On Quantum Computation, Anyons, and Categories
Andreas Blass and Yuri Gurevich
Abstract We explain the use of category theory in describing certain sorts of anyons. Yoneda’s lemma leads to a simplification of that description. For the particular case of Fibonacci anyons, we also exhibit some calculations that seem to be known to the experts but not explicit in the literature.
Keywords Anyon model • Fibonacci anyons • Quantum computing • Categories • Yoneda Lemma • Mathematical foundations
This paper attempts to explain the use of category theory in describing certain sorts of anyons. These are rather mysterious physical phenomena which, one hopes, will provide a basis for quantum computing needing far less error correction than other approaches.
The first author of this paper has long been a fan of category theory; even as a graduate student, he was described by one of his professors as “functorized”. The second author has been far more skeptical about the value of category theory in computer science, because of its distance from applications and because of the peril of potential (and in some cases actual) over-abstraction. In 2012, both authors began working with the Quantum Architectures and Computing (QuArC) Group at Microsoft Research and found anyons to be near the top of the group’s agenda. Seeing calculations and applications that use unitary matrices to represent braiding of anyons, we naturally wondered what Hilbert space these matrices are intended to operate on. We made rather a nuisance of ourselves by asking different people, on
A. Blass (B)
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Computational Logic, and Mathematical Foundations,
Outstanding Contributions to Logic 10, DOI 10.1007/978-3-319-41842-1_8
different occasions, what anyons actually are, from a mathematical point of view. Are they Hilbert spaces? Are they vectors in a Hilbert space? Are they something else? It turned out that the only mathematically sound answer in the literature involved a special sort of categories, modular tensor categories. So the second author agreed that categories can be quite relevant to important applications in computer science.
Our purpose in this paper is to describe some of the ideas surrounding categories and anyons in general and the special case of Fibonacci anyons and their category description. We hope that our presentation will be accessible and useful for mathematicians and computer scientists who have some acquaintance with the basics of category theory. Where we need to go beyond the basics, we explain, albeit briefly, the concepts from category theory that we use. We have also included a section describing the physical background that this mathematics is intended to formalize.
To describe more of our motivation for studying anyons, we need to presuppose some general information that will be explained in later sections of this paper. In particular, we shall refer to the fusion rule т ® т = т ® 1 for Fibonacci anyons т (and the vacuum 1). We hope that the following paragraphs will give the reader a rough idea of what we are looking at, and that re-reading them after the rest of the paper will provide a less rough idea.
In contrast to what occurs elsewhere in quantum theory, the states (represented, as usual, by vectors in Hilbert spaces, up to scalar multiples) in the modular tensor category picture are ways in which one configuration can fuse to form another configuration. They are not the configurations themselves. For example, in the Fibonacci case, there is a 2-dimensional Hilbert space of ways for three anyons to be regarded as (or to fuse into) one anyon; this is the Hilbert space Hom^ ® т ® т, т).
When we first heard about Fibonacci anyons, we thought that the fusion rule т ® т = т ® 1 meant that, if we put two т anyons together, then the result might look like one т anyon or like the vacuum (this much is true in the modular tensor category model) and that the general result would be a superposition of these two alternatives. But the model doesn’t allow such superpositions. Nor does the model say anything about the probabilities of the two possible outcomes.
Instead, we get superpositions of the following sort. Start with three т’s. Fuse the first two to get one т or vacuum. If you got vacuum, then the overall result is one т, namely the third of the original ones, which you haven’t yet fused. If, on the other hand, fusing the first two т’s gives a т, then fusing that with the third т might produce a т. (It might also produce vacuum, but that’s irrelevant for the present discussion.) So we have two ways to end up with one т, according to whether the first two т’s fused to vacuum or to т. And it is these two ways that the model allows superpositions of. Another possibility for getting two ways here is to fuse the last two т ’s first and then fuse the result with the first т. These two form another basis of the same 2-dimensional Hilbert space of “ways”. The relation between the two bases is
(part of) the associativity isomorphism of the modular tensor category. Yet another possibility would begin by fusing the first and third т’s. The modular tensor category representation of this possibility would use a braiding isomorphism to move the first anyon to be adjacent to the third (or vice versa), and it would depend on the path along which that anyon is moved around the second one.
In Sect. 8.2, we give a general introduction to anyons from the point of view of physics and quantum computation. That section is intended to give the reader a rough idea of what anyons are and why researchers in quantum computation would be interested in them. The treatment here is quite superficial, and we give references for more detailed treatments.
In Sect. 8.3, we gradually introduce modular tensor categories, and we explain how they are intended to be used to describe anyons. This section borrows heavily from the axiomatization given in , but with some modifications and rearrangements.
Section8.4 is devoted to an application of one of the central theorems of category theory, known as Yoneda’s Lemma, to producing a simplified view of modular tensor categories.
Finally, in Sect. 8.5, we consider the special case of Fibonacci anyons. This special case is unusually simple in some respects. Nevertheless (or perhaps therefore) it occupies a prominent place in quantum computing research. Section8.5 begins with a general description of Fibonacci anyons and then exhibits some calculations, whose results seem to be well known to some in the quantum computing community but which we have not been able to find written down in the literature.
More detailed treatments of modular tensor categories are available in the papers  of Panangaden and Paquette and  of Wang. Much of our exposition is based on the former. For other aspects of anyons and topological quantum computation, see, for example,  and the references there.
Remark 1 We encountered numerous explanations of the notion offusion of anyons, and they seemed to contradict each other. At one extreme was the picture of fusion as a physical process in which anyons are brought into spatial proximity with each other and energy is released as they form a new anyon (or perhaps annihilate each other). A minor modification of this picture is that energy need not be released; it might actually be consumed in the process. Another picture, however, did not insist that the anyons be brought together. They could remain far apart, and a suitable global measurement of the system’s quantum numbers could reveal how they “fused”. A path to reconciling these apparently contradictory pictures is suggested by a comment at the end of Sect. II.A of ; the idea is as follows. Consider several anyons, which we intend to fuse. As long as they are far apart, the various possible results of their fusion have energies that are very close together. (In technical terms, the ground state of the system is very nearly degenerate.) So the different fusion results can be distinguished in principle but not practically. When the anyons are brought closer together, though, the energy differences between the fusion possibilities become larger, and so it becomes practical to distinguish these possibilities. Thus, the discrepancy between various views of fusion seems to be largely a discrepancy between what can be observed in principle (or what is “really” happening) and what can be detected in practice.
-  Other answers explained the physics, in terms of excitations, but these matters are not the subject ofthis paper, which is specifically about mathematics except for the introductory material summarizedin Sect. 8.2.
-  For more on the notion of fusion, see Remark 1 at the end of this introduction.