The ZX-calculus is a graphical language which allows for reasoning about suitably represented tensor networks -namely ZX-diagrams -in terms of rewrite rules. Here, we focus on problems which amount to exactly computing a scalar encoded as a closed tensor network. In general, such problems are #P-hard. However, there are families of such problems which are known to be in P when the dimension is below a certain value. By expressing problem instances from these families as ZX-diagrams, we see that the easy instances belong to the stabilizer fragment of the ZX-calculus. Building on previous work on efficient simplification of qubit stabilizer diagrams, we present simplifying rewrites for the case of qutrits, which are of independent interest in the field of quantum circuit optimisation. Finally, we look at the specific examples of evaluating the Jones polynomial and of counting graph-colourings. Our exposition further champions the ZX-calculus as a suitable and unifying language for studying the complexity of a broad range of classical and quantum problems.

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The temporal organisation of biological systems into phases and subphases is often crucial to their functioning. Identifying this multiscale organisation can yield insight into the underlying biological mechanisms at play. To date, however, this identification requires a priori biological knowledge of the system under study. Here, we recover the temporal organisation of the cell cycle of budding yeast into phases and subphases, in an automated way. To do so, we model the cell cycle as a partially temporal network of protein-protein interactions (PPIs) by combining a traditional static PPI network with protein concentration or RNA expression time series data. Then, we cluster the snapshots of this temporal network to infer phases, which show good agreement with our biological knowledge of the cell cycle. We systematically test the robustness of the approach and investigate the effect of having only partial temporal information. The generality of the method makes it suitable for application to other, less well-known biological systems for which the temporal organisation of processes plays an important role.

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