Simulating the Measurement Calculus by Rewriting

The one-way model of quantum computing [RB01] has recently emerged as one of the most promising approaches to quantum computing. It is relatively straight forward to demonstrate one-way patterns which simulate the standard unitary gates of the more traditional quantum circuit model and hence realise any circuit computation by composing the appropriate gates. In contrast, the opposite translation is less obvious: measurement-based computations may be very compact and their logical structure can become hopelessly tangled, exacerbated by the intrinsic parallelism of the one-way model. Understanding and verifying a non-trivial measurement pattern typically requires computing its matrix representation which, aside from its potentially enormous dimension, will give little insight as to where any programming error may lie. In this work I demonstrate a diagrammatic notation which can faithfully represent one-way computations and other models of quantum computing, measurement based or otherwise. This graphical language is equipped with a sound rewriting theory, which can reduce a large class of measurement patterns to equivalent quantum circuits. The reduction of a pattern to a circuit constitutes a proof that the original pattern has flow in the sense of Danos and Kashefi [DK06]. Indeed, we prove an equivalence between the existence of a circuit-like reduct for a given pattern and the presence of flow in that pattern’s underlying geometry. The reduction technique is based on the pattern itself rather than the geometry hence the reduct itself is a proof that the given pattern is deterministic, rather than a statement about the existence of a deterministic pattern. This work is derived from the category theoretic approach to quantum mechanics developed in [AC04, Dun07]. However, although these results hold in a more general algebraic framework, no knowledge of category theory is required for the present work.