A graphical approach to measurement-based quantum computing

Quantum computations are easily represented in the graphical notation known as the zx-calculus, a.k.a. the red-green calculus. We demonstrate its use in reasoning about measurement-based quantum computing, where the graphical syntax directly captures the structure of the entangled states usesd to represent computations, and show that the notion of information flow within the entangled states gives rise to rewriting strategies for proving the correctness of quantum programs. Quantum computation, at least for the finite dimensional systems usually considered, lives in the setting of finite dimensional Hilbert spaces. Even ignoring the possibly enormous dimension of the spaces involved, a Hilbert space is a very rich mathematical environment which often hides the structure of the states and conceals the behaviour of their maps, making it difficult to analyse quantum programs. Can this difficulty be circumvented? In this chapter, we will present an abstract formulation of quantum theory, based on algebraic features present in the Hilbert space theory, but making no reference to Hilbert spaces themselves. The reader will perhaps be unsurprised to learn that the tool of choice for this reformulation of quantum mechanics is category theory, and in particular the theory of symmetric monoidal categories (smcs). In a seminal paper [1], Abramsky and Coecke introduced the notions of †-symmetric monoidal category (†-smc) and †-compact category, and, exploiting the fact that the category of finite dimensional Hilbert spaces and linear maps (henceforth called fdHilb) forms a †-compact category, gave a high-level proof of correctness of the quantum teleportation protocol [4]. In so doing, they showed that quantum protocols do not necessarily rely upon the full apparatus of Hilbert spaces: a more abstract presentation of quantum mechanics can suffice. We will use such a high level presentation to analyse measurement-based quantum programs. As discussed earlier in this volume, †-compact categories admit a graphical notation where the morphisms of the category are represented by diagrams. Sequential composition of morphisms is represented by plugging together diagrams, and parallel composition (i.e. the tensor product) is represented by juxtaposition 1 ar X iv :1 20 3. 62 42 v1 [ qu an tph ] 2 8 M ar 2 01 2