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John van de Wetering (Mar 12 2021 at 11:02):Hi all, @Aleks Kissinger , @Miriam Backens , @Hector Miller-Bakewell, Sal Wolffs and me have just put a new paper on the arxiv: Completeness of the ZH-calculus
People here might be familiar with the ZX-calculus, which is a graphical language of string diagrams for reasoning about qubit quantum computation. While the ZX-calculus is very useful when it comes to gate sets consisting of Clifford gates and phase gates, it is less suitable for talking about circuits containing non-linear Boolean gates, such as the Toffoli gate (i.e. the controlled-controlled-NOT gate).
This is where the ZH-calculus comes in, which does allow for easy representation and reasoning about Toffoli gates.
We use this to find a simple complete diagrammatic axiomatisation of the PROP of matrices over the ring , which corresponds to qubit computation with the approximately universal Toffoli+Hadamard gate set.
We argue that this is the simplest complete axiomatisation of a universal fragment of quantum computing found so far. Our rule set consists of just 8 rules:
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John van de Wetering (Mar 12 2021 at 11:03):We also found an extension of these rules that gives a complete axiomatisation of the PROP of matrices over any ring as long as is cancellable in .