe, Asia, Australia and North America. ZXW calculus was developed by our scientists as they work on a series of foundational problems relating to quantum computing and compositional intelligence.